{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Magenta E mphasis" -1 263 "Times" 1 12 200 0 200 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Orange Emphasis" -1 264 "Times" 1 12 225 100 10 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 265 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }2 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {PARA 3 "" 0 "" {TEXT -1 70 "Interpolation for Verner's \"most efficient\" order 8 Runge-Kutta scheme" }}{PARA 0 "" 0 "" {TEXT -1 47 "by Peter Stone, Gabriola Island, B.C., Canada " }}{PARA 0 "" 0 " " {TEXT -1 18 "Version: 13.4.2010" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "lo ad procedures for constructing Runge-Kutta schemes " }}{PARA 0 "" 0 " " {TEXT -1 17 "The Maple m-file " }{TEXT 262 9 "butcher.m" }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 " It can be read into a Maple session by a command similar to the one th at follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"C:\\\\Maple/procdrs/butcher.m\";" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "The pu rpose of this worksheet " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 100 "In this worksheet the interpolation scheme for Ve rner's \"most efficient\" order 8 Runge-Kutta scheme." }}{PARA 0 "" 0 "" {TEXT -1 113 "In this worksheet an interpolation scheme is construc ted for the underlying 12 stage, order 8 Runge-Kutta method " }}{PARA 0 "" 0 "" {TEXT -1 82 "taken from the \"most robust\" Runge-Kutta 7-8 pair given on Jim Verner's website. " }}{PARA 0 "" 0 "" {TEXT -1 5 "S ee: " }{URLLINK 17 "http://www.math.sfu.ca/~jverner/" 4 "http://www.ma th.sfu.ca/~jverner/" "" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "The scheme (scheme A) was construct ed to provide a lower order interpolation scheme as well as a higher o rder scheme." }}{PARA 0 "" 0 "" {TEXT -1 136 "An alternative interpola tion scheme (scheme B) is also obtained where the construction of a hi gh order scheme is the only consideration." }}{PARA 0 "" 0 "" {TEXT -1 135 "Although this scheme appears to have a principal error curve w hich is marginally improved, it still has a maximum that is considerab ly " }}{PARA 0 "" 0 "" {TEXT -1 64 "greater than the value at the end \+ of the interpolation interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 123 "The third interpolation scheme (scheme C ) has an improved principal error curve at the expense of requiring an extra stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 43 "#------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 70 "convert/polynom_order_conditions,convert/i nterpolation_order_condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Procedures for converting a standard order cond ition into: " }}{PARA 0 "" 0 "" {TEXT -1 147 "(i) a list of order cond itions for coefficients of an (approximate) weight polynomial associat ed with the construction of an interpolation scheme, " }}{PARA 0 "" 0 "" {TEXT -1 98 "(ii) a single related order condition associated with \+ the construction of an interpolation scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1402 "`convert/polynom _order_conditions` := proc(ordcon::`=`,ord::posint)\n local LS,RS,dg ,st,step,i,trm;\n LS := lhs(ordcon);\n if not type(LS,polynom) the n\n error \"the left side of the argument must be a polynomial\" \n end if;\n RS := rhs(ordcon);\n if not type(1/RS,posint) then \n error \"the right side of the argument must be the reciprocal \+ of a positive integer\"\n end if;\n dg := degree(LS,indets(LS));\n if dg>ord then\n error \"the second argument must be greater t han or equal to the order,%1, of the order condition\", dg;\n end if ;\n st := max(op(map(_u->op(1,_u),select(has,indets(LS),b))));\n s tep := [];\n for i from 1 to ord do\n trm := subs(\{seq(b[j]=d[ j,i],j=1..st)\},ordcon);\n if i=dg then\n step := [op(ste p),trm];\n else\n step := [op(step),lhs(trm)=0];\n e nd if;\n end do:\nend proc:\n\n`convert/interpolation_order_conditio n` := proc(ord::`=`)\n local LS,RS,st,deg;\n LS := lhs(ord);\n i f not type(LS,polynom) then\n error \"the left side of the argume nt must be a polynomial\"\n end if;\n RS := rhs(ord);\n if not t ype(1/RS,posint) then\n error \"the right side of the argument mu st be the reciprocal of a positive integer\"\n end if;\n deg := de gree(LS,indets(LS));\n st := max(op(map(_u->op(1,_u),select(has,inde ts(LS),b))));\n LS := subs(\{seq(b[i]=a[st+1,i],i=1..st)\},LS);\n \+ RS := RS*c[st+1]^deg;\n LS=RS;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 43 "#------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 96 "Stage by stage construction of th e interpolation scheme A .. [7 stage scheme] .. (longer method)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Start wi th linking coefficients using the weights of the 12 stage scheme as th e linking coefficients for the first new stage." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5614 "e1 := \{c[2] = 1/20,\nc[3] = 341/3200,\nc[4] = 1023/6400,\nc[5 ] = 39/100,\nc[6] = 93/200,\nc[7] = 31/200,\nc[8] = 943/1000,\nc[9] = \+ 7067558016280/7837150160667,\nc[10] = 909/1000,\nc[11] = 47/50,\nc[12] = 1,\nc[13] = 1,\na[2,1] = 1/20,\na[3,1] = -7161/1024000,\na[3,2] = 1 16281/1024000,\na[4,1] = 1023/25600,\na[4,2] = 0,\na[4,3] = 3069/25600 ,\na[5,1] = 4202367/11628100,\na[5,2] = 0,\na[5,3] = -3899844/2907025, \na[5,4] = 3982992/2907025,\na[6,1] = 5611/114400,\na[6,2] = 0,\na[6,3 ] = 0,\na[6,4] = 31744/135025,\na[6,5] = 923521/5106400,\na[7,1] = 211 73/343200,\na[7,2] = 0,\na[7,3] = 0,\na[7,4] = 8602624/76559175,\na[7, 5] = -26782109/689364000,\na[7,6] = 5611/283500,\na[8,1] = -1221101821 869329/690812928000000,\na[8,2] = 0,\na[8,3] = 0,\na[8,4] = -125/2,\na [8,5] = -1024030607959889/168929280000000,\na[8,6] = 1501408353528689/ 265697280000000,\na[8,7] = 6070139212132283/92502016000000,\na[9,1] = \+ -147251426448621580388138470887726424634604443330709420782905197804453 1801133057155/\n 12468948016200320011570596216439860248033015 58393487900440453636168046069686436608,\na[9,2] = 0,\na[9,3] = 0,\na[9 ,4] = -517229431108566845837517565524698123003902533693369911413831527 0772319372469280000/\n 12461938100480914589727863057121529836 5257079410236252921850936749076487132995191,\na[9,5] = -12070679258469 254807978936441733187949484571516120469966534514296406891652614970375/ \n 2722031154761657221710478184531100699497284085048389015085 076961673446140398628096,\na[9,6] = 7801251558438936413230905525304310 36567795592568497182701460674803126770111481625/\n 1831104254 1273197219788987450715878685922610298086185950524144307362914310080537 6,\na[9,7] = 664113122959911642134782135839106469928140328160577035357 155340392950009492511875/\n 151784655985862481363330231072953 49175279765150089078301139943253016877823170816,\na[9,8] = 10332848184 452015604056836767286656859124007796970668046446015775000000/\n \+ 1312703550036033648073834248740727914537972028638950165249582733679 393783,\na[10,1] = -29055573360337415088538618442231036441314060511/\n 22674759891089577691327962602370597632000000000,\na[10,2] = \+ 0,\na[10,3] = 0,\na[10,4] = -20462749524591049105403365239069/45425191 3499893469596231268750,\na[10,5] = -1802692598031722811637246632249810 97/38100922558256871086579832832000000,\na[10,6] = 2112767021417280287 0128286992003940810655221489/\n 4679473877997892906145822697 976708633673728000,\na[10,7] = 318607235173649312405151265849660869927 653414425413/\n 67147167155589653031329380729354654239109120 00000,\na[10,8] = 212083202434519082281842245535894/200224260447756725 63822865371173879,\na[10,9] = -269840492940084251872116648508712979856 2269848229517793703413951226714583/\n 4695456749139343150770 00442080871141884676035902717550325616728175875000000,\na[11,1] = -234 2659845814086836951207140065609179073838476242943917/\n 1358 480961351056777022231400139158760857532162795520000,\na[11,2] = 0,\na[ 11,3] = 0,\na[11,4] = -996286030132538159613930889652/1635306888599616 4905464325675,\na[11,5] = -26053085959256534152588089363841/4377552804 565683061011299942400,\na[11,6] = 209808223450967602922240867949781053 12644533925634933539/\n 377588999200755080387872783911549464 1972212962174156800,\na[11,7] = 89072299375637918641892962209583383526 4322635782294899/\n 1392124200139511265750194195559401382283 0119803764736,\na[11,8] = 161021426143124178389075121929246710833125/1 0997207722131034650667041364346422894371443,\na[11,9] = 30076066976810 2517834232497565452434946672266195876496371874262392684852243925359864 884962513/\n 46554433375013464555850653366045056037608247796 15521285751892810315680492364106674524398280000,\na[11,10] = -31155237 437111730665923206875/392862141594230515010338956291,\na[12,1] = -2866 556991825663971778295329101033887534912787724034363/\n 86822 6711619262703011213925016143612030669233795338240,\na[12,2] = 0,\na[12 ,3] = 0,\na[12,4] = -16957088714171468676387054358954754000/1436904151 19654683326368228101570221,\na[12,5] = -458349397448457291294931467335 6033540575/451957703655250747157313034270335135744,\na[12,6] = 2346305 388553404258656258473446184419154740172519949575/\n 25672671 6407895402892744978301151486254183185289662464,\na[12,7] = 16571215593 19846802171283690913610698586256573484808662625/\n 134314804 11255146477259155104956093505361644432088109056,\na[12,8] = 3456853795 54677052215495825476969226377187500/7477116743693007722166720317955134 7546362089,\na[12,9] = \n -320589096271707254279143431215272753400810 2774023210240571361570757249056167015230160352087048674542196011/\n 9 4756954968396581478301512445127360498465774712725761537244920597319265 7306017239103491074738324033259120,\na[12,10] = 4027954583270623343310 0438588458933210937500/8896460842799482846916972126377338947215101,\na [12,11] = -6122933601070769591613093993993358877250/105051700151023551 3198246721302027675953,\na[13,1] = 44901867737754616851973/10140464099 80231013380680,\na[13,2] = 0,\na[13,3] = 0,\na[13,4] = 0,\na[13,5] = 0 ,\na[13,6] = 791638675191615279648100000/2235604725089973126411512319, \na[13,7] = 3847749490868980348119500000/15517045062138271618141237517 ,\na[13,8] = -13734512432397741476562500000/87513289292499590774692878 3,\na[13,9] = 12274765470313196878428812037740635050319234276006986398 294443554969616342274215316330684448207141/\n 489345147493715517 6503858341435109348888292806866096544828965267965233530521667572994528 52166040,\na[13,10] = -9798363684577739445312500000/308722986341456031 822630699,\na[13,11] = 282035543183190840068750/1229540762987304042599 1,\na[13,12] = -306814272936976936753/1299331183183744997286\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "subs(e1,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(13 -i)],i=2..13)])):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7.7/$\"+++++]!#6F(%!GF+F+F+F+F+F+F+F+F+F+7/$\"++]il5!#5$!+iS;$*p !#7$\"+TmbN6F/F+F+F+F+F+F+F+F+F+F+7/$\"++vV)f\"F/$\"+]P4'*RF*$\"\"!F;$ \"+D\"G))>\"F/F+F+F+F+F+F+F+F+F+7/$\"+++++RF/$\"+Gc(Rh$F/F:$!+nS_T8!\" *$\"+/l7q8FEF+F+F+F+F+F+F+F+7/$\"++++]YF/$\"+!G?Z!\\F*F:F:$\"+U?(4N#F/ $\"+Ifb3=F/F+F+F+F+F+F+F+7/$\"++++]:F/$\"+W!*GphF*F:F:$\"+JolB6F/$!+rg /&)QF*$\"+8()=z>F*F+F+F+F+F+F+7/$\"++++I%*F/$!+S-jnB3l&FE$\"+U'p@c'F]oF+F+F+F+F+7/$\"+$QWV%FE$\"+*=3/E%FE$\"+ASOvVF]o$\"+!\\D9(yF2F+F +F+F+7/$\"++++!4*F/$!+**fS\"G\"FEF:F:$!+'*Rr/XF]o$!+p?OJZFE$\"+,$)R'F]o$\"+DG?k9F*$\"+s(3 /Y'F*$!+pJKIzF*F+F+7/$\"\"\"F;$!+oEi,LFEF:F:$!+CF6!=\"!\"($!+RA995F]o$ \"+K8JR\"*FE$\"+G%fPB\"F^s$\"+zVCBYFE$!+QxF$Q$FE$\"++@fFXFE$!+'[&\\GeF EF+7/Fhr$\"+>%*)zU%F*F:F:F:F:$\"+#R\\5a$F/$\"+b@pzCF/$!+/-Up:F]o$\"+( \\1%3DF]o$!+zn$Q<$F]o$\"+F$GQH#F]o$!+LYKhBF/Q(pprint46\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 14" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each sta ndard (simple) order condition gives rise to a group \{list) of equat ions to be satisfied by the \"d\" coefficients of the weight polynomia ls for a given stage (corresponding to an \"approximate\" interpolatio n scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "SO7_13 := SimpleOrderConditions(7,13,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "whch := [1,2,3,6,7,8,12,15,16,27,31,32,64]:\nordeqns1 := []: \nfor ct in whch do\n eqn_group := convert(SO7_13[ct],'polynom_order _conditions',7):\n ordeqns1 := [op(ordeqns1),op(eqn_group)];\nend do :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Sub stitute for all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns1 := []:\nfor ct to nops(ordeqns1) do\n eqns1 \+ := [op(eqns1),expand(subs(e1,ordeqns1[ct]))];\nend do:\nnops(eqns1);\n nops(indets(eqns1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d1 := solve(\{op(eqns1)\}) :\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs(d1,matrix([seq([seq(d[j,i],j=1 ..13)],i=1..7)])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7)7/$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+F+F+7/$!'.sk!\"&F+F+F +F+$!'OQ*)F/$\"'-65!\"%$!'/E@!\"\"$\"'&[S*!\"#$!'Lb8F7$\"'yRDF7$\"'-\" R\"F4$\"'4XCF/7/$\"'-V>F4F+F+F+F+$\"'DIkF4$!'ivVF4$\"'Ly;F*$!'$\\S(F7$ \"'un5F*$!'r/?F*$!'-p5!\"$$!'Rj>F47/$!'p_JF4F+F+F+F+$!'[l:FT$\"'3)G)F4 $!'8iZF*$\"'P*4#F*$!'#*GIF*$\"'_*o&F*$\"'m)o#FT$\"'jn^F47/$\"'YkGF4F+F +F+F+$\"'t0=FT$!':)>)F4$\"'%eH'F*$!'o!y#F*$\"'k8SF*$!'yEvF*$!'$3u#FT$! '7BbF47/$!'%fP\"F4F+F+F+F+$!'T:5FT$\"'.`TF4$!'qcRF*$\"'sc " 0 " " {MPLTEXT 1 0 318 "nm := NULL:\nfor ct to nops(SO7_13) do\n eqn_gro up := convert(SO7_13[ct],'polynom_order_conditions',7):\n tt := expa nd(subs(\{op(e1),op(d1)\},eqn_group));\n tt := map(_Z->`if`(lhs(_Z)= rhs(_Z),0,1),tt);\n if add(op(i,tt),i=1..nops(tt))=0 then nm := nm,c t end if;\nend do:\nnm;\nnops([%]);\nop(\{seq(i,i=1..64)\} minus \{nm \});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6H\"\"\"\"\"#\"\"$\"\"%\"\"&\"\" '\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#=\"#>\"#?\"#A\"#B\"#C\" #D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#Z\"#^\"#`\"#d\"#f\"#g\"#i\"#k" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6<\"#<\"#@\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#[ \"#\\\"#]\"#_\"#a\"#b\"#c\"#e\"#h\"#j" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "Evaluate the \"weight\" polynomials a t the node " }{XPPEDIT 18 0 "c[14]=1/2-sqrt(7)/14" "6#/&%\"cG6#\"#9,&* &\"\"\"F*\"\"#!\"\"F**&-%%sqrtG6#\"\"(F*F'F,F," }{TEXT -1 83 " to obt ain the linking coefficients in the next stage in the interpolation sc heme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "eqs14 := \{seq(a[1 4,j]=add(expand(subs(\{op(d1),c[14]=1/2-7^(1/2)/14\},d[j,i]*c[14]^i)), i=1..7),j=1..13)\}:\ne2 := `union`(eqs14,\{c[14]=1/2-7^(1/2)/14\},e1): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e2 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "e2;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#]Pf3jKD`#)\\^T[/$\">^U!*3$fbMJ2LEu7:\"\"\"*(\"@]iS;G j#[CKQUR!y)eF4\"?*H3j81C$f$e?!*QCT(!\"\"\"\"(#F4\"\"#F4/&F&6$F.\"\"$F* /&%\"cG6#F.,&F:F4*&F.F8F9F:F8/&F&6$\"#7F?F*/&F&6$\"\"%F;F*/&FB6#F(F4/& F&6$\"#6F?F*/&F&6$FIF;F*/&F&6$FTF;F*/&FB6#FIF4/&F&6$F/F?F*/&F&6$F.F(,& #F?\"$#RF4*(F?F4F`oF8F9F:F8/&F&6$F/F;F*/&F&6$\"\")F;F*/&F&6$FhoF?F*/&F &6$\"\"*F;F*/&F&6$F.F;F*/&F&6$\"\"'F?F*/&F&6$F_pF?F*/&F&6$F.F)F*/&F&6$ FfpF;F*/&F&6$F9F;F*/&F&6$F9F?F*/&F&6$F)F;F*/&F&6$F(F;F*/&F&6$F.FMF*/&F &6$F(FMF*/&F&6$F(F?F*/&F&6$F.FT,&#\":+vNoBf_X#[em9\"9fN(3)*yP'Q(\\Z-'F 8*(\"]i!* GrK)zGgT.2?u\">n.^*fz%*zCLv6:)G%F4*(\"A]P%['f3T5E<#*>ZmnGF4\"@$)z+w.]u ^\"H\"f2%>,@F8F9F:F4/&F&6$F.F4,&#\":*oQk?zk#zcl5%*)\"F 4*(\"<(G[**[mYLxT4QG:F4\"?![CPF&=9ZIH&eE7J$F8F9F:F4/&F&6$F.F_p,&#\"^qZ \\2ttOk1DD2rJrJ;BnF@q@0vN*=[/>a'z(4![\\t\"=8>N#zaTd*>(G\"\"]qSQaC!fFpI WoC#)>x&=@D>xkyA\\vX,R0@QK9G@\\BcZf9Co(3*[;\"f*F8*(\"aq\")>\\[P$4RlY?F >fmS.?\"3'HXJJX]OY!*fZWOcbsa+0wy1Hix\"o?3&=#))F4\"aqSu.!*[`OT`hUKUAh\\ &HSF+rAO,x#G)*zve[j'[&)yt)Q'[r-*y+#32))yf\"F8F9F:F8/&F&6$F_pF4#!]pbr0L 6!=`W!y>0Hy?%42LVWgMYUEx)3ZQ\")Q!e@'[kU^s9\"]p3mV'opg/ohj`/W+z[$Re:I.[ -')Rk@'fq:,?.?;![*oC\"/&F&6$FhoF9#\"1$GK@@R,2'\"/+++;?]#*/&F&6$FhoFfp# \"1*oGNN39]\"\"0+++!G(pl#/&F&6$FhoF)#!1*))fzgIS-\"\"0+++!GH*o\"/&FB6#F fp#\"#$*\"$+#/&F&6$F_pF)#!^pv.(\\h_;*oS'H9X`m*p/7;:d%[\\z=LBxq_JQT6*pLpLD!R+B\")pCbc&*H8([w!\\n$4& =#HDO-TzqDl$)H:7dI'ys*e94[+\"Q>Y7/&F&6$F_pFho#\"bo+++vd,Yk/o1(pz2S7fol 'Gnn$o0/c,_W=[GL5\"do$y$RzOt#e\\_;]*Q'G?(z`9zsS([U$Q2[O.O+b.FJ\"/&F&6$ F_pF9#\"\\pv=^#\\4+&HRS`:d`.x0;G.9G*pk5Re8#yM@k6*fH78Tm\"[p;3XV-K37#\"DzQDa%/&F&6$F/F4#!P60198WO5BU%='Q& )3:uLgLdb!H\"P++++?j(fqBgizK\"px&*3\"*)fZnA/&F&6$FTF4#!Xk%\\:\"RysyQ!3b2?***)ex$/&F&6$FTF)#!ATQO*3)e_T`c#ff3`g#\"@+C%**H 651$olX!GbxV/&F&6$FTFM#!?_'*)3$Rhf\"QD8IgG'**\">vcKka!\\;'*f))oIN;/&F& 6$F)FM#\"(#*H)R\"(Dq!H/&F&6$FIF4#!XjV.Cxy7\\`()Q.,\"H`Hy<(RmD=*pbmG\"W S#Q`zL#p1.7O9;]#R@6Iqi#>;rE#o)/&F&6$FTF/#!>vo?BfmI<6PuBb6$\"?\"Hc*Q.,: 0B%fT@'GR/&F&6$FTF_p#\"hp8D'\\)[')f`#RC_[o#RiU(=P'\\we>mAnY\\V_ac(\\KU $y^-\"o(p1w+$\"ip++G)RCXn1TO#\\!o:.\"G*=v&G@bhzZ#3w.c]/mLl]ebkM,vLVal% /&F&6$FTFho#\"KDJ$3rY#H>7v!*QyT7VhU@5;\"MV9P%*GUYVOTqm]Y.J@s2s*4\"/&F& 6$FTF9#\"W**[H#yNEKk_$Q$e4A'H*=k=zjv$*Hs!*)\"VOZw.)>,$G#Q,%fb>%>]dE6&R ,?C@R\"/&F&6$FIFfp#\"Xv&\\*>DW=YMZeileUS`&)Q0jM#\"WkCm*G&=$=ai[^6 Iy\\u#*GS&*yS;nsc#/&F&6$FIF)#!Iv0aLgNtYJ\\H\"Hd%[uR\\$e%\"HWd8N.FMIJdr u]_l.x&>X/&F&6$FfpF4#\"%6c\"'+W6/&F&6$FIFM#!G+Sva*eV0(Qw'o9<9()3dp\"\" E@-d,\"G#ojK$oa'>^T!pV\"/&F&6$FIF_p#!eq6g>UXn[q3_.;I_,nh0\\svq:Or0C5K- uF53S`FF:7VV\"zUD2tf?\\CP:wDFrud Y)\\gt7XC^,$y9e'Ro\\&pv%*/&F&6$FIFho#\"N+v=xjAppZDe\\:A0xYbz`oX$\"M*3i jaZ8bzJ?n;Ax+$pVn6xu/&F&6$FIF9#\"enDEm3[[tlD'e)p5O\"4p$Gr@!o%)>$f:7d; \"Yc!4\")3KWkh`]$4c\\5b\"fsZY^D6/[JM\"/&F&6$F(F4#\"8t>&ohaxtn=!\\%\":! 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#S/&F%6$F'\"\"($\"Ikl7@-Ml5q)y5WU8xp;oL#FD/&F%6$F'\"\")$\"I]TDb='3AQ.y 5u1@%o8!Ry$!#Q/&F%6$F'\"\"*$!IN3P(R2.\\RhE5YUX*G8\"\\f\"FQ/&F%6$F'\"#5 $\"IS>4!fHjf5D9&G5;=No$GI#FQ/&F%6$F'\"#6$!IZF+[p,)*48;[_7%px]yb[%FQ/&F %6$F'\"#7$!IB8yS,L-u14&4SWZ'o(e)zjF+/&F%6$F'\"#8$!ILD;U)>XYK5Coi;'QaN] f7F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 " #---------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 39 "The corresponding published values are:" }}{PARA 0 "" 0 " " {TEXT -1 548 "a[14,1] = .4620700646754963101730413150238116432863e- 1\na[14,2] = 0.\na[14,3] = 0.\na[14,4] = 0.\na[14,5] = 0.\na[14,6] = .4503904160842480866828520384400679697151e-1\na[14,7] = .23368166 97713424410788701065340221126565\na[14,8] = 37.8390136842106741078033 8220861855254153\na[14,9] = -15.94911328945424610266139490307397370835 \na[14,10] = 23.02836835181610285142510596329590091940\na[4,11] = -44 .85578507769412524816130998016948002745\na[14,12] = -.6379858768647444 009509067402330140781326e-1\na[14,13] = -.1259503554386166268241032464 519842162533e-1\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 15" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each sta ndard (simple) order condition gives rise to a group \{list) of equat ions to be satisfied by the \"d\" coefficients of the weight polynomia ls for a given stage (corresponding to an \"approximate\" interpolatio n scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "SO7_14 := SimpleOrderConditions(7,14,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "whch := [1,2,3,6,7,8,12,15,16,21,27,31,32,64]:\nordeqns2 := [ ]:\nfor ct in whch do\n eqn_group := convert(SO7_14[ct],'polynom_ord er_conditions',7):\n ordeqns2 := [op(ordeqns2),op(eqn_group)];\nend \+ do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "S ubstitute for all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns2 := []:\nfor ct to nops(ordeqns2) do\n eqns2 \+ := [op(eqns2),expand(subs(e2,ordeqns2[ct]))];\nend do:\nnops(eqns2);\n nops(indets(eqns2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#)*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#)*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d2 := solve(\{op(eqns2)\}): \ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs(d2,matrix([seq([seq(d[j,i],j=1 ..14)],i=1..7)])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7)70$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+F+F+F+70$!'8js!\"&F+F +F+F+$\"',#Q%F/$\"'e0:!\"%$!'*p!\\!\"#$\"'$G9#F7$!'e3JF7$\"'oseF7F+$\" '3dA!\"'$!'oM7F470$\"'UqCF4F+F+F+F+$!'**\\CF4$!'tswF4$\"'1\")e!\"\"$!' ejDFL$\"'>9PFL$!'NIqFL$!'z;9F4$!'1Q[F/$\"'AJ#)F470$!'LRVF4F+F+F+F+$\"' rDVF4$\"'mq:!\"$$!'74BF*$\"'255F*$!'@i9F*$\"'wgFF*$\"'Q@gF4$\"'cQ=F4$! '-_=Fjn70$\"',sRF4F+F+F+F+$!%7fFjn$!'@7:Fjn$\"'O1SF*$!'+k$F*$\"'$yT\"F*$!'f[?F*$\"'QCQF*$\"'&eI$F4$\"'mQ5F4$!''=w&F470$ \"'uFFF/F+F+F+F+$\"'g]AF4$!'kN&)F/$\"'2r&*FL$!'A(G%FL$\"'x(='FL$!'LZ6F *F+F+F+Q(pprint66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 67 "We can check which of the groups of order conditions ar e satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "nm := NULL: \nfor ct to nops(SO7_14) do\n eqn_group := convert(SO7_14[ct],'polyn om_order_conditions',7):\n tt := expand(subs(\{op(e2),op(d2)\},eqn_g roup));\n tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op (i,tt),i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([ %]);\nop(\{seq(i,i=1..64)\} minus \{nm\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6J\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6 \"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\" #H\"#I\"#J\"#K\"#Z\"#^\"#`\"#d\"#f\"#g\"#i\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6:\"#L\"#M\"#N\"# O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#[\"#\\\"#]\"#_\"#a\"#b\"#c \"#e\"#h\"#j" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "Evaluate the \"weight\" polynomials at the node to obtai n the linking coefficients in the next stage in the interpolation sche me." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "eqs15 := \{seq(a[15, j]=add(expand(subs(\{op(d2),c[15]=69/400\},d[j,i]*c[15]^i)),i=1..7),j= 1..14)\}:\ne3 := `union`(eqs15,\{c[15]=69/400\},e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "e3;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#]Pf3jKD`# )\\^T[/$\">^U!*3$fbMJ2LEu7:\"\"\"*(\"@]iS;Gj#[CKQUR!y)eF4\"?*H3j81C$f$ e?!*QCT(!\"\"\"\"(#F4\"\"#F4/&F&6$F.\"\"$F*/&%\"cG6#F.,&F:F4*&F.F8F9F: F8/&F&6$\"#7F?F*/&F&6$\"\"%F;F*/&FB6#F(F4/&F&6$\"#6F?F*/&F&6$FIF;F*/&F &6$FTF;F*/&FB6#FIF4/&F&6$F/F?F*/&F&6$F.F(,&#F?\"$#RF4*(F?F4F`oF8F9F:F8 /&F&6$F/F;F*/&F&6$\"\")F;F*/&F&6$FhoF?F*/&F&6$\"\"*F;F*/&F&6$F.F;F*/&F &6$\"\"'F?F*/&F&6$F_pF?F*/&F&6$F.F)F*/&F&6$FfpF;F*/&F&6$F9F;F*/&F&6$F9 F?F*/&F&6$F)F;F*/&F&6$F(F;F*/&F&6$F.FMF*/&F&6$F(FMF*/&F&6$F(F?F*/&F&6$ \"#:F)F*/&F&6$FhrFMF*/&F&6$FhrF?F*/&F&6$FhrF;F*/&F&6$F.FT,&#\":+vNoBf_ X#[em9\"9fN(3)*yP'Q(\\Z-'F8*(\"]i!*GrK)zGgT.2?u\">n.^*fz%*zCLv6:)G%F4*(\"A]P%[' f3T5E<#*>ZmnGF4\"@$)z+w.]u^\"H\"f2%>,@F8F9F:F4/&F&6$F.F4,&#\":*oQk?zk# zcl5%*)\"F4*(\"<(G[**[mYLxT4QG:F4\"?![CPF&=9ZIH&eE7J$F 8F9F:F4/&F&6$F.F_p,&#\"^qZ\\2ttOk1DD2rJrJ;BnF@q@0vN*=[/>a'z(4![\\t\"=8 >N#zaTd*>(G\"\"]qSQaC!fFpIWoC#)>x&=@D>xkyA\\vX,R0@QK9G@\\BcZf9Co(3*[; \"f*F8*(\"aq\")>\\[P$4RlY?F>fmS.?\"3'HXJJX]OY!*fZWOcbsa+0wy1Hix\"o?3&= #))F4\"aqSu.!*[`OT`hUKUAh\\&HSF+rAO,x#G)*zve[j'[&)yt)Q'[r-*y+#32))yf\" F8F9F:F8/&FB6#Fhr#\"#p\"$+%/&F&6$FhrFfp,&#\"E8OB`B,=mA;')y)o)\\J5>\"G+ +++c9(R@b#GY#oSeZdp\"F8*(\"E6-ag1XG,DXR]fWuh&>#F4\"G++++gl!R335TSi$yvO 46F8F9F:F4/&F&6$FhrF9,&#\"G6'fDaeC].xiuovNnQsT\"\"G+++gP\")pyVo]^S2#3) oz!*F4*(\"D2`Hb?^Xgs3gr%fo[#R&F4\"F++++C)\\zgm*)*)=fia$y2*F8F9F:F4/&F& 6$FhrF4,&#\"EV=Q@Gv$oxww-Dq&fk,I\"F++++++Gh(4)za#z&f\">&o&F4*(\"D.C(=H %)zQ\"e^E0U+s)RY>3b>c/7g=dF8*(\"B@]Wvgo'[#)3SB-FRJ^F4\"A+!oZ9i ^ZlsGMVN^!oF8F9F:F4/&F&6$FhrF_p,&#\"jq$fm[F]./w_n`GpfA)*z-K?%oy$*zrB9e I)R>;[b(=CS\"elNWD\"*f$*3:tq[3bX\"iq++++++orW\\9Lf@ivlc-w$*[]dY^$R(Gb8^dY ^<7!)G[7MZ]]k4u8\"zE.*R*)3)F4\"_q++++++#R$[nx>FQ'**)pxaJ31h)*)=a\\ZN#* >&**oy)))Rwbn9GH#ew2.&p#F8F9F:F8/&F&6$FhrF/,&#\"AZ^R.aLC%HN&o8!>([\") \"@+#*p.E@7LAqv!4*e\"))F8*(\"A\"yHe\"))GQNkTnXG0Hy?%42LVWgMYUEx)3ZQ\")Q!e@'[k U^s9\"]p3mV'opg/ohj`/W+z[$Re:I.[-')Rk@'fq:,?.?;![*oC\"/&F&6$FhoF9#\"1$ GK@@R,2'\"/+++;?]#*/&F&6$FhoFfp#\"1*oGNN39]\"\"0+++!G(pl#/&F&6$FhoF)#! 1*))fzgIS-\"\"0+++!GH*o\"/&FB6#Ffp#\"#$*\"$+#/&F&6$F_pF)#!^pv.(\\h_;*o S'H9X`m*p/7;:d%[\\z=L@)4\")F4*(\"D\\/(\\qs&4z(fyA;(>L(>5F4\"C++++S-`Jj(4**z'\\PN6F8F9F :F8/&F&6$F_pFM#!]p++GpCP>Bxq_JQT6*pLpLD!R+B\")pCbc&*H8([w!\\n$4&=#HDO-TzqDl$)H:7dI'ys*e94[+\"Q>Y7/&F&6$F_pFho#\"bo+++vd ,Yk/o1(pz2S7fol'Gnn$o0/c,_W=[GL5\"do$y$RzOt#e\\_;]*Q'G?(z`9zsS([U$Q2[O .O+b.FJ\"/&F&6$F_pF9#\"\\pv=^#\\4+&HRS`:d`.x0;G.9G*pk5Re8#yM@k6*fH78Tm \"[p;3s>t7aU5J=/&F&6$F/F9#\"T8aU9MlF*p3m\\eE^^S7$\\OXV-K37#\"DzQDa%/&F&6$F/F4#!P 60198WO5BU%='Q&)3:uLgLdb!H\"P++++?j(fqBgizK\"px&*3\"*)fZnA/&F&6$FTF4#! 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I2O68lN'\\\\W#\\tMcQ&fKb1$F+/&F%6$F'\"#8$!IFh5yyP%4>&\\0d8'H%z*zWD&!#U /&F%6$F'\"#9$!Icrpe!eUY`'Qv*HzCUk%>*R)F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "#--------------------------------- ------------" }}{PARA 0 "" 0 "" {TEXT -1 39 "The corresponding publish ed values are:" }}{PARA 0 "" 0 "" {TEXT -1 603 "a[15,1] = .5037946855 482040993065158747220696112586e-1\na[15,2] = 0.\na[15,3] = 0.\na[15, 4] = 0.\na[15,5] = 0.\na[15,6] = .410983613104607933991653061402884 8248545e-1\na[15,7] = .1718054153348195783296309209549424619697\na[15 ,8] = 4.61410531998151886974342237185977124648\na[15,9] = -1.79166788 3085396449712744996746836471721\na[15,10] = 2.53165893048504140846224 3518792913614971\na[15,11] = -5.32497786020573071925718815977276269909 \na[15,12] = -.3065532595385634734924449496356513113607e-1\na[15,13] = -.5254479979429613570549519094377878106127e-2\na[15,14] = -.839919464 4224792997538653464258058697156e-1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 24 "calculation for stage 16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order condit ion gives rise to a group \{list) of equations to be satisfied by the \"d\" coefficients of the weight polynomials for a given stage (corre sponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_15 := SimpleOrderConditions(7,15,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "whch := [1,2,3,6,7,8 ,12,15,16,21,27,31,32,36,64]:\nordeqns3 := []:\nfor ct in whch do\n \+ eqn_group := convert(SO7_15[ct],'polynom_order_conditions',7):\n ord eqns3 := [op(ordeqns3),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known \+ coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns3 := [ ]:\nfor ct to nops(ordeqns3) do\n eqns3 := [op(eqns3),expand(subs(e3 ,ordeqns3[ct]))];\nend do:\nnops(eqns3);\nnops(indets(eqns3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$0\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$0\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "d3 := solve(\{op(eqns3)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs(d3,matrix([seq([seq(d[j,i],j=1..15)],i=1..7)])):\nevalf[6 ](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)71$\"\"\"\"\"!$ F*F*F+F+F+F+F+F+F+F+F+F+F+F+F+71$!'NIx!\"&F+F+F+F+$\"'Z5nF/$\"'\"p%H! \"%$!'>B!yFP$\"'p.$*FP$!'h))=F7$!'>MMF4 $!'4l=F7$!'KGrF771$\"'_*z(F4F+F+F+F+$!'jm>F7$!')>L\"FP$\"'mw7!\"\"$!'1 !4\"Fip$\"'Py:Fip$!'^-=Fip$\"'f8NF7$\"'m,nF4$\"':^6F7$\"'U,5FP$!'th6Fip$\"'aZ5Fip$!'`&\\\"Fip$\"'(Rk\"F ip$!'f'3$F7$!'I&>'F4$!'2TfF4$!'oz(*F771$\"'Fy6F4F+F+F+F+$!'7iAF4$!'tyG F7$\"'#)\\RFP$!'+>PFP$\"'?__FP$!'\"**f&FP$\"'$*=5F7$\"'-d@F4$\"'0W`FB$ \"'5;HF7Q(pprint76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions a re satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "nm := NULL :\nfor ct to nops(SO7_15) do\n eqn_group := convert(SO7_15[ct],'poly nom_order_conditions',7):\n tt := expand(subs(\{op(e3),op(d3)\},eqn_ group));\n tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(o p(i,tt),i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops( [%]);\nop(\{seq(i,i=1..64)\} minus \{nm\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6R\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6 \"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\" #H\"#I\"#J\"#K\"#O\"#R\"#V\"#Y\"#Z\"#\\\"#^\"#`\"#a\"#c\"#d\"#f\"#g\"# i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"#L\"#M\"#N\"#P\"#Q\"#S\"#T\"#U\"#W\"#X\"#[\"#]\"#_\"# b\"#e\"#h" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate the \"weight\" polynomials at the node " }{XPPEDIT 18 0 "c[16] = 3923/5000;" "6#/&%\"cG6#\"#;*&\"%BR\"\"\"\"%+]!\"\"" } {TEXT -1 83 " to obtain the linking coefficients in the next stage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "eqs16 := \{seq(a[16,j]=add(expand(subs(\{op(d3),c[16]=3923/5000\} ,d[j,i]*c[16]^i)),i=1..7),j=1..15)\}:\ne4 := `union`(eqs16,\{c[16]=392 3/5000\},e3):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e4 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "e4;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#]Pf3jKD`#)\\^T[/$\">^U!*3$fbMJ2LEu7:\"\"\"*(\"@]iS;G j#[CKQUR!y)eF4\"?*H3j81C$f$e?!*QCT(!\"\"\"\"(#F4\"\"#F4/&F&6$F.\"\"$F* /&%\"cG6#F.,&F:F4*&F.F8F9F:F8/&F&6$\"#7F?F*/&F&6$\"\"%F;F*/&FB6#F(F4/& F&6$\"#6F?F*/&F&6$FIF;F*/&F&6$FTF;F*/&FB6#FIF4/&F&6$F/F?F*/&F&6$F.F(,& #F?\"$#RF4*(F?F4F`oF8F9F:F8/&F&6$F/F;F*/&F&6$\"\")F;F*/&F&6$FhoF?F*/&F &6$\"\"*F;F*/&F&6$F.F;F*/&F&6$\"\"'F?F*/&F&6$F_pF?F*/&F&6$F.F)F*/&F&6$ FfpF;F*/&F&6$F9F;F*/&F&6$F9F?F*/&F&6$F)F;F*/&F&6$F(F;F*/&F&6$F.FMF*/&F &6$F(FMF*/&F&6$F(F?F*/&F&6$\"#:F)F*/&F&6$FhrFMF*/&F&6$FhrF?F*/&F&6$Fhr F;F*/&F&6$F.FT,&#\":+vNoBf_X#[em9\"9fN(3)*yP'Q(\\Z-'F8*(\"]i!*GrK)zGgT.2?u\">n.^*fz%*zCLv6:)G%F4*(\"A]P%['f3T5E<#*> ZmnGF4\"@$)z+w.]u^\"H\"f2%>,@F8F9F:F4/&F&6$F_tFMF*/&F&6$F_tF?F*/&F&6$F _tF4,&#\"jq,+,!**fgIzw9)yeITEL0,\\ivDM;Lu/kvJOcH)Gv-i7mNueio\"R>a*)*Rj*Q^i\"4rX$> v&)*o.z(F 4\"hq++++++++DcT)368&>0x#GFg>+y:U5G6@msh?l#3*>#)[m*pjZF8F9F: F8/&F&6$F_tF)F*/&F&6$F_tFfp,&#\"iq,#ohxH#G,Wnri_/kl0&*p#3nX.0TG[H4_qJ! 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6$F9FM#\"(CEg)\")v\"fl(/&F&6$FMF4#F]emFiem/&F&6$F;F4Fibm/&F&6$F?F4#!%h rF][n/&FB6#F)#\"#R\"$+\"/&F&6$F.FI,&#\"8h)fm9o')4G@N6\"9c!o%>S,/>\"*oY DF4*(\"7,[#zgG*RE%e@&F4\"9GSt4q+_fXMt7F8F9F:F8/&FB6#FT#\"#Z\"#]/&F&6$F fpF)#\"'@N#*\"(+k5&/&F&6$F9F4#\"&t6#\"'+KM/&F&6$FfpFM#\"&W<$\"'D]8" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[16,i]=subs(e4,a[16,i]),i=1..15):\nevalf[40](%); " }}{PARA 12 "" 1 "" {XPPMATH 20 "61/&%\"aG6$\"#;\"\"\"$\"I'Q'z`ECc(=r ??'zq*H8(*G3%!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F %6$F'\"\"&F0/&F%6$F'\"\"'$\"I4c[KBtdm3#*)=AjZU^zWC%!#S/&F%6$F'\"\"($\" I\"3'[X['p4+^YRXBv7`\"4EBFD/&F%6$F'\"\")$\"I3H'f.95()G0yig!=r?D)zn#!#R /&F%6$F'\"\"*$\"Id?ijH-Wqgxk@X*QtlE3U(FD/&F%6$F'\"#5$\"I@gH78tRB*4#R>h 9%z%yPg9FD/&F%6$F'\"#6$!IV0oL5l!zn7F+/&F%6$F' \"#9$!I#QyCbhD.&y_(HWnY*\\jVVuF+/&F%6$F'\"#:$\"I)Qjw)p(QZ6bda:&yvz![Fy %F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "# ---------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 39 "The corresponding published values are:" }}{PARA 0 "" 0 " " {TEXT -1 654 "a[16,1] = .4082897132997079620207118756242653796386e- 1\na[16,2] = 0.\na[16,3] = 0.\na[16,4] = 0.\na[16,5] = 0.\na[16,6] = .4244479514247632218892086657732332485609\na[16,7] = .23260915312 75234539465100096964845486081\na[16,8] = 2.67798252071180606278052887 1014035962908\na[16,9] = .7420826657338945216477607044022963622057\na [16,10] = .1460377847941461193920992339731312296021\na[16,11] = -3.57 9344509890565218033356743825917680543\na[16,12] = .113884438960017370 4531638716149985665239\na[16,13] = .126779065103319004737869353761568 7232109e-1\na[16,14] = -.7443436349946674429752785032561552478382e-1\n a[16,15] = .4782748079757851554575511473876987663388e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 17" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each sta ndard (simple) order condition gives rise to a group \{list) of equat ions to be satisfied by the \"d\" coefficients of the weight polynomia ls for a given stage (corresponding to an \"approximate\" interpolatio n scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "SO7_16 := SimpleOrderConditions(7,16,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "whch := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nordeqns 4 := []:\nfor ct in whch do\n eqn_group := convert(SO7_16[ct],'polyn om_order_conditions',7):\n ordeqns4 := [op(ordeqns4),op(eqn_group)]; \nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns4 := []:\nfor ct to nops(ordeqns4) do\n e qns4 := [op(eqns4),expand(subs(e4,ordeqns4[ct]))];\nend do:\nnops(eqns 4);\nnops(indets(eqns4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d4 := solve(\{op(eqns4) \}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs(d4,matrix([seq([seq(d[j,i],j=1 ..16)],i=1..7)])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7)72$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+F+F+F+F+F+72$!'aQs!\" &F+F+F+F+$\"'L:6!\"%$\"&)[BF2$!'JF5!\"#$\"'bo:F7$!')3+#F7$\"'i'\\\"F7$ !'LT;F2$!&nH%F2$!'iT?F2$\"',`;F2$!'0j=F272$\"'#4g#F2F+F+F+F+$!&h<*!\"$ $!&s;\"FM$\"&()>*!\"\"$!'a*R\"FR$\"'V'y\"FR$!'vR8FR$\"'5w9FM$\"&X'QFM$ \"'@N:FM$!&(o'*FM$\"'*>k\"FM72$!'oB]F2F+F+F+F+$\"'2!)*F772$!'[1FF2F+F+F+F+$\"'a%* HFM$!&2V\"F7$!'%*pZFR$\"&%\\rF*$!&4:*F*$\"&+%pF*$!&Wz(F7$!'Ne?FM$!'z() RFM$\"&ea$F7$!&A'yF772$\"']aaF/F+F+F+F+$!'#*yzF2$\"&(4hFM$\"';&\\\"FR$ !'ZKAFR$\"'XfGFR$!'#[<#FR$\"'QaCFM$\"' " 0 "" {MPLTEXT 1 0 285 "nm := NULL:\n for ct to nops(SO7_16) do\n eqn_group := convert(SO7_16[ct],'polynom _order_conditions',7):\n tt := expand(subs(\{op(e4),op(d4)\},eqn_gro up));\n tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op(i ,tt),i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%] );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"&\"\" '\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\" #B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S \"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d \"#e\"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Evalua te the \"weight\" polynomials at the node " }{XPPEDIT 18 0 "c[17] = 3 7/100;" "6#/&%\"cG6#\"#<*&\"#P\"\"\"\"$+\"!\"\"" }{TEXT -1 83 " to ob tain the linking coefficients in the next stage in the interpolation s cheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "eqs17 := \{seq(a[ 17,j]=add(expand(subs(\{op(d4),c[17]=37/100\},d[j,i]*c[17]^i)),i=1..7) ,j=1..16)\}:\ne5 := `union`(eqs17,\{c[17]=37/100\},e4):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "e5;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
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I,&#\"8h)fm9o')4G@N6\"9c!o%>S,/>\"*oYDF4*(\"7,[#zgG*RE%e@&F4\"9GSt4q+_ fXMt7F8F9F:F8/&FB6#FT#\"#Z\"#]/&F&6$FfpF)#\"'@N#*\"(+k5&/&F&6$F9F4#\"& t6#\"'+KM/&F&6$FfpFM#\"&W<$\"'D]8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[17,i]=subs(e 5,a[17,i]),i=1..16):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 2/&%\"aG6$\"#<\"\"\"$\"I2mn9X*z#p8G*HOToOR#o7_!#T/&F%6$F'\"\"#$\"\"!F1 /&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"I\\'Gc1ZB 'o54#=xzWnR3DR&F+/&F%6$F'\"\"($\"I=DG**fI>aG3kMu4e2m\"F+/&F%6$F'\" \")$!I(erIY)H$*p$*=alzn#zv&[aW!#R/&F%6$F'\"\"*$\"Ip9j_\"oz\"oH6=A[L6()FP/&F%6$F'\"#6$\"Il%H+xA&)*p?UYV%)4=bsqF+/&F%6$F'\"#8 $!ISk'H?Tz$H%=6Hk@$*>\\JS&=F+/&F%6$F'\"#9$\"IZ1>if/(3Ul6XYQNa5-/N#F+/& F%6$F'\"#:$\"IhkxuFS8yPc04AyS.^zWB!#S/&F%6$F'\"#;$!I^mwox4)p*3Be)))*G: ,D2T#)F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "#---------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 39 "The corresponding published values are:" }}{PARA 0 "" 0 " " {TEXT -1 720 "a[17,1] = .5212682393668413629928136927994514676607e- 1\na[17,2] = 0.\na[17,3] = 0.\na[17,4] = 0.\na[17,5] = 0.\na[17,6] = .5392508396744797718209106862347065628649e-1\na[17,7] = .16607580 97434640828541930599928251901718e-1\na[17,8] = -4.45448575792677965541 8936993298463071587\na[17,9] = 6.835218278632146381711296817968152631 469\na[17,10] = -8.711334822181993739847172734848837971169\na[17,11] = 6.491635839232917053651267142703105653517\na[17,12] = -.707255180984 4346422069985227700294651922e-1\na[17,13] = -.185403149199321642911184 2937941202966440e-1\na[17,13] = .235040210543538464511654208704596219 0647e-1\na[17,15] = .2344795103407822090556377813402774776461\na[17,1 6] = -.8241072501152898885823089698097768766651e-1\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order condition gives r ise to a group \{list) of equations to be satisfied by the \"d\" coef ficients of the weight polynomials for a given stage (corresponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_17 := SimpleOr derConditions(7,17,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "whch := [1,2,3,6,7,8,12,15, 16,21,27,31,32,33,36,64]:\nordeqns5 := []:\nfor ct in whch do\n eqn_ group := convert(SO7_17[ct],'polynom_order_conditions',7):\n ordeqns 5 := [op(ordeqns5),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known coeffi cients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns5 := []:\nfo r ct to nops(ordeqns5) do\n eqns5 := [op(eqns5),expand(subs(e5,ordeq ns5[ct]))];\nend do:\nnops(eqns5);\nnops(indets(eqns5));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$>\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "d5 := solve(\{op(eqns5)\},indets(eqns5) minus \{seq(d [1,i],i=1..7)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We would like to ensure that " } {XPPEDIT 18 0 "a[18,17] = 0;" "6#/&%\"aG6$\"#=\"#<\"\"!" }{TEXT -1 29 " as in the published scheme. " }}{PARA 0 "" 0 "" {TEXT -1 22 "We use \+ the fact that " }{XPPEDIT 18 0 "a[18,17] = Sum(d[17,i]*c[18]^i,i = 1 \+ .. 7);" "6#/&%\"aG6$\"#=\"#<-%$SumG6$*&&%\"dG6$F(%\"iG\"\"\")&%\"cG6#F 'F0F1/F0;F1\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "We se t " }{XPPEDIT 18 0 "c[18] = 1/2;" "6#/&%\"cG6#\"#=*&\"\"\"F)\"\"#!\" \"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "c_18 : = 1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c_18G#\"\"\"\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "eq := add(subs(\{op(d5),c[18]=c_18\},d[17,i]*c[18]^i),i=1..7)=0: \ndd := \{d[1,1]=1,seq(d[1,i]=0,i=2..6)\}:\nsol := \{d[1,7]=expand(rat ionalize(solve(subs(dd,eq))))\}:\nsol;\ndd_5 := `union`(subs(sol,dd),s ol):\nd_5 := `union`(subs(dd_5,d5),dd_5):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/&%\"dG6$\"\"\"\"\"(,&#\"Q$HceP#QE@TT.t*F( \"PjtM8V!=RS7x$ftH`2X(R$)*[!z " 0 "" {MPLTEXT 1 0 67 "subs(d_5,mat rix([seq([seq(d[j,i],j=1..17)],i=1..7)])):\nevalf[5](%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7)73$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F +F+F+F+F+F+F+F+73F+F+F+F+F+$\"&KI*!\"#$\"&fp(F/$!&.(HF*$\"&n!\\F*$!&I< 'F*$\"&?N%F*$!&DE%F/$!&V;\"F/$\"&r1#!\"\"$!&m9\"F@$!&2/(F/$!&d_#F@73F+ F+F+F+F+$!&YR$F@$!&%oFF@$\"&C7\"F)$!&m%=F)$\"&[K#F)$!&Sk\"F)$\"&.i\"F@ $\"&bT%F/$!&uM(F@$\"&C3%F@$\"&si#F@$\"&_2*F@73F+F+F+F+F+$\"&4n'F@$\"&< K&F@$!&@K#F)$\"&!*z$F)$!&uy%F)$\"&)*R$F)$!&-Q$F@$!&_=*F/$\"&_S\"F*$!&F !yF@$!&jL&F@$!&Hv\"F*73F+F+F+F+F+$!&!\\qF@$!&'HaF@$\"&Dk#F)$!&)*G%F)$ \"&HT&F)$!&m'QF)$\"&$*)QF@$\"&Q0\"F@$!&LW\"F*$\"&`%zF@$\"&c\"fF@$\"&'= =F*73F+F+F+F+F+$\"&kt$F@$\"&bs#F@$!&#\\:F)$\"&3\\#F)$!&$[JF)$\"&`E#F)$ !&=J#F@$!&6D'F/$\"&kS(F@$!&T*RF@$!&!\\LF@$!&PW*F@73$!&uv&!\"$F+F+F+F+$ !&L3)F@$!&'>mF@$\"&lk#F)$!&*eVF)$\"&m[&F)$!&p(QF)$\"&S\"QF@$\"&=/\"F@$ !&s!=F*$\"&8+\"F*$\"&$3iF@$\"&#*>#F*Q(pprint96\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 286 "nm := NULL:\nfor ct to nops(SO7_17) do\n eqn_gro up := convert(SO7_17[ct],'polynom_order_conditions',7):\n tt := expa nd(subs(\{op(e5),op(d_5)\},eqn_group));\n tt := map(_Z->`if`(lhs(_Z) =rhs(_Z),0,1),tt);\n if add(op(i,tt),i=1..nops(tt))=0 then nm := nm, ct end if;\nend do:\nnm;\nnops([%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6\\o\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9 \"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\" #K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"# \\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate the \"weight\" polynomials a t the node " }{XPPEDIT 18 0 "c[18] = 1/2;" "6#/&%\"cG6#\"#=*&\"\"\"F) \"\"#!\"\"" }{TEXT -1 83 " to obtain the linking coefficients in the \+ next stage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "eqs18 := \{seq(a[18,j]=add(expand(subs(\{op(d_5),c[1 8]=c_18\},d[j,i]*c[18]^i)),i=1..7),j=1..17)\}:\ne6 := `union`(eqs18,\{ c[18]=c_18\},e5):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e6 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "e6;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#]Pf3jKD`#)\\^T[/$\">^U!*3$fbMJ2LEu7:\"\"\"*(\"@ ]iS;Gj#[CKQUR!y)eF4\"?*H3j81C$f$e?!*QCT(!\"\"\"\"(#F4\"\"#F4/&F&6$F.\" \"$F*/&%\"cG6#F.,&F:F4*&F.F8F9F:F8/&F&6$\"#7F?F*/&F&6$\"\"%F;F*/&FB6#F (F4/&F&6$\"#6F?F*/&F&6$FIF;F*/&F&6$FTF;F*/&FB6#FIF4/&F&6$F/F?F*/&F&6$F .F(,&#F?\"$#RF4*(F?F4F`oF8F9F:F8/&F&6$F/F;F*/&F&6$\"\")F;F*/&F&6$FhoF? 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\"G;\"\"(+S-\"/&F&6$F9FM#\"(CEg)\")v\"fl(/&F&6$FMF4#FicnF^en/&F&6$F;F4 F_an/&F&6$F?F4#!%hrFf[o/&FB6#F)#\"#RF[s/&F&6$F.FI,&#\"8h)fm9o')4G@N6\" 9c!o%>S,/>\"*oYDF4*(\"7,[#zgG*RE%e@&F4\"9GSt4q+_fXMt7F8F9F:F8/&FB6#FT# \"#Z\"#]/&F&6$FfpF)#\"'@N#*\"(+k5&/&F&6$F9F4#\"&t6#\"'+KM/&F&6$FfpFM# \"&W<$\"'D]8/&F&6$F`vFho,&*(\"R]7.dMH..r)>C2\"*3'3MSx(H?'*Q5F4\"Q>;:[g \"=$G)='ozLKFPdVMgOZl:F8F9F:F8#\"Sv=<^*fPSvL*o0Z:68A!pl5GWc;\"\"Q9(4*) G'*3*pHr6y-%ROU9m?'>%GR*F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[18,i]=subs(e6,a[18 ,i]),i=1..17):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "63/&%\" aG6$\"#=\"\"\"$\"Ii8Y$)y(*>W'**p)f8dNqG5?]!#T/&F%6$F'\"\"#$\"\"!F1/&F% 6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"IRBkn0q/hAK\\ 6)\\&zM!4Ab\"!#S/&F%6$F'\"\"($\"I+j]ZZ'[8\"48Z\"\\B*3C%oUE\"FD/&F%6$F' \"\")$!H^\\&=s0YT<\\q,Z)RNI1#\\^!#Q/&F%6$F'\"\"*$\"H^:J%\\Jj<`ugEHp.** 4Mo%)FQ/&F%6$F'\"#5$!II%)\\&42i$)4UaF&\\\"3\"oI@m5FQ/&F%6$F'\"#6$\"H.R L=!p:?'*4HOG(f\\AL=a(FQ/&F%6$F'\"#7$!HuP0d$fC\\mS%RC9KQ6opV(FD/&F%6$F' \"#8$!IkVw@6Af<#)R$>EQ=mo()e0#F+/&F%6$F'\"#9$\"HWeRiyx$*Hyhs!)H5ZE&z`x FD/&F%6$F'\"#:$\"Ixte()RLr>w8jHWDN?#fi/\"FD/&F%6$F'\"#;$!I6^X8I1(oA]9_ $z>X1L@z6FD/&F%6$F'\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculatio n for stage 19" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order condition gives rise to a \+ group \{list) of equations to be satisfied by the \"d\" coefficients o f the weight polynomials for a given stage (corresponding to an \"appr oximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_18 := SimpleOrderConditi ons(7,18,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "whch := [1,2,3,6,7,8,12,15,16,21,2 7,31,32,33,36,64]:\nordeqns6 := []:\nfor ct in whch do\n eqn_group : = convert(SO7_18[ct],'polynom_order_conditions',7):\n ordeqns6 := [o p(ordeqns6),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known coefficients ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns6 := []:\nfor ct t o nops(ordeqns6) do\n eqns6 := [op(eqns6),expand(subs(e6,ordeqns6[ct ]))];\nend do:\nnops(eqns6);\nnops(indets(eqns6));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$E\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "d6 := solve(\{op(eqns6)\},indets(eqns6) minus \{seq(d[1,i],i=1..7 ),seq(d[9,i],i=1..7)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We would like to ensure t hat " }{XPPEDIT 18 0 "a[19, 18] = 0;" "6#/&%\"aG6$\"#>\"#=\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[19,17] = 0;" "6#/&%\"aG6$\"#>\" #<\"\"!" }{TEXT -1 29 " as in the published scheme." }}{PARA 0 "" 0 " " {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[19] = 7/10;" "6#/&%\"cG6#\"# >*&\"\"(\"\"\"\"#5!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "c_19 := 7/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c _19G#\"\"(\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 292 "eq1 := add(subs(\{op(d6),c[19]=c_19\},d[17,i] *c[19]^i),i=1..7)=0:\neq2 := add(subs(\{op(d6),c[19]=c_19\},d[18,i]*c[ 19]^i),i=1..7)=0:\ndd := \{d[1,1]=1,seq(d[1,i]=0,i=2..6),seq(d[9,i]=0, i=1..6)\}:\nsol := solve(subs(dd,\{eq1,eq2\}));\ndd_6 := `union`(subs( sol,dd),sol):\nd_6 := `union`(subs(dd_6,d6),dd_6):" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%$solG<$/&%\"dG6$\"\"*\"\"(,&#\"er&zgL%Q1>?=q3)*3@%f $QeFQoV=\\=tv!4&*H***4**z$G7$oj09?d#zFd$R\"\\=!oL3S8fdlE\"cr.$[\"\"\"F/F0F+#F3\"\"#F3/&F(6$F3F+,&#\"U0/^qzO$**zR&*)H#)QZli#ROU*zv Xks\"TB)*Q(G$Gl^z6(R:dh+KW'G)=zG9q*)F0*(\"Q+17ay%e$prMv%fe&yYHn(yORZe' F3\"S\\b7#\\5]*QZP\"o6-V2!41,WS^FALF0F+F4F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "subs(d_6,matrix([s eq([seq(d[j,i],j=1..18)],i=1..7)])):\nevalf[6](%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7)74$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+F+ F+F+F+F+F+F+74F+F+F+F+F+$!'/oH!\"$$!'^`NF/$!'rS7!\"#F+$!'v$Q%F/$\"'M\" o\"F4$!'_\\W!\"%$!'Y@x!\"&$!'ygAF/$\"'KBVF/$\"'B]\\F;$\"'z'\\#F/$\"'xm >F/74F+F+F+F+F+$\"'%QA\"F4$\"'Cl9F4$\"')e6&F4F+$\"'f2=F4$!'$G$pF4$\"'q M=F/$\"'%HC$F;$\"'>$G\"F4$!'sf=F4$!'***3#F/$!')*p8F4$!'&>S(F/74F+F+F+F +F+$!'?IGF4$!'Y)Q$F4$!'4$=\"!\"\"F+$!'9!=%F4$\"'D.;Fco$!'!HC%F/$!'?*o( F;$!'&Rq$F4$\"'e@WF4$\"']#)\\F/$\"'qfRF4$\"'zA:F474F+F+F+F+F+$\"'uzOF4 $\"'e0WF4$\"'@Q:FcoF+$\"'$\\V&F4$!']%3#Fco$\"'X;bF/$\"'\"Q.\"F/$\"'?;c F4$!'*y&eF4$!'BFnF/$!'\\ygF4$!'^>F+F+F+F+$\"''fw*F/$\"'7q6F4$\"'WoTF4$ !':d7F/$\"'6/;F4$!'DdcF4$\"'by9F/$\"'w7HF;$\"'R)Q\"F4$!'S5:F4$!'bL=F/$ !'$eb\"F4$!'ySWF/Q)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "nm := NULL:\nfor ct to nops(SO7_18) do\n eqn_group := convert(S O7_18[ct],'polynom_order_conditions',7):\n tt := expand(subs(\{op(e6 ),op(d_6)\},eqn_group));\n tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),t t);\n if add(op(i,tt),i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"# \"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"# =\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N \"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_ \"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 47 "Evaluate the \"weight\" polynomials at the node " } {XPPEDIT 18 0 "c[19] = 7/10;" "6#/&%\"cG6#\"#>*&\"\"(\"\"\"\"#5!\"\"" }{TEXT -1 83 " to obtain the linking coefficients in the next stage i n the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "eqs19 := \{seq(a[19,j]=add(expand(subs(\{op(d_6),c[19]=c_19\},d[j ,i]*c[19]^i)),i=1..7),j=1..18)\}:\ne7 := `union`(eqs19,\{c[19]=c_19\}, e6):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e7 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "e7;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#\"#7,&#\"Ozr]GE8)G1:;30km`'G&H`y`V\"\"O++5(>kVF$yS8J/-)z3%e$RO 1)\\\"\"\"*(\"K<4\\lhq&=Bu]Uv&*y$p1-xE?F4\"L+D=+L^%)*f(pF8F9F:F8/&F&6$\"#9 \"#5,&#\">]Pf3jKD`#)\\^T[/$\">^U!*3$fbMJ2LEu7:F4*(\"@]iS;Gj#[CKQUR!y)e F4\"?*H3j81C$f$e?!*QCT(F8F9F:F4/&F&6$FI\"\"$F*/&%\"cG6#FI,&F:F4*&FIF8F 9F:F8/&F&6$F/FUF*/&F&6$\"\"%F;F*/&FX6#F(F4/&F&6$\"#6FUF*/&F&6$F/F;F*/& F&6$FcoF;F*/&FX6#F/F4/&F&6$F.FI,&#\"Dr*3D;lLRv\"Q!G^\\(pFN\"\"E]PW%=SD 5s&e)o**\\QjjO#F8*(\"A)*=pQ\">t>y>!ycQ&[5$F4\"Bvxtg,T)GMa()**RNXl%*F8F 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2\"*3'3MSx(H?'*Q5F4\"Q>;:[g\"=$G)='ozLKFPdVMgOZl:F8F9F:F8#\"Sv=<^*fPSv L*o0Z:68A!pl5GWc;\"\"Q9(4*)G'*3*pHr6y-%ROU9m?'>%GR*F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[19,i]=subs(e7,a[19,i]),i=1..18):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "64/&%\"aG6$\"#>\"\"\"$\"Ix\\8%4+)[l]dFpDyXY9MPP!# T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F %6$F'\"\"'$\"IC#*3r!R$ou3nnS;$Q`qI\\]$!#S/&F%6$F'\"\"($\"It`![[tT#)*)* )HLa-t$>GlA\\FD/&F%6$F'\"\")$\"Iz`&eJDn+uD`[8fI!*H<`.\"!#Q/&F%6$F'\"#5$\"I$\\tZaga(G3^.*\\\"HDF/K$Q \"FX/&F%6$F'\"#6$!I7![!>Xy$eB&HP'=YyIV#4G7FX/&F%6$F'\"#7$\"I7rIW'yL65o ui(4ll&f^\">'HF+/&F%6$F'\"#:$!IZ+.D%QPi:wk-q1F'QRz^EFD/&F %6$F'\"#;$\"HIcZR(e.(z+<`l1Q<'R]H%*FD/&F%6$F'\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 24 "calculation for stage 20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order condit ion gives rise to a group \{list) of equations to be satisfied by the \"d\" coefficients of the weight polynomials for a given stage (corre sponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_19 := SimpleOrderConditions(7,19,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "whch := [1,2,3,6,7,8 ,12,15,16,21,27,31,32,33,36,64]:\nordeqns7 := []:\nfor ct in whch do\n eqn_group := convert(SO7_19[ct],'polynom_order_conditions',7):\n \+ ordeqns7 := [op(ordeqns7),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all know n coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns7 := []:\nfor ct to nops(ordeqns7) do\n eqns7 := [op(eqns7),expand(subs( e7,ordeqns7[ct]))];\nend do:\nnops(eqns7);\nnops(indets(eqns7));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$L\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 103 "d7 := solve(\{op(eqns7)\},indets(eqns7) minus \{se q(seq(d[j,i],i=1..7),j=[1,9,13])\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We would like t o ensure that " }{XPPEDIT 18 0 "a[20,19]=0" "6#/&%\"aG6$\"#?\"#>\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[20,18] = 0;" "6#/&%\"aG6$\"#?\"# =\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[20,17] = 0;" "6#/&%\"a G6$\"#?\"#<\"\"!" }{TEXT -1 29 " as in the published scheme." }} {PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[20] = 9/10;" " 6#/&%\"cG6#\"#?*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "c_20 := 9/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c_20G#\"\"*\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "eq1 := add(subs(\{op(d7),c[ 20]=c_20\},d[17,i]*c[20]^i),i=1..7)=0:\neq2 := add(subs(\{op(d7),c[20] =c_20\},d[18,i]*c[20]^i),i=1..7)=0:\neq3 := add(subs(\{op(d7),c[20]=c_ 20\},d[19,i]*c[20]^i),i=1..7)=0:\ndd := \{d[1,1]=1,seq(d[1,i]=0,i=2..6 ),seq(d[9,i]=0,i=1..6),seq(d[13,i]=0,i=1..6)\}:\nsol := solve(subs(dd, \{eq1,eq2,eq3\}));\ndd_7 := `union`(subs(sol,dd),sol):\nd_7 := `union` (subs(dd_7,d7),dd_7):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<%/&% \"dG6$\"\"*\"\"(,&*(\"_r+*ypHa\"4])*GCrt)p*y6-@DVijwA>M!)z7G/gz%y$\\5o e/c0=2kcj>!>0]'H0qNJ\"\"\"\"^rH\\G0-pGJ3?VVc'[j0&)eP8N_0Zo3fF1/&F(6$F/F+,&*(\"Q+z#o%>TpD V\"Rw)\\NW^swy\"ft(pvF/\"THf9;'*H%)=jC$*>Y#>:xJ-7-9@q,NF1F+F2F/#\"T&Q[ m/FO'4u\\w)QSF^-t#*e%=872K'F=F1/&F(6$\"#8F+,&*(\"?+CE)))*od0*HmBx4_\"F /\"Ax\"4NYlsM59+\\\"RmH6F1F+F2F1#\"?+ " 0 "" {MPLTEXT 1 0 67 "subs( d_7,matrix([seq([seq(d[j,i],j=1..19)],i=1..7)])):\nevalf[5](%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)75$\"\"\"\"\"!$F*F*F+F+F +F+F+F+F+F+F+F+F+F+F+F+F+F+F+75F+F+F+F+F+$\"&4'H!\"#$\"&Za$F/$\"&xB\"! \"\"F+$\"&JP%F/$!&wn\"F4$\"&)QW!\"$F+$\"&V\\\"F4$!&\\%eF/$\"&OI\"F/$!& !3>F4$\"&*p`F/$!&,i$F/75F+F+F+F+F+$!&iE\"F4$!&g^\"F4$!&FH&F4F+$!&,(=F4 $\"&D<(F4$!&\")*=F/F+$!%UfF*$\"&3T#F4$!&e[&F/$\"&;p(F4$!&%p@F4$\"&(>:F 475F+F+F+F+F+$\"&O2$F4$\"&)zOF4$\"&\\G\"F*F+$\"&(RXF4$!&;u\"F*$\"&%3YF /F+$\"&GM\"F*$!&Tq&F4$\"&NI\"F4$!&Ev\"F*$\"&=\"\\F4$!&]g$F475F+F+F+F+F +$!&#eUF4$!&%)4&F4$!&*zp%F/F+$\"&\"37F*$!&Ci&F4$\"&LC\"F4$!& V$F475$!&$*z\"!\"%F+F+F+F+$!&w+\"F4$!&j?\"F4$!&J@%F 4$\"%!)=F;$!&0\\\"F4$\"&(4dF4$!&2^\"F/$\"&gy#!\"'$!&P\"QF4$\"&9\"=F4$! &q'QF/$\"&S'\\F4$!&zG\"F4$\"&92\"F4Q)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 286 "nm := NULL:\nfor ct to nops(SO7_19) do\n eqn_gro up := convert(SO7_19[ct],'polynom_order_conditions',7):\n tt := expa nd(subs(\{op(e7),op(d_7)\},eqn_group));\n tt := map(_Z->`if`(lhs(_Z) =rhs(_Z),0,1),tt);\n if add(op(i,tt),i=1..nops(tt))=0 then nm := nm, ct end if;\nend do:\nnm;\nnops([%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6\\o\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9 \"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\" #K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"# \\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate the \"weight\" polynomials a t the node " }{XPPEDIT 18 0 "c[20] = 9/10;" "6#/&%\"cG6#\"#?*&\"\"*\" \"\"\"#5!\"\"" }{TEXT -1 82 " to obtain the linking coefficients in th e next stage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "eqs20 := \{seq(a[20,j]=add(expand(subs(\{op(d_7),c[2 0]=c_20\},d[j,i]*c[20]^i)),i=1..7),j=1..19)\}:\ne8 := `union`(eqs20,\{ c[20]=c_20\},e7):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e8 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "e8;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<]x/&%\"aG6$\"#8\"\"&\" \"!/&F&6$\"#>\"#7,&#\"Ozr]GE8)G1:;30km`'G&H`y`V\"\"O++5(>kVF$yS8J/-)z3 %e$RO1)\\\"\"\"*(\"K<4\\lhq&=Bu]Uv&*y$p1-xE?F4\"L+D=+L^%)*f(pF8F9F:F8/&F&6 $\"#9\"#5,&#\">]Pf3jKD`#)\\^T[/$\">^U!*3$fbMJ2LEu7:F4*(\"@]iS;Gj#[CKQU 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>3^EX@M`wtTp<,]F8F9F:F8/&F&6$FUF;#\"'\"G;\"\"(+S-\"/&F&6$F9F\\o#\"(CEg )\")v\"fl(/&F&6$F\\oF4#FeboFjco/&F&6$F;F4Fj^o/&F&6$FUF4#!%hrF\\[p/&F&6 $F\\xFgq,&*(\"P]P\\wQg(f$H/0;K8$*=dLr\\oLz#F4\"P\"Hhk6NN?4-a)3P\"33(Rg \"y^TRS, />\"*oYDF4*(\"7,[#zgG*RE%e@&F4\"9GSt4q+_fXMt7F8F9F:F8/&FX6#Fco#\"#Z\"# ]/&F&6$FerF)#\"'@N#*\"(+k5&/&F&6$F9F4#\"&t6#\"'+KM/&F&6$FerF\\o#\"&W<$ \"'D]8/&F&6$Fh]lFgq,&*(\"R]7.dMH..r)>C2\"*3'3MSx(H?'*Q5F4\"Q>;:[g\"=$G )='ozLKFPdVMgOZl:F8F9F:F8#\"Sv=<^*fPSvL*o0Z:68A!pl5GWc;\"\"Q9(4*)G'*3* pHr6y-%ROU9m?'>%GR*F4/&F&6$F\\xFer,&*(\"O)R)G`srkn?iJ,$yGq%*Gqpi@A#F4 \"Q:y;k*RvgYm!\\uRY04WRT/`u@AF8F9F:F8#\"P`DAbT%))y " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[20,i]=subs(e 8,a[20,i]),i=1..19):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 5/&%\"aG6$\"#?\"\"\"$\"IKr)>_B\\jq1TV4DGbMe!RR!#T/&F%6$F'\"\"#$\"\"!F1 /&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"Ij]rB`vAt pOJ=CWBTh^eN!#S/&F%6$F'\"\"($\"IeiOl+sEbAs$H+h_fA#Q(>%FD/&F%6$F'\"\")$ \"Hg52OS?DD<$HmT>2y(\\/s)!#R/&F%6$F'\"\"*$\"H6O/<9H=.6v'p<(>+C^>#)H%F+/&F%6$F'\"#8$\"IkTc! >*)=*e\\8qq:R(ocdDL\"F+/&F%6$F'\"#9$\"I+Qx(4G>H,hWM![T'R0Fi(=F+/&F%6$F '\"#:$!I*p8lf#fo$z`^qb5AH86%f=FD/&F%6$F'\"#;$\"IU++hj$)HZ1E_u-Y#>F9Ox \"FD/&F%6$F'\"#F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "#------------------------ ---------------------" }}{PARA 0 "" 0 "" {TEXT -1 39 "The correspondin g published values are:" }}{PARA 0 "" 0 "" {TEXT -1 753 "a[20,1] = .3 939058345528250943410670634923521987132e-1\na[20,2] = 0.\na[20,3] = \+ 0.\na[20,4] = 0.\na[20,5] = 0.\na[20,6] = .355851614123442418313669 7322755323715063\na[20,7] = .4197382225952610029372225526720065366258 \na[20,8] = .872044977807194166293172525204036071060\na[20,9] = .898 952083487659486126627160171417043611\na[20,10] = -.6305806161059883590 23456649527853470403\na[20,11] = -1.1218872205954835507366816454252150 81433\na[20,12] = .4298219512400197176967511031829197714867e-1\na[20, 13] = .1332557566873915707013495891889190564164e-1\na[20,14] = .1876 227053964148034446101291928097773800e-1\na[20,15] = -.1859411132922105 570515379368592596513699\na[20,16] = .1773614271924602745226064729836 361000042\na[20,17] = 0.\na[20,18] = 0.\na[20,19] = 0.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "calculation of the interpolation coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (s imple) order condition gives rise to a group \{list) of equations to \+ be satisfied by the \"d\" coefficients of the weight polynomials for a given stage (corresponding to an \"approximate\" interpolation scheme )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO8_20 := SimpleOrderConditions(8,20,'expanded'):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,3,7,8,12,15,16,31,32,58,63,64,102,117,121,123,125, 127,128]:\nordeqns8 := []:\nfor ct in whch do\n eqn_group := convert (SO8_20[ct],'polynom_order_conditions',8):\n ordeqns8 := [op(ordeqns 8),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known coefficients." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns8 := []:\nfor ct to nop s(ordeqns8) do\n eqns8 := [op(eqns8),expand(subs(e8,ordeqns8[ct]))]; \nend do:\nnops(eqns8);\nnops(indets(eqns8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$g\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$g\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Solve the system of equations. 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\"\"$!)'*f4&)FJ7-$!)b7>JF2F+F+F+F+$\")[p5:F:$\")\\*y0\"F:$!)^^&p'FJ$\" )^9q5Feo$!)L.a8Feo$\")0,'y*FJ7-$\"))fgu\"F2F+F+F+F+$!)'*['**)F7$!)c(** H'F7$\")IJ()RFJ$!)!GHP'FJ$\")!RN1)FJ$!)YwFeFJ7-$!)i1PSF/F+F+F+F+$\")kx q@F7$\")08?:F7$!)r/@'*F:$\")LtP:FJ$!)MmX>FJ$\")+>19FJQ)pprint126\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+$\"\"!F)F(F(F(F(F(F(F( F(7+$!)/$\\0\"!\"&$!)8O'[\"!\"'$\")'o\\V\"F0$!)CXD5F-$!)i?8QF0$!)'z#Qm F0$!)w)=/*F0$!)W)Q(fF0$!)Q0?fF07+$\"),eH5!\"%$\")gjd9F-$!)%\\H]\"F-$\" )F.u5FB$\")Y&Q*RF-$\")xHefF-$\")Q]>$*F-$\"),()[aF-$\"),mW>%FB$\")TB\")[FB7+$!)mR25!\"$$!)J\\%R \"FB$\")p*ed\"FB$!)A;E6Ffp$!):q(=%FB$!)d.%[%FB$!)3fW5Ffp$!)&pLg%FB$!)) =,m&FB7+$\")yC**fFB$\")Gco\")F-$!)i(yY*F-$\")D!fw'FB$\")7%f^#FB$\")$[1 ]#FB$\"))=iU'FB$\")9?>EFB$\")nq3MFB7+$!)ucZ9FB$!)izH>F-$\")%H()H#F-$!) .rU;FB$!)m^3hF-$!)Bi6dF-$!)8E#f\"FB$!)wo\\gF-$!)!zVL)F-Q)pprint136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions are satisfied." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 285 "nm := NULL:\nfor ct to nops (SO8_20) do\n eqn_group := convert(SO8_20[ct],'polynom_order_conditi ons',8):\n tt := expand(subs(\{op(e8),op(d8)\},eqn_group));\n tt : = map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op(i,tt),i=1..nops (tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\s\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\") \"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D \"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\" #V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"# g\"#h\"#i\"#j\"#k\"#l\"#m\"#n\"#o\"#p\"#q\"#r\"#s\"#t\"#u\"#v\"#w\"#x \"#y\"#z\"#!)\"#\")\"##)\"#$)\"#%)\"#&)\"#')\"#()\"#))\"#*)\"#!*\"#\"* \"##*\"#$*\"#%*\"#&*\"#'*\"#(*\"#)*\"#**\"$+\"\"$,\"\"$-\"\"$.\"\"$/\" \"$0\"\"$1\"\"$2\"\"$3\"\"$4\"\"$5\"\"$6\"\"$7\"\"$8\"\"$9\"\"$:\"\"$; \"\"$<\"\"$=\"\"$>\"\"$?\"\"$@\"\"$A\"\"$B\"\"$C\"\"$D\"\"$E\"\"$F\"\" $G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$G\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "princip le error graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "The interpolation scheme amounts to having a Runge-Kutta \+ method for each value of the parameter " }{TEXT 266 1 "u" }{TEXT -1 8 " where " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<1 " "6#2%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "The no des and linking coefficients are ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "e_u := map(_U->lhs(_U)=rhs(_U)/u,e8):" }}}{PARA 0 "" 0 "" {TEXT -1 43 " ... and the weight polynomials (of degree " } {XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 33 " and re-using th e weight symbol " }{XPPEDIT 18 0 "b[j]" "6#&%\"bG6#%\"jG" }{TEXT -1 10 ") are ... " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "pols := [s eq(b[j]=add(simplify(subs(d8,d[j,i]))*u^(i-1),i=1..8),j=1..20)]:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "The whol e interpolation scheme (Runge-Kutta scheme with a parameter), includin g the weights, is given by the set of equations:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "eu := `union`(e_u,\{op(pols)\}):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can calculate \+ the principal error norm, that is, the root mean square of the residue s of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "errterms8_20 := PrincipalErrorTerms(8,20,'expanded'):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "sm := 0:\nfor ct to nops(errterms8_20) do\n sm := sm+expand(subs (eu,errterms8_20[ct]))^2;\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 28 "Because the step has width " }{XPPEDIT 18 0 "u*h" "6#*&%\"uG\"\"\"%\"hGF%" }{TEXT -1 17 " we multiply by " } {XPPEDIT 18 0 "u^9;" "6#*$%\"uG\"\"*" }{TEXT -1 45 " in order to provi de appropriate weighting. 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(d[1,i]=subs(d8, d[1,i]),i=1..8):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6*/&% \"dG6$\"\"\"F'$F'\"\"!/&F%6$F'\"\"#$!IQe,FPb4]u!G)*=Xb]Y:R+\"!#Q/&F%6$ F'\"\"$$\"I1igh_GZX]P\\RJB'e\\5#z`F0/&F%6$F'\"\"%$!IF$H'GIv#z'=#4n@ZNs 0z0l\"!#P/&F%6$F'\"\"&$\"I?E9wF#=gW(*[-,hMack-)HF=/&F%6$F'\"\"'$!I*p') zi\">$GK5v*o+zq[a7>JF=/&F%6$F'\"\"($\"INVGE6Nhg/hFar6p_)fgu\"F=/&F%6$F '\"\")$!I+3=%*eNj'exlH%f>@jh1PSF0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 394 "bi8[1,1] = 1.\nbi8[1,2] = -10.039154650 55451898280745009553727015838\nbi8[1,3] = 53.792104958623313949375045 47285261606206\nbi8[1,4] = -165.0579057235472167092186792753028629327 \nbi8[1,5] = 298.0264565434610102489744601822776142620\nbi8[1,6] = -3 11.9125448707900689751032283191627986699\nbi8[1,7] = 174.605985269117 1542761046061351126284335\nbi8[1,8] = -40.3706616321195942965775866335 5894180800\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "seq(d[18,i ]=subs(d8,d[18,i]),i=1..8):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6*/&%\"dG6$\"#=\"\"\"$\"\"!F*/&F%6$F'\"\"#$!I/f3s3XoG*RM( yeI7Q)='\\1P:[S)Q]>$*!#P/&F%6$F'\"\"% $!I./\"Rq*zpKo%)f;cJrx$)*G'R!#O/&F%6$F'\"\"&$\"I'*y4my\"**QW#H$Q7bD+U< Lt)F?/&F%6$F'\"\"'$!Igt&z(3n&Q7_``gw))*=3fW5!#N/&F%6$F'\"\"($\"I_)H'z* fV+$zZOp)f " 0 "" {MPLTEXT 1 0 1 ":" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 97 " Stage by stage construction of the interpolation scheme A .. [7 stage \+ scheme] .. (shorter method)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Start with linking coefficients using the weig hts of the 12 stage scheme as the linking coefficients for the first n ew stage." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5614 "e1 := \{c[2] = 1/20,\nc[3] = 341/3200,\nc[4] = 1023/6400,\nc[5] = 39/100,\nc[6] = 93/200,\nc[7] \+ = 31/200,\nc[8] = 943/1000,\nc[9] = 7067558016280/7837150160667,\nc[10 ] = 909/1000,\nc[11] = 47/50,\nc[12] = 1,\nc[13] = 1,\na[2,1] = 1/20, \na[3,1] = -7161/1024000,\na[3,2] = 116281/1024000,\na[4,1] = 1023/256 00,\na[4,2] = 0,\na[4,3] = 3069/25600,\na[5,1] = 4202367/11628100,\na[ 5,2] = 0,\na[5,3] = -3899844/2907025,\na[5,4] = 3982992/2907025,\na[6, 1] = 5611/114400,\na[6,2] = 0,\na[6,3] = 0,\na[6,4] = 31744/135025,\na [6,5] = 923521/5106400,\na[7,1] = 21173/343200,\na[7,2] = 0,\na[7,3] = 0,\na[7,4] = 8602624/76559175,\na[7,5] = -26782109/689364000,\na[7,6] = 5611/283500,\na[8,1] = -1221101821869329/690812928000000,\na[8,2] = 0,\na[8,3] = 0,\na[8,4] = -125/2,\na[8,5] = -1024030607959889/1689292 80000000,\na[8,6] = 1501408353528689/265697280000000,\na[8,7] = 607013 9212132283/92502016000000,\na[9,1] = -14725142644862158038813847088772 64246346044433307094207829051978044531801133057155/\n 1246894 8016200320011570596216439860248033015583934879004404536361680460696864 36608,\na[9,2] = 0,\na[9,3] = 0,\na[9,4] = -51722943110856684583751756 55246981230039025336933699114138315270772319372469280000/\n 1 2461938100480914589727863057121529836525707941023625292185093674907648 7132995191,\na[9,5] = -12070679258469254807978936441733187949484571516 120469966534514296406891652614970375/\n 272203115476165722171 0478184531100699497284085048389015085076961673446140398628096,\na[9,6] = 7801251558438936413230905525304310365677955925684971827014606748031 26770111481625/\n 1831104254127319721978898745071587868592261 02980861859505241443073629143100805376,\na[9,7] = 66411312295991164213 4782135839106469928140328160577035357155340392950009492511875/\n \+ 151784655985862481363330231072953491752797651500890783011399432530 16877823170816,\na[9,8] = 10332848184452015604056836767286656859124007 796970668046446015775000000/\n 131270355003603364807383424874 0727914537972028638950165249582733679393783,\na[10,1] = -2905557336033 7415088538618442231036441314060511/\n 22674759891089577691327 962602370597632000000000,\na[10,2] = 0,\na[10,3] = 0,\na[10,4] = -2046 2749524591049105403365239069/454251913499893469596231268750,\na[10,5] \+ = -180269259803172281163724663224981097/381009225582568710865798328320 00000,\na[10,6] = 21127670214172802870128286992003940810655221489/\n \+ 4679473877997892906145822697976708633673728000,\na[10,7] = 31 8607235173649312405151265849660869927653414425413/\n 6714716 715558965303132938072935465423910912000000,\na[10,8] = 212083202434519 082281842245535894/20022426044775672563822865371173879,\na[10,9] = -26 9840492940084251872116648508712979856226984822951779370341395122671458 3/\n 4695456749139343150770004420808711418846760359027175503 25616728175875000000,\na[11,1] = -234265984581408683695120714006560917 9073838476242943917/\n 1358480961351056777022231400139158760 857532162795520000,\na[11,2] = 0,\na[11,3] = 0,\na[11,4] = -9962860301 32538159613930889652/16353068885996164905464325675,\na[11,5] = -260530 85959256534152588089363841/4377552804565683061011299942400,\na[11,6] = 20980822345096760292224086794978105312644533925634933539/\n \+ 3775889992007550803878727839115494641972212962174156800,\na[11,7] = 8 90722993756379186418929622095833835264322635782294899/\n 139 21242001395112657501941955594013822830119803764736,\na[11,8] = 1610214 26143124178389075121929246710833125/1099720772213103465066704136434642 2894371443,\na[11,9] = 30076066976810251783423249756545243494667226619 5876496371874262392684852243925359864884962513/\n 4655443337 5013464555850653366045056037608247796155212857518928103156804923641066 74524398280000,\na[11,10] = -31155237437111730665923206875/39286214159 4230515010338956291,\na[12,1] = -2866556991825663971778295329101033887 534912787724034363/\n 86822671161926270301121392501614361203 0669233795338240,\na[12,2] = 0,\na[12,3] = 0,\na[12,4] = -169570887141 71468676387054358954754000/143690415119654683326368228101570221,\na[12 ,5] = -4583493974484572912949314673356033540575/4519577036552507471573 13034270335135744,\na[12,6] = 2346305388553404258656258473446184419154 740172519949575/\n 25672671640789540289274497830115148625418 3185289662464,\na[12,7] = 16571215593198468021712836909136106985862565 73484808662625/\n 134314804112551464772591551049560935053616 44432088109056,\na[12,8] = 3456853795546770522154958254769692263771875 00/74771167436930077221667203179551347546362089,\na[12,9] = \n -32058 9096271707254279143431215272753400810277402321024057136157075724905616 7015230160352087048674542196011/\n 9475695496839658147830151244512736 0498465774712725761537244920597319265730601723910349107473832403325912 0,\na[12,10] = 40279545832706233433100438588458933210937500/8896460842 799482846916972126377338947215101,\na[12,11] = -6122933601070769591613 093993993358877250/1050517001510235513198246721302027675953,\na[13,1] \+ = 44901867737754616851973/1014046409980231013380680,\na[13,2] = 0,\na[ 13,3] = 0,\na[13,4] = 0,\na[13,5] = 0,\na[13,6] = 79163867519161527964 8100000/2235604725089973126411512319,\na[13,7] = 384774949086898034811 9500000/15517045062138271618141237517,\na[13,8] = -1373451243239774147 6562500000/875132892924995907746928783,\na[13,9] = 1227476547031319687 8428812037740635050319234276006986398294443554969616342274215316330684 448207141/\n 489345147493715517650385834143510934888829280686609 654482896526796523353052166757299452852166040,\na[13,10] = -9798363684 577739445312500000/308722986341456031822630699,\na[13,11] = 2820355431 83190840068750/12295407629873040425991,\na[13,12] = -30681427293697693 6753/1299331183183744997286\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "subs(e1,matrix([se q([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..13)])):\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7/$\"+++++]!#6F(%!GF+F+F +F+F+F+F+F+F+F+7/$\"++]il5!#5$!+iS;$*p!#7$\"+TmbN6F/F+F+F+F+F+F+F+F+F+ F+7/$\"++vV)f\"F/$\"+]P4'*RF*$\"\"!F;$\"+D\"G))>\"F/F+F+F+F+F+F+F+F+F+ 7/$\"+++++RF/$\"+Gc(Rh$F/F:$!+nS_T8!\"*$\"+/l7q8FEF+F+F+F+F+F+F+F+7/$ \"++++]YF/$\"+!G?Z!\\F*F:F:$\"+U?(4N#F/$\"+Ifb3=F/F+F+F+F+F+F+F+7/$\"+ +++]:F/$\"+W!*GphF*F:F:$\"+JolB6F/$!+rg/&)QF*$\"+8()=z>F*F+F+F+F+F+F+7 /$\"++++I%*F/$!+S-jnB3l&FE$\"+U'p @c'F]oF+F+F+F+F+7/$\"+$QWV%FE$ \"+*=3/E%FE$\"+ASOvVF]o$\"+!\\D9(yF2F+F+F+F+7/$\"++++!4*F/$!+**fS\"G\" FEF:F:$!+'*Rr/XF]o$!+p?OJZFE$\"+,$)R'F]o$\"+DG?k9F*$\"+s(3/Y'F*$!+pJKIzF*F+F+7/$\"\"\"F;$ !+oEi,LFEF:F:$!+CF6!=\"!\"($!+RA995F]o$\"+K8JR\"*FE$\"+G%fPB\"F^s$\"+z VCBYFE$!+QxF$Q$FE$\"++@fFXFE$!+'[&\\GeFEF+7/Fhr$\"+>%*)zU%F*F:F:F:F:$ \"+#R\\5a$F/$\"+b@pzCF/$!+/-Up:F]o$\"+(\\1%3DF]o$!+zn$Q<$F]o$\"+F$GQH# F]o$!+LYKhBF/Q)pprint146\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "convert(ListTools[Enumerate](Simpl eOrderConditions(7)),matrix):\nlinalg[augment](linalg[delcols](%,2..2) ,matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7\\o7%\"\"\"%#~~G/*&%\"bGF (%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"$F)/*&F,F(-%!G6#*&%\"aGF(F 2F(F(#F(\"\"'7%\"\"%F)/*&F,F()F2F/F(#F(F57%\"\"&F)/*&F,F(-F96#*&FF)/*(F,F(F2F(F8F(#F(\"\")7%\"\"(F)/*&F,F(-F96#*&FCF(FF)/*&F,F(-F96#*(F< F(F2F(FIF(F(#F(\"$!=7%F]rF)/*(F,F(F2F(F_oF(#F(\"$W\"7%\"#@F)/*&F,F(-F9 6#*&FF(#F(FTQ)pprin t236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=subs(e8,c[i]),i=2..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "65/&%\"cG6#\"\"##\"\"\"\"#?/&F%6#\"\"$#\"$T$\"%+K/&F%6#\" \"%#\"%B5\"%+k/&F%6#\"\"&#\"#R\"$+\"/&F%6#\"\"'#\"#$*\"$+#/&F%6#\"\"(# \"#JFF/&F%6#\"\")#\"$V*\"%+5/&F%6#\"\"*#\".!G;!ev1(\".n1;]r$y/&F%6#\"# 5#\"$4*FS/&F%6#\"#6#\"#Z\"#]/&F%6#\"#7F)/&F%6#\"#8F)/&F%6#\"#9,&#F)F'F )*&F]p!\"\"FJF_pFap/&F%6#\"#:#\"#p\"$+%/&F%6#\"#;#\"%BR\"%+]/&F%6#\"#< #\"#PF?/&F%6#\"#=F_p/&F%6#\"#>#FJFhn/&F%6#F*#FWFhn" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 14" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The stan dard (simple) order conditions can be adapted to give a method of stag e by stage construction for an interpolation scheme that avoids dealin g with the weight polynomials for a given stage (corresponding to an \+ \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_13 := SimpleOrderCon ditions(7,13,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "whch := [1,2,4,8,16,17,25,32,64]: \ninterp_order_eqns14 := []:\nfor ct in whch do\n temp_eqn := conver t(SO7_13[ct],'interpolation_order_condition'):\n interp_order_eqns14 := [op(interp_order_eqns14),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order \+ conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 347 "interp_order_eqns14 := [add(a[14,i],i=1..13 )=c[14],seq(op(StageOrderConditions(i,14..14,'expanded')),i=2..7),\n \+ add(a[14,i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*add(a[l,m]*c[m],\n \+ m=2..l-1),l=2..k-1),k=2..j-1),j=2..i-1),i=2..13)=c[14]^6/720, ##17\n add(a[14,i]*add(a[i,j]*c[j]^2*add(a[j,k]*c[k],k=2..j-1),j=2..i-1), i=2..13)=c[14]^6/60]: ##25" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "e2 := `union`(e1,\{seq(a[14 ,i]=0,i=2..5)\}):\neqs_14 := expand(subs(e2,interp_order_eqns14)):\nno ps(eqs_14);\nindets(eqs_14);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<,&%\"aG6$\"#9\"\")&%\" cG6#F'&F%6$F'\"\"*&F%6$F'\"\"(&F%6$F'\"#8&F%6$F'\"#7&F%6$F'\"#5&F%6$F' \"\"\"&F%6$F'\"#6&F%6$F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We s olve for the linking coefficients in terms of " }{XPPEDIT 18 0 "c[14] ;" "6#&%\"cG6#\"#9" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "sol_14 := solve(\{op(eqs_14)\},indets(eqs_14) minus \+ \{c[14]\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 20 "We choose the node " }{XPPEDIT 18 0 "c[1 4];" "6#&%\"cG6#\"#9" }{TEXT -1 63 " so that an additional (adapted) \+ order condition is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 6 "EITHER" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "extra_eqn := add(a[14,i]*ad d(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..13)=c[14]^6/120:" }}}{PARA 0 "" 0 "" {TEXT -1 6 "OR ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "extra_eqn := convert(SO7_13[27],'interpolation_order_ condition'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "expand(subs(e2,extra_eqn)):\neq_14 := subs(sol_1 4,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eq_14G/,,*&#\"APBpMi4C#)37 g9'[,.$\"C])RR#HzYro67#[D&HUB\"\"\"*$)&%\"cG6#\"#9\"\"&F+F+!\"\"*&#\"B LL;JU16;_ph1qaVb$\"D+#ys3::wX-aay0Vv5GF+*$)F.\"\"'F+F+F+*&#\"@\">8yu8? .%eW^t$yGV\"C+(zy%ee$HuLUU'40f%o%F+*$)F.\"\"#F+F+F+*&#\"@(R/Ee/SMh[r6z #HW\"\"B&)RR#HzYro67#[D&HUBF+*$)F.\"\"$F+F+F3*&#\"At&RMCTg4_PV07N')H\" \"BSfdprre[n%[G>5=p$*F+*$)F.\"\"%F+F+F+,$*&#F+\"$?\"F+F8F+F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lhs(eq_14)-rhs(eq_14);\nfactor(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*&#\"APBpMi4C#)37g9'[,.$\"C])RR#HzYro67#[D&HUB\"\"\"*$)&%\"cG6 #\"#9\"\"&F(F(!\"\"*&#\"Az2By?.3%HS+#Q&\\+,\"F'F(*$)F+\"\"'F(F(F(*&#\" @\">8yu8?.%eW^t$yGV\"C+(zy%ee$HuLUU'40f%o%F(*$)F+\"\"#F(F(F(*&#\"@(R/E e/SMh[r6z#HW\"\"B&)RR#HzYro67#[D&HUBF(*$)F+\"\"$F(F(F0*&#\"At&RMCTg4_P V07N')H\"\"BSfdprre[n%[G>5=p$*F(*$)F+\"\"%F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"@(R/Ee/SMh[r6z#HW\"\"C+(zy%ee$HuLUU'40f%o%\"\"\" *()&%\"cG6#\"#9\"\"#F(,(*&F.F(F*F(F(*&F.F(F+F(!\"\"\"\"$F(F(),&F+F(F(F 3F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "14*c[14]^2-14*c[14]+3;\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"#9\"\"\")&%\"cG6#F%\"\"#F&F&*&F%F&F(F&!\"\"\" \"$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"\"\"\"\"#F%*&\"#9!\"\"\" \"(F$F%,&F$F%*&F(F)F*F$F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We can now obtain values for the linking coeffici ents for this stage." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "e3 : = \{c[14]=1/2-1/14*7^(1/2)\}:\ne4 := solve(\{op(subs(e3,eqs_14))\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "e5 := `union`(e2,e3,e4):\n[c[14]=subs(e5,c[14]),seq(a[14,i]=s ubs(e5,a[14,i]),i=1..13)]:\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&%\"cG6#\"#9$\"I@fp*4H)=tTFR'Q'Q&\\jx,6$!#S/&%\"aG6$ F(\"\"\"$\"IiGV;\"Q-:8/t,J'\\vY1q?Y!#T/&F.6$F(\"\"#$\"\"!F9/&F.6$F(\" \"$F8/&F.6$F(\"\"%F8/&F.6$F(\"\"&F8/&F.6$F(\"\"'$\"H:(pz1S%Q?&Go'3[U3; /R]%F+/&F.6$F(\"\"($\"Ikl7@-Ml5q)y5WU8xp;oL#F+/&F.6$F(\"\")$\"I]TDb='3 AQ.y5u1@%o8!Ry$!#Q/&F.6$F(\"\"*$!IN3P(R2.\\RhE5YUX*G8\"\\f\"FX/&F.6$F( \"#5$\"IS>4!fHjf5D9&G5;=No$GI#FX/&F.6$F(\"#6$!IZF+[p,)*48;[_7%px]yb[%F X/&F.6$F(\"#7$!IB8yS,L-u14&4SWZ'o(e)zjF3/&F.6$F(\"#8$!ILD;U)>XYK5Coi;' QaN]f7F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "These linking coefficients essentially agree to digits with those \+ of the published scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 70 "#------------------------------------------------- --------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check whic h of the (adapted) simple order conditions are satisfied at this stage ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "recd := []:\nfor ct to nops(SO7_13) do\n tt := con vert(SO7_13[ct],'interpolation_order_condition'):\n if expand(subs(e 5,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop(r ecd);\nnops(recd);\nop(\{seq(i,i=1..nops(SO7_13))\} minus \{op(recd)\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6J\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"' \"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"# B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#Z\"#^\"#`\"#d\"#f\"#g\"#i\"#k " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6:\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#[\"# \\\"#]\"#_\"#a\"#b\"#c\"#e\"#h\"#j" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 70 "#--------------------------------------- ------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6888 "e5 := \{a[13,5] = 0, a[14,10] = 30448415149 825325326308593750/15127426330731345559308904251+588780394238322448263 2816406250/741243890205835932406136308299*7^(1/2), a[14,3] = 0, c[14] \+ = 1/2-1/14*7^(1/2), a[12,3] = 0, a[4,2] = 0, c[13] = 1, a[11,3] = 0, a [12,2] = 0, a[11,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/392-3/39 2*7^(1/2), a[10,2] = 0, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[14,2] = \+ 0, a[6,3] = 0, a[9,3] = 0, a[14,5] = 0, a[6,2] = 0, a[7,2] = 0, a[7,3] = 0, a[5,2] = 0, a[13,2] = 0, a[14,4] = 0, a[13,4] = 0, a[13,3] = 0, \+ a[14,11] = -1466584824552592368357500/602474973863778980873559-9466779 79546641857718938375/59042547438650340125608782*7^(1/2), a[14,8] = 742 00703416028798327128906250/42881511753324799479599510367+2867664719921 7261041085964843750/2101194075912915174500376007983*7^(1/2), a[14,1] = 8941065567926479206438689/198753096356125278622613280+152838094177334 666489948287/331122658529304714185273724480*7^(1/2), a[14,9] = -128719 9574154792351913181734948009779654190448189357505217021276723163171317 10725250664367373074947/9591164890876824145947562349212814323821053901 4575492278647719252118577198224684430692759024543840-88218508206817762 2906787605005472555636444759904636504531314529608120034066591927204665 390933748491981/159788807082007890271486388737885486634858757998282770 136227100274029549612242324261534136534890037440*7^(1/2), a[9,1] = -14 7251426448621580388138470887726424634604443330709420782905197804453180 1133057155/12468948016200320011570596216439860248033015583934879004404 53636168046069686436608, a[8,7] = 6070139212132283/92502016000000, a[8 ,6] = 1501408353528689/265697280000000, a[8,5] = -1024030607959889/168 929280000000, c[6] = 93/200, a[9,5] = -1207067925846925480797893644173 3187949484571516120469966534514296406891652614970375/27220311547616572 21710478184531100699497284085048389015085076961673446140398628096, a[9 ,4] = -517229431108566845837517565524698123003902533693369911413831527 0772319372469280000/12461938100480914589727863057121529836525707941023 6252921850936749076487132995191, a[9,8] = 1033284818445201560405683676 7286656859124007796970668046446015775000000/13127035500360336480738342 48740727914537972028638950165249582733679393783, a[9,7] = 664113122959 911642134782135839106469928140328160577035357155340392950009492511875/ 1517846559858624813633302310729534917527976515008907830113994325301687 7823170816, a[9,6] = 7801251558438936413230905525304310365677955925684 97182701460674803126770111481625/1831104254127319721978898745071587868 59226102980861859505241443073629143100805376, a[10,7] = 31860723517364 9312405151265849660869927653414425413/67147167155589653031329380729354 65423910912000000, a[10,8] = 212083202434519082281842245535894/2002242 6044775672563822865371173879, a[10,6] = 211276702141728028701282869920 03940810655221489/4679473877997892906145822697976708633673728000, a[10 ,5] = -180269259803172281163724663224981097/38100922558256871086579832 832000000, a[10,4] = -20462749524591049105403365239069/454251913499893 469596231268750, a[10,1] = -290555733603374150885386184422310364413140 60511/22674759891089577691327962602370597632000000000, a[11,1] = -2342 659845814086836951207140065609179073838476242943917/135848096135105677 7022231400139158760857532162795520000, a[10,9] = -26984049294008425187 21166485087129798562269848229517793703413951226714583/4695456749139343 15077000442080871141884676035902717550325616728175875000000, a[11,6] = 20980822345096760292224086794978105312644533925634933539/377588999200 7550803878727839115494641972212962174156800, a[11,5] = -26053085959256 534152588089363841/4377552804565683061011299942400, a[11,4] = -9962860 30132538159613930889652/16353068885996164905464325675, a[5,4] = 398299 2/2907025, a[12,1] = -286655699182566397177829532910103388753491278772 4034363/868226711619262703011213925016143612030669233795338240, a[11,1 0] = -31155237437111730665923206875/392862141594230515010338956291, a[ 11,9] = 30076066976810251783423249756545243494667226619587649637187426 2392684852243925359864884962513/46554433375013464555850653366045056037 60824779615521285751892810315680492364106674524398280000, a[11,8] = 16 1021426143124178389075121929246710833125/10997207722131034650667041364 346422894371443, a[11,7] = 8907229937563791864189296220958338352643226 35782294899/13921242001395112657501941955594013822830119803764736, a[1 2,6] = 2346305388553404258656258473446184419154740172519949575/2567267 16407895402892744978301151486254183185289662464, a[12,5] = -4583493974 484572912949314673356033540575/451957703655250747157313034270335135744 , a[6,1] = 5611/114400, a[12,4] = -16957088714171468676387054358954754 000/143690415119654683326368228101570221, a[12,9] = -32058909627170725 4279143431215272753400810277402321024057136157075724905616701523016035 2087048674542196011/94756954968396581478301512445127360498465774712725 7615372449205973192657306017239103491074738324033259120, a[12,8] = 345 685379554677052215495825476969226377187500/747711674369300772216672031 79551347546362089, a[12,7] = 16571215593198468021712836909136106985862 56573484808662625/1343148041125514647725915510495609350536164443208810 9056, a[13,1] = 44901867737754616851973/1014046409980231013380680, a[1 2,11] = -6122933601070769591613093993993358877250/10505170015102355131 98246721302027675953, a[12,10] = 4027954583270623343310043858845893321 0937500/8896460842799482846916972126377338947215101, c[3] = 341/3200, \+ a[13,6] = 791638675191615279648100000/2235604725089973126411512319, a[ 13,8] = -13734512432397741476562500000/875132892924995907746928783, a[ 13,7] = 3847749490868980348119500000/15517045062138271618141237517, a[ 13,12] = -306814272936976936753/1299331183183744997286, c[2] = 1/20, a [13,11] = 282035543183190840068750/12295407629873040425991, a[13,10] = -9798363684577739445312500000/308722986341456031822630699, a[13,9] = \+ 1227476547031319687842881203774063505031923427600698639829444355496961 6342274215316330684448207141/48934514749371551765038583414351093488882 9280686609654482896526796523353052166757299452852166040, a[8,1] = -122 1101821869329/690812928000000, c[10] = 909/1000, c[4] = 1023/6400, a[5 ,3] = -3899844/2907025, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100 , a[8,4] = -125/2, a[7,6] = 5611/283500, a[14,6] = 3037913416047823635 649583750/15649233075629811884880586233-302465625814318865951896498250 /5367686944941025476514041077919*7^(1/2), a[7,5] = -26782109/689364000 , a[14,7] = 183874328794901398385760606250/760335208044775309288920638 333-114787229090554407592495836250/37256425194193990155157111278317*7^ (1/2), c[7] = 31/200, c[8] = 943/1000, c[9] = 7067558016280/7837150160 667, a[3,2] = 116281/1024000, a[7,4] = 8602624/76559175, a[4,1] = 1023 /25600, a[2,1] = 1/20, a[3,1] = -7161/1024000, c[5] = 39/100, a[14,12] = 11352128098668146659861/254668911904014019468056-521584263992860792 4801/127334455952007009734028*7^(1/2), c[11] = 47/50, a[6,5] = 923521/ 5106400, a[7,1] = 21173/343200, a[6,4] = 31744/135025\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 " calculation for stage 15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 253 "The standard (simple) order conditions can be ada pted to give a method of stage by stage construction for an interpolat ion scheme that avoids dealing with the weight polynomials for a given stage (corresponding to an \"approximate\" interpolation scheme)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_14 := SimpleOrderConditions(7,14,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "whch : = [1,2,4,8,16,17,25,27,32,64]:\ninterp_order_eqns15 := []:\nfor ct in \+ whch do\n temp_eqn := convert(SO7_14[ct],'interpolation_order_condit ion'):\n interp_order_eqns15 := [op(interp_order_eqns15),temp_eqn]; \nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 439 "interp_orde r_eqns15 := [add(a[15,i],i=1..14)=c[15],seq(op(StageOrderConditions(i, 15..15,'expanded')),i=2..7),\n add(a[15,i]*add(a[i,j]*add(a[j,k]*ad d(a[k,l]*add(a[l,m]*c[m],\n m=2..l-1),l=2..k-1),k=2..j-1),j=2..i -1),i=2..14)=c[15]^6/720, ##17\n add(a[15,i]*add(a[i,j]*c[j]^2*add( a[j,k]*c[k],k=2..j-1),j=2..i-1),i=2..14)=c[15]^6/60, ##25\n add(a[1 5,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..14)=c[15]^6 /120]: ##27" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We specify the node for this stage immediately, namely \+ " }{XPPEDIT 18 0 "c[15] = 69/400;" "6#/&%\"cG6#\"#:*&\"#p\"\"\"\"$+%! \"\"" }{TEXT -1 80 ", and have enough equations to determine the corre sponding linking coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "e6 := `union`(e5,\{c[15]=69/400,seq(a[15,i]=0,i=2..5)\}):\neqs_ 15 := expand(subs(e6,interp_order_eqns15)):\nnops(eqs_15);\nindets(eqs _15);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<,&%\"aG6$\"#:\"\"(&F%6$F'\"\")&F%6$F'\"\"'&F%6$ F'\"\"*&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\"#7&F%6$F'\"\"\"&F%6$F'\"#8&F%6$F '\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "e7 := solve( \{op(eqs_15)\}):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "e8 := `union`(e6,e7):\n[seq(a[15,i]=subs(e8,a[15,i]), i=1..14)]:\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&%\"aG 6$\"#:\"\"\"$\"I'e7hp?sue^1$*4/#[bo%z.&!#T/&F&6$F(\"\"#$\"\"!F2/&F&6$F (\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F&6$F(\"\"'$\"IX&[#[)GShIl\"*R $zg/Jh$)4TF,/&F&6$F(\"\"($\"I(p>YU\\&4#4jH$y&>[L:a!=`59Y!#Q/&F&6$F(\"\"*$!I@2t0-'y(\\K&FR/&F&6$F(\"#7$!I2O68lN'\\\\W#\\tMcQ&fKb1$F,/&F&6$F(\" #8$!IFh5yyP%4>&\\0d8'H%z*zWD&!#U/&F&6$F(\"#9$!Icrpe!eUY`'Qv*HzCUk%>*R) F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Th ese linking coefficients agree to 40 digits with those of the publishe d scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#----------------------------------------------------------------- ----" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check which of the (adapte d) simple order conditions are satisfied at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "recd : = []:\nfor ct to nops(SO7_14) do\n tt := convert(SO7_14[ct],'interpo lation_order_condition'):\n if expand(subs(e8,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop(recd);\nnops(recd);\nop( \{seq(i,i=1..nops(SO7_14))\} minus \{op(recd)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6J\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6 \"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\" #H\"#I\"#J\"#K\"#Z\"#^\"#`\"#d\"#f\"#g\"#i\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6:\"#L\"#M\"#N\"# O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#[\"#\\\"#]\"#_\"#a\"#b\"#c \"#e\"#h\"#j" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#-------------------------------------------------------- -------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8580 "e8 := \{a[13,5] = 0, a[14,10] = 30448415149825325326308593750/15 127426330731345559308904251+5887803942383224482632816406250/7412438902 05835932406136308299*7^(1/2), a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), a [12,3] = 0, a[4,2] = 0, c[13] = 1, a[11,3] = 0, a[12,2] = 0, a[11,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[10,2] = \+ 0, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[14,2] = 0, a[6,3] = 0, a[9,3] = 0, a[14,5] = 0, a[6,2] = 0, a[7,2] = 0, a[7,3] = 0, a[5,2] = 0, a[1 3,2] = 0, a[14,4] = 0, a[13,4] = 0, a[13,3] = 0, a[15,5] = 0, a[15,4] \+ = 0, a[15,3] = 0, a[15,2] = 0, a[14,11] = -1466584824552592368357500/6 02474973863778980873559-946677979546641857718938375/590425474386503401 25608782*7^(1/2), a[14,8] = 74200703416028798327128906250/428815117533 24799479599510367+28676647199217261041085964843750/2101194075912915174 500376007983*7^(1/2), a[14,1] = 8941065567926479206438689/198753096356 125278622613280+152838094177334666489948287/33112265852930471418527372 4480*7^(1/2), a[14,9] = -128719957415479235191318173494800977965419044 818935750521702127672316317131710725250664367373074947/959116489087682 4145947562349212814323821053901457549227864771925211857719822468443069 2759024543840-88218508206817762290678760500547255563644475990463650453 1314529608120034066591927204665390933748491981/15978880708200789027148 6388737885486634858757998282770136227100274029549612242324261534136534 890037440*7^(1/2), c[15] = 69/400, a[15,6] = -191031498688788616226618 012353233613/16957475840682462825521397145600000000+219561744459503945 250128450660540211/11093675783624041100808390656000000000*7^(1/2), a[1 5,7] = 14172386735756874627703502458542559611/907968808207405150684378 69813760000000+53924868594716008726045512055295307/9077835462591889896 660794982400000000*7^(1/2), a[15,1] = 30016459570250276767768375282138 1843/5685191595792547980976128000000000000-484972017420526515813879842 91872403/53061788227397114489110528000000000000*7^(1/2), a[15,8] = -87 7010740240762900922180472417/57186012045619550819463987200+51313927022 3400882486686075445021/68051354334287265475162144768000*7^(1/2), a[15, 9] = 45550848707315089359912544356558140241875548161939830581423717993 7868420320279982259692853675276040350274866593/74074089804961736793547 1008762629422134325227668913118716370672910772046081053818353815239174 47168000000000000-8088939903267911374096450504734124828801217514657511 355287393514657504893760256657562215933144919629/269503077658229281467 5576398887868995199235474954188986106083154776989963827197767483392000 000000000*7^(1/2), a[15,10] = -81487190136853529424335403395147/881589 0907570223312212603699200+41281728456741643538288815829781/92757826525 27467499145003008000*7^(1/2), a[9,1] = -147251426448621580388138470887 7264246346044433307094207829051978044531801133057155/12468948016200320 01157059621643986024803301558393487900440453636168046069686436608, a[8 ,7] = 6070139212132283/92502016000000, a[8,6] = 1501408353528689/26569 7280000000, a[8,5] = -1024030607959889/168929280000000, c[6] = 93/200, a[9,5] = -12070679258469254807978936441733187949484571516120469966534 514296406891652614970375/272203115476165722171047818453110069949728408 5048389015085076961673446140398628096, a[15,11] = 14952684189753476847 7167673223553/8109821199993554522521600000000-101973319716227859779095 72704970449/1135374967999097633153024000000000*7^(1/2), a[9,4] = -5172 2943110856684583751756552469812300390253369336991141383152707723193724 69280000/1246193810048091458972786305712152983652570794102362529218509 36749076487132995191, a[9,8] = 103328481844520156040568367672866568591 24007796970668046446015775000000/1312703550036033648073834248740727914 537972028638950165249582733679393783, a[9,7] = 66411312295991164213478 2135839106469928140328160577035357155340392950009492511875/15178465598 586248136333023107295349175279765150089078301139943253016877823170816, a[9,6] = 780125155843893641323090552530431036567795592568497182701460 674803126770111481625/183110425412731972197889874507158786859226102980 861859505241443073629143100805376, a[10,7] = 3186072351736493124051512 65849660869927653414425413/6714716715558965303132938072935465423910912 000000, a[10,8] = 212083202434519082281842245535894/200224260447756725 63822865371173879, a[10,6] = 21127670214172802870128286992003940810655 221489/4679473877997892906145822697976708633673728000, a[10,5] = -1802 69259803172281163724663224981097/38100922558256871086579832832000000, \+ a[10,4] = -20462749524591049105403365239069/45425191349989346959623126 8750, a[10,1] = -29055573360337415088538618442231036441314060511/22674 759891089577691327962602370597632000000000, a[11,1] = -234265984581408 6836951207140065609179073838476242943917/13584809613510567770222314001 39158760857532162795520000, a[10,9] = -2698404929400842518721166485087 129798562269848229517793703413951226714583/469545674913934315077000442 080871141884676035902717550325616728175875000000, a[15,12] = -22716443 91263030621748072222279/74102764220560004300800000000000, a[11,6] = 20 980822345096760292224086794978105312644533925634933539/377588999200755 0803878727839115494641972212962174156800, a[15,13] = -288610253631/128 000000000000-4643982156663/4096000000000000*7^(1/2), a[11,5] = -260530 85959256534152588089363841/4377552804565683061011299942400, a[11,4] = \+ -996286030132538159613930889652/16353068885996164905464325675, a[5,4] \+ = 3982992/2907025, a[12,1] = -2866556991825663971778295329101033887534 912787724034363/868226711619262703011213925016143612030669233795338240 , a[11,10] = -31155237437111730665923206875/39286214159423051501033895 6291, a[11,9] = 300760669768102517834232497565452434946672266195876496 371874262392684852243925359864884962513/465544333750134645558506533660 4505603760824779615521285751892810315680492364106674524398280000, a[11 ,8] = 161021426143124178389075121929246710833125/109972077221310346506 67041364346422894371443, a[11,7] = 89072299375637918641892962209583383 5264322635782294899/13921242001395112657501941955594013822830119803764 736, a[12,6] = 2346305388553404258656258473446184419154740172519949575 /256726716407895402892744978301151486254183185289662464, a[12,5] = -45 83493974484572912949314673356033540575/4519577036552507471573130342703 35135744, a[6,1] = 5611/114400, a[15,14] = -32507875096641/10240000000 00000*7^(1/2), a[12,4] = -16957088714171468676387054358954754000/14369 0415119654683326368228101570221, a[12,9] = -32058909627170725427914343 1215272753400810277402321024057136157075724905616701523016035208704867 4542196011/94756954968396581478301512445127360498465774712725761537244 9205973192657306017239103491074738324033259120, a[12,8] = 345685379554 677052215495825476969226377187500/747711674369300772216672031795513475 46362089, a[12,7] = 16571215593198468021712836909136106985862565734848 08662625/13431480411255146477259155104956093505361644432088109056, a[1 3,1] = 44901867737754616851973/1014046409980231013380680, a[12,11] = - 6122933601070769591613093993993358877250/10505170015102355131982467213 02027675953, a[12,10] = 40279545832706233433100438588458933210937500/8 896460842799482846916972126377338947215101, c[3] = 341/3200, a[13,6] = 791638675191615279648100000/2235604725089973126411512319, a[13,8] = - 13734512432397741476562500000/875132892924995907746928783, a[13,7] = 3 847749490868980348119500000/15517045062138271618141237517, a[13,12] = \+ -306814272936976936753/1299331183183744997286, c[2] = 1/20, a[13,11] = 282035543183190840068750/12295407629873040425991, a[13,10] = -9798363 684577739445312500000/308722986341456031822630699, a[13,9] = 122747654 7031319687842881203774063505031923427600698639829444355496961634227421 5316330684448207141/48934514749371551765038583414351093488882928068660 9654482896526796523353052166757299452852166040, a[8,1] = -122110182186 9329/690812928000000, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = -38 99844/2907025, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100, a[8,4] \+ = -125/2, a[7,6] = 5611/283500, a[14,6] = 3037913416047823635649583750 /15649233075629811884880586233-302465625814318865951896498250/53676869 44941025476514041077919*7^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 183874328794901398385760606250/760335208044775309288920638333-11478 7229090554407592495836250/37256425194193990155157111278317*7^(1/2), c[ 7] = 31/200, c[8] = 943/1000, c[9] = 7067558016280/7837150160667, a[3, 2] = 116281/1024000, a[7,4] = 8602624/76559175, a[4,1] = 1023/25600, a [2,1] = 1/20, a[3,1] = -7161/1024000, c[5] = 39/100, a[14,12] = 113521 28098668146659861/254668911904014019468056-5215842639928607924801/1273 34455952007009734028*7^(1/2), c[11] = 47/50, a[6,5] = 923521/5106400, \+ a[7,1] = 21173/343200, a[6,4] = 31744/135025\}: " }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculatio n for stage 16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order conditions can be adapted to give a method of stage by stage construction for an interpolation sch eme that avoids dealing with the weight polynomials for a given stage \+ (corresponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO 7_15 := SimpleOrderConditions(7,15,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "whch := [1, 2,4,8,16,17,25,27,32,63,64]:\ninterp_order_eqns16 := []:\nfor ct in wh ch do\n temp_eqn := convert(SO7_15[ct],'interpolation_order_conditio n'):\n interp_order_eqns16 := [op(interp_order_eqns16),temp_eqn];\ne nd do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 509 "interp_order_e qns16 := [add(a[16,i],i=1..15)=c[16],seq(op(StageOrderConditions(i,16. .16,'expanded')),i=2..7),\n add(a[16,i]*add(a[i,j]*add(a[j,k]*add(a [k,l]*add(a[l,m]*c[m],\n m=2..l-1),l=2..k-1),k=2..j-1),j=2..i-1), i=2..15)=c[16]^6/720, ##17\n add(a[16,i]*add(a[i,j]*c[j]^2*add(a[j, k]*c[k],k=2..j-1),j=2..i-1),i=2..15)=c[16]^6/60, ##25\n add(a[16,i] *add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..15)=c[16]^6/120 , ##27\n add(a[16,i]*add(a[i,j]*c[j]^5,j=2..i-1),i=2..15)=c[16]^7/ 42]: ##63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the node " }{XPPEDIT 18 0 "c[16] = 3923/5000;" "6#/ &%\"cG6#\"#;*&\"%BR\"\"\"\"%+]!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "e9 := `union`(e8,\{c[16]=3923/5000,seq(a[1 6,i]=0,i=2..5)\}):\neqs_16 := expand(subs(e9,interp_order_eqns16)):\nn ops(eqs_16);\nindets(eqs_16);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<-&%\"aG6$\"#;\"\")&F%6$ F'\"#6&F%6$F'\"#5&F%6$F'\"\"*&F%6$F'\"#:&F%6$F'\"#9&F%6$F'\"#8&F%6$F' \"#7&F%6$F'\"\"'&F%6$F'\"\"(&F%6$F'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "e10 := solve(\{op(eqs_16)\}):\ninfolevel[sol ve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "e11 := `union`(e9,e10):\nseq(a[16,i]=subs(e11,a[16,i] ),i=1..15):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "61/&%\"aG6 $\"#;\"\"\"$\"I'Q'z`ECc(=r??'zq*H8(*G3%!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F '\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"I4c[KBtdm3#*)=Aj ZU^zWC%!#S/&F%6$F'\"\"($\"I\"3'[X['p4+^YRXBv7`\"4EBFD/&F%6$F'\"\")$\"I 3H'f.95()G0yig!=r?D)zn#!#R/&F%6$F'\"\"*$\"Id?ijH-Wqgxk@X*QtlE3U(FD/&F% 6$F'\"#5$\"I@gH78tRB*4#R>h9%z%yPg9FD/&F%6$F'\"#6$!IV0oL5l!zn7F+/&F%6$F'\"#9$!I#QyCbhD.&y_(HWnY*\\jVVuF+/&F%6$F'\"#:$ \"I)Qjw)p(QZ6bda:&yvz![Fy%F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "These linking coefficients agree to 40 di gits with those of the published scheme." }}{PARA 0 "" 0 "" {TEXT -1 70 "#----------------------------------------------------------------- ----" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check which of the (adapte d) simple order conditions are satisfied at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "recd : = []:\nfor ct to nops(SO7_15) do\n tt := convert(SO7_15[ct],'interpo lation_order_condition'):\n if expand(subs(e11,lhs(tt)=rhs(tt))) the n recd := [op(recd),ct] end if: \nend do:\nop(recd);\nnops(recd);\nop( \{seq(i,i=1..nops(SO7_15))\} minus \{op(recd)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6R\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6 \"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\" #H\"#I\"#J\"#K\"#O\"#R\"#V\"#Y\"#Z\"#\\\"#^\"#`\"#a\"#c\"#d\"#f\"#g\"# i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"#L\"#M\"#N\"#P\"#Q\"#S\"#T\"#U\"#W\"#X\"#[\"#]\"#_\"# b\"#e\"#h" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#-------------------------------------------------------------- -------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13669 "e11 := \{a[13,5] = 0, a[14,10] = 30448415149825325326308593750/ 15127426330731345559308904251+5887803942383224482632816406250/74124389 0205835932406136308299*7^(1/2), a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[4,2] = 0, c[13] = 1, a[11,3] = 0, a[12,2] = 0, a[11,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[10,2] \+ = 0, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[14,2] = 0, a[6,3] = 0, a[9, 3] = 0, a[14,5] = 0, a[6,2] = 0, a[7,2] = 0, a[7,3] = 0, a[5,2] = 0, a [13,2] = 0, a[14,4] = 0, a[13,4] = 0, a[13,3] = 0, a[15,5] = 0, a[15,4 ] = 0, a[15,3] = 0, a[15,2] = 0, a[14,11] = -1466584824552592368357500 /602474973863778980873559-946677979546641857718938375/5904254743865034 0125608782*7^(1/2), a[16,2] = 0, a[14,8] = 742007034160287983271289062 50/42881511753324799479599510367+28676647199217261041085964843750/2101 194075912915174500376007983*7^(1/2), a[16,4] = 0, a[16,3] = 0, a[16,1] = 3184241137993004885982493411533410452435081234662965417047433163425 75624901053326413058788147679306059990010001/7717193457109162513896339 9895419391686258743566126202752882956363175641980882416112396323331250 00000000000000000-7790368985751938923820674667096775602063482288285741 29617032755363065443176248630443242600895553442815357/4763699664882199 0826520617219394686226085644176621112810421578001960272827705195131108 84156250000000000000000*7^(1/2), a[16,5] = 0, a[16,6] = 38677873603874 0234948212246635605205337571290317052092948284105034567082699505656404 52627167440128229776168201/1000800821189181857363609351052680617003947 57237804533541386478622349830693961360202657479143973437500000000000+2 6603884961798645943608158417608557665087305230552993954979509880880776 475635566879209893579250983975560997/185333485405404047659927657602348 2624081384393292676547062712567080552420258543707456619984147656250000 000000*7^(1/2), a[14,1] = 8941065567926479206438689/198753096356125278 622613280+152838094177334666489948287/331122658529304714185273724480*7 ^(1/2), a[16,7] = 3773085879033406777926055420852886774791582751698542 8125843490240154695455951816700929871689234244950950501493/13892859740 6614710422701561505536918813454339785587147820728065583231442155198771 083100889637109062500000000000-102330014535795077906358306091556548197 8351900676151998097343801718266872083464334323592817694003399143529013 /694642987033073552113507807527684594067271698927935739103640327916157 21077599385541550444818554531250000000000*7^(1/2), c[16] = 3923/5000, \+ a[14,9] = -12871995741547923519131817349480097796541904481893575052170 2127672316317131710725250664367373074947/95911648908768241459475623492 128143238210539014575492278647719252118577198224684430692759024543840- 8821850820681776229067876050054725556364447599046365045313145296081200 34066591927204665390933748491981/1597888070820078902714863887378854866 34858757998282770136227100274029549612242324261534136534890037440*7^(1 /2), a[16,9] = 1408267893588902172954752885879200378495596410425522130 3813118180540289540909500823975046055485025984309922595887619652662278 78689854153082563752624774515387959169654573113399136914397/3724061476 2200483599041922126398403139352465609300058294686034773950283950518320 2930223886132655198287480093051834125673996613031952728485414449188704 0986350427874375000000000000000000+36589487248952850975626989635036036 9840439730384597110234875444563167708871855196974974526134564130489860 8718025716162435543460343413409884815320619233885431694117641997082749 518637/266004391158574882850299443759988593852517611495000416390614534 0996448853608451449501599186661822844910572093227386611957118664513948 0606101032084907435616788770531250000000000000000*7^(1/2), a[16,13] = \+ 1667795459666994596006593734612718364094871113097496394713308473411640 245893712505210071/148176942057718746344268989730526651074011467617513 077874633459904607921875000000000000000+995832251155619761130552690028 2303722213821417596901636742244109953434503977764414401/18522117757214 8432930336237163158313842514334521891347343291824880759902343750000000 00000*7^(1/2), a[16,14] = 10091733461247592876816891204420595023183753 7138211695125551197160193984809290378367269/99638921083071752167902327 4600361612169126960875165356735545085748281214843750000000000000-14035 0519207811981108980209355304116732499169570091599701547604838639510566 88692206659/4981946054153587608395116373001808060845634804375826783677 72542874140607421875000000000*7^(1/2), c[15] = 69/400, a[16,11] = -692 8011549454881761699481713711235043591502295679423604869246153519670533 666503210900114933332845376150551/220168689298819394894713630837330164 6572872307345365971075591622778810160246298613549949543750000000000000 -333373184604500792272379654433507180957103057593155834003047471420894 2484430219726989269407196212710461/20385989749890684712473484336789830 060859928771716351584033255766470464446724987162499532812500000000000* 7^(1/2), a[15,6] = -191031498688788616226618012353233613/1695747584068 2462825521397145600000000+219561744459503945250128450660540211/1109367 5783624041100808390656000000000*7^(1/2), a[15,7] = 1417238673575687462 7703502458542559611/90796880820740515068437869813760000000+53924868594 716008726045512055295307/9077835462591889896660794982400000000*7^(1/2) , a[15,1] = 300164595702502767677683752821381843/568519159579254798097 6128000000000000-48497201742052651581387984291872403/53061788227397114 489110528000000000000*7^(1/2), a[15,8] = -8770107402407629009221804724 17/57186012045619550819463987200+513139270223400882486686075445021/680 51354334287265475162144768000*7^(1/2), a[15,9] = 455508487073150893599 1254435655814024187554816193983058142371799378684203202799822596928536 75276040350274866593/7407408980496173679354710087626294221343252276689 1311871637067291077204608105381835381523917447168000000000000-80889399 0326791137409645050473412482880121751465751135528739351465750489376025 6657562215933144919629/26950307765822928146755763988878689951992354749 54188986106083154776989963827197767483392000000000000*7^(1/2), a[16,12 ] = 687623078790282776716076720521458751380729771175496461512364339062 2098477381787285420842315969989227509309/72708051904986754072892470543 4890757902995174274827662746719926470473225148913521710030117187500000 00000000+7862136496466487636203045219318977472234187207775060875131229 872919912547716368298063580928728400258823/107715632451832228256136993 3977615937634066924851596537402548039215515889109501513644489062500000 000000000*7^(1/2), a[15,10] = -81487190136853529424335403395147/881589 0907570223312212603699200+41281728456741643538288815829781/92757826525 27467499145003008000*7^(1/2), a[9,1] = -147251426448621580388138470887 7264246346044433307094207829051978044531801133057155/12468948016200320 01157059621643986024803301558393487900440453636168046069686436608, a[8 ,7] = 6070139212132283/92502016000000, a[8,6] = 1501408353528689/26569 7280000000, a[8,5] = -1024030607959889/168929280000000, c[6] = 93/200, a[9,5] = -12070679258469254807978936441733187949484571516120469966534 514296406891652614970375/272203115476165722171047818453110069949728408 5048389015085076961673446140398628096, a[15,11] = 14952684189753476847 7167673223553/8109821199993554522521600000000-101973319716227859779095 72704970449/1135374967999097633153024000000000*7^(1/2), a[9,4] = -5172 2943110856684583751756552469812300390253369336991141383152707723193724 69280000/1246193810048091458972786305712152983652570794102362529218509 36749076487132995191, a[9,8] = 103328481844520156040568367672866568591 24007796970668046446015775000000/1312703550036033648073834248740727914 537972028638950165249582733679393783, a[9,7] = 66411312295991164213478 2135839106469928140328160577035357155340392950009492511875/15178465598 586248136333023107295349175279765150089078301139943253016877823170816, a[9,6] = 780125155843893641323090552530431036567795592568497182701460 674803126770111481625/183110425412731972197889874507158786859226102980 861859505241443073629143100805376, a[10,7] = 3186072351736493124051512 65849660869927653414425413/6714716715558965303132938072935465423910912 000000, a[10,8] = 212083202434519082281842245535894/200224260447756725 63822865371173879, a[10,6] = 21127670214172802870128286992003940810655 221489/4679473877997892906145822697976708633673728000, a[10,5] = -1802 69259803172281163724663224981097/38100922558256871086579832832000000, \+ a[10,4] = -20462749524591049105403365239069/45425191349989346959623126 8750, a[10,1] = -29055573360337415088538618442231036441314060511/22674 759891089577691327962602370597632000000000, a[11,1] = -234265984581408 6836951207140065609179073838476242943917/13584809613510567770222314001 39158760857532162795520000, a[10,9] = -2698404929400842518721166485087 129798562269848229517793703413951226714583/469545674913934315077000442 080871141884676035902717550325616728175875000000, a[15,12] = -22716443 91263030621748072222279/74102764220560004300800000000000, a[11,6] = 20 980822345096760292224086794978105312644533925634933539/377588999200755 0803878727839115494641972212962174156800, a[15,13] = -288610253631/128 000000000000-4643982156663/4096000000000000*7^(1/2), a[11,5] = -260530 85959256534152588089363841/4377552804565683061011299942400, a[11,4] = \+ -996286030132538159613930889652/16353068885996164905464325675, a[5,4] \+ = 3982992/2907025, a[12,1] = -2866556991825663971778295329101033887534 912787724034363/868226711619262703011213925016143612030669233795338240 , a[11,10] = -31155237437111730665923206875/39286214159423051501033895 6291, a[11,9] = 300760669768102517834232497565452434946672266195876496 371874262392684852243925359864884962513/465544333750134645558506533660 4505603760824779615521285751892810315680492364106674524398280000, a[11 ,8] = 161021426143124178389075121929246710833125/109972077221310346506 67041364346422894371443, a[11,7] = 89072299375637918641892962209583383 5264322635782294899/13921242001395112657501941955594013822830119803764 736, a[12,6] = 2346305388553404258656258473446184419154740172519949575 /256726716407895402892744978301151486254183185289662464, a[12,5] = -45 83493974484572912949314673356033540575/4519577036552507471573130342703 35135744, a[6,1] = 5611/114400, a[15,14] = -32507875096641/10240000000 00000*7^(1/2), a[12,4] = -16957088714171468676387054358954754000/14369 0415119654683326368228101570221, a[12,9] = -32058909627170725427914343 1215272753400810277402321024057136157075724905616701523016035208704867 4542196011/94756954968396581478301512445127360498465774712725761537244 9205973192657306017239103491074738324033259120, a[12,8] = 345685379554 677052215495825476969226377187500/747711674369300772216672031795513475 46362089, a[12,7] = 16571215593198468021712836909136106985862565734848 08662625/13431480411255146477259155104956093505361644432088109056, a[1 3,1] = 44901867737754616851973/1014046409980231013380680, a[16,15] = - 8818028336501383042724502163418043935242930059774292246092303626208484 1482480653958/11852361034459769519597506546144613989755568051785365407 543288652815101388446044921875+198068155659884577440583331314306224717 8836198896959594619834515105995312818424208/94818888275678156156780052 369156911918044544414282923260346309222520811107568359375*7^(1/2), a[1 2,11] = -6122933601070769591613093993993358877250/10505170015102355131 98246721302027675953, a[12,10] = 4027954583270623343310043858845893321 0937500/8896460842799482846916972126377338947215101, c[3] = 341/3200, \+ a[13,6] = 791638675191615279648100000/2235604725089973126411512319, a[ 13,8] = -13734512432397741476562500000/875132892924995907746928783, a[ 13,7] = 3847749490868980348119500000/15517045062138271618141237517, a[ 13,12] = -306814272936976936753/1299331183183744997286, c[2] = 1/20, a [13,11] = 282035543183190840068750/12295407629873040425991, a[13,10] = -9798363684577739445312500000/308722986341456031822630699, a[13,9] = \+ 1227476547031319687842881203774063505031923427600698639829444355496961 6342274215316330684448207141/48934514749371551765038583414351093488882 9280686609654482896526796523353052166757299452852166040, a[8,1] = -122 1101821869329/690812928000000, c[10] = 909/1000, c[4] = 1023/6400, a[5 ,3] = -3899844/2907025, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100 , a[8,4] = -125/2, a[7,6] = 5611/283500, a[14,6] = 3037913416047823635 649583750/15649233075629811884880586233-302465625814318865951896498250 /5367686944941025476514041077919*7^(1/2), a[16,8] = 586899259341979493 9656405512179634168543838567661624521691517151343133691358404104159327 002855091833886481/250730181931602953351962624259572198106967555130814 4546055132040021979206324088409943760373235100000000000+59180686976880 8939000154715250591718172649305577148018849844766255673325814572178653 5132299414153253117/46431515172519065435548634122142999649438436135336 010112132074815221837154149785369328895800650000000000*7^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 183874328794901398385760606250/76033 5208044775309288920638333-114787229090554407592495836250/3725642519419 3990155157111278317*7^(1/2), c[7] = 31/200, c[8] = 943/1000, c[9] = 70 67558016280/7837150160667, a[16,10] = 36668490233102499502202105336704 2762257672081921157863737233079069665848901352651139887527426189000273 823/884507612016992413045942215510719856000077549488758852211462407717 486163006277913807113787900300000000000-554145776168200641359928749044 9451392743826453759482574311165248687435785616082275654917223189127650 23/5459923530969088969419396392041480592593071293140486742046064245169 667672878258727204406098150000000000*7^(1/2), a[3,2] = 116281/1024000, a[7,4] = 8602624/76559175, a[4,1] = 1023/25600, a[2,1] = 1/20, a[3,1] = -7161/1024000, c[5] = 39/100, a[14,12] = 11352128098668146659861/25 4668911904014019468056-5215842639928607924801/127334455952007009734028 *7^(1/2), c[11] = 47/50, a[6,5] = 923521/5106400, a[7,1] = 21173/34320 0, a[6,4] = 31744/135025\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 17" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The stan dard (simple) order conditions can be adapted to give a method of stag e by stage construction for an interpolation scheme that avoids dealin g with the weight polynomials for a given stage (corresponding to an \+ \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_16 := SimpleOrderCon ditions(7,16,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,17,25,27,32,61 ,63,64]:\ninterp_order_eqns17 := []:\nfor ct in whch do\n temp_eqn : = convert(SO7_16[ct],'interpolation_order_condition'):\n interp_orde r_eqns17 := [op(interp_order_eqns17),temp_eqn];\nend do:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 585 "interp_order_eqns17 := [add(a[17,i ],i=1..16)=c[17],seq(op(StageOrderConditions(i,17..17,'expanded')),i=2 ..7),\n add(a[17,i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*add(a[l,m]*c[m ],\n m=2..l-1),l=2..k-1),k=2..j-1),j=2..i-1),i=2..16)=c[17]^6/720 , ##17\n add(a[17,i]*add(a[i,j]*c[j]^2*add(a[j,k]*c[k],k=2..j-1),j= 2..i-1),i=2..16)=c[17]^6/60, ##25\n add(a[17,i]*add(a[i,j]*add(a[j, k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..16)=c[17]^6/120, ##27\n add(a[1 7,i]*c[i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..16)=c[17]^7/35, ##61\n \+ add(a[17,i]*add(a[i,j]*c[j]^5,j=2..i-1),i=2..16)=c[17]^7/42]: ##63" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We spec ify the node " }{XPPEDIT 18 0 "c[17] = 37/100;" "6#/&%\"cG6#\"#<*&\"# P\"\"\"\"$+\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "e12 := `union`(e11,\{c[17]=37/100,seq(a[17,i]=0,i=2. .5)\}):\neqs_17 := expand(subs(e12,interp_order_eqns17)):\nnops(eqs_17 );\nindets(eqs_17);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#<\"#6&F%6$F'\"\"\"&F% 6$F'\"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$F'\"#5&F%6$F'\"#7&F% 6$F'\"#8&F%6$F'\"#9&F%6$F'\"#:&F%6$F'\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "e13 := solve(\{op(eqs_17)\}):\ninfolevel[sol ve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e14 := `union`(e12,e13):\nseq(a[17,i]=subs(e14,a[17,i ]),i=1..16):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "62/&%\"aG 6$\"#<\"\"\"$\"I2mn9X*z#p8G*HOToOR#o7_!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F' \"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"I\\'Gc1ZB'o54#=xz WnR3DR&F+/&F%6$F'\"\"($\"I=DG**fI>aG3kMu4e2m\"F+/&F%6$F'\"\")$!I(er IY)H$*p$*=alzn#zv&[aW!#R/&F%6$F'\"\"*$\"Ip9j_\"oz\"oH6=A[L6()FP/&F%6$F'\"#6$\"Il%H+xA&)*p?UYV%)4=bsqF+/&F%6$F'\"#8$!ISk'H?T z$H%=6Hk@$*>\\JS&=F+/&F%6$F'\"#9$\"IZ1>if/(3Ul6XYQNa5-/N#F+/&F%6$F'\"# :$\"IhkxuFS8yPc04AyS.^zWB!#S/&F%6$F'\"#;$!I^mwox4)p*3Be)))*G:,D2T#)F+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Thes e linking coefficients agree to 40 digits with those of the published \+ scheme." }}{PARA 0 "" 0 "" {TEXT -1 70 "#----------------------------- ----------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check which of the (adapted) simple order conditions are sa tisfied at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_16 ) do\n tt := convert(SO7_16[ct],'interpolation_order_condition'):\n \+ if expand(subs(e14,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end \+ if: \nend do:\nop(recd);\nnops(recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"# 9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J \"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\" #\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "#------------------------------------ ---------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e14" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16462 "e14 := \{a[13,5] = 0, a[14,10] = 304484151 49825325326308593750/15127426330731345559308904251+5887803942383224482 632816406250/741243890205835932406136308299*7^(1/2), a[14,3] = 0, c[14 ] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[4,2] = 0, c[13] = 1, a[11,3] = 0, a[12,2] = 0, a[11,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/392-3/ 392*7^(1/2), a[10,2] = 0, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[14,2] \+ = 0, a[6,3] = 0, a[9,3] = 0, a[14,5] = 0, a[6,2] = 0, a[7,2] = 0, a[7, 3] = 0, a[5,2] = 0, a[13,2] = 0, a[14,4] = 0, a[13,4] = 0, a[13,3] = 0 , a[17,1] = 259918413834103868653305362543252668375436100480694679123/ 5337146950192352259892078131372117541294004208900000000000+76201610563 498999758072388412964731305853858164539/588309849007093503074523603546 30925278813979375000000*7^(1/2), a[17,2] = 0, a[17,3] = 0, a[15,5] = 0 , a[15,4] = 0, a[15,3] = 0, a[15,2] = 0, a[17,12] = 375570851428850058 39199344190346713535658856538249/1743222737754430793071508969274146027 468985000000000-154646441255718575659310956019446499029993293/44343272 73490106819982470923062032019406250000*7^(1/2), a[17,13] = 89396876305 5007240865092535772471/318849293214150122419952455000000000-5023926712 398881127692249781201/622752525808886957851469638671875*7^(1/2), a[17, 15] = 34256025405285245733900192129792/2420024525218555187013580150796 75+7819897427138839913759197790208/222642256320107077205249373873301*7 ^(1/2), a[17,14] = -44728197763861767267686691630511059789/98598474374 9870244050427250768515625000+51329695014236122164753612289240491/19719 69487499740488100854501537031250*7^(1/2), a[14,11] = -1466584824552592 368357500/602474973863778980873559-946677979546641857718938375/5904254 7438650340125608782*7^(1/2), c[17] = 37/100, a[16,2] = 0, a[14,8] = 74 200703416028798327128906250/42881511753324799479599510367+286766471992 17261041085964843750/2101194075912915174500376007983*7^(1/2), a[16,4] \+ = 0, a[16,3] = 0, a[16,1] = 318424113799300488598249341153341045243508 123466296541704743316342575624901053326413058788147679306059990010001/ 7717193457109162513896339989541939168625874356612620275288295636317564 198088241611239632333125000000000000000000-779036898575193892382067466 7096775602063482288285741296170327553630654431762486304432426008955534 42815357/4763699664882199082652061721939468622608564417662111281042157 800196027282770519513110884156250000000000000000*7^(1/2), a[16,5] = 0, a[16,6] = 38677873603874023494821224663560520533757129031705209294828 410503456708269950565640452627167440128229776168201/100080082118918185 7363609351052680617003947572378045335413864786223498306939613602026574 79143973437500000000000+2660388496179864594360815841760855766508730523 0552993954979509880880776475635566879209893579250983975560997/18533348 5405404047659927657602348262408138439329267654706271256708055242025854 3707456619984147656250000000000*7^(1/2), a[14,1] = 8941065567926479206 438689/198753096356125278622613280+152838094177334666489948287/3311226 58529304714185273724480*7^(1/2), a[16,7] = 377308587903340677792605542 0852886774791582751698542812584349024015469545595181670092987168923424 4950950501493/13892859740661471042270156150553691881345433978558714782 0728065583231442155198771083100889637109062500000000000-10233001453579 5077906358306091556548197835190067615199809734380171826687208346433432 3592817694003399143529013/69464298703307355211350780752768459406727169 892793573910364032791615721077599385541550444818554531250000000000*7^( 1/2), c[16] = 3923/5000, a[14,9] = -1287199574154792351913181734948009 77965419044818935750521702127672316317131710725250664367373074947/9591 1648908768241459475623492128143238210539014575492278647719252118577198 224684430692759024543840-882185082068177622906787605005472555636444759 904636504531314529608120034066591927204665390933748491981/159788807082 0078902714863887378854866348587579982827701362271002740295496122423242 61534136534890037440*7^(1/2), a[16,9] = 140826789358890217295475288587 9200378495596410425522130381311818054028954090950082397504605548502598 4309922595887619652662278786898541530825637526247745153879591696545731 13399136914397/3724061476220048359904192212639840313935246560930005829 4686034773950283950518320293022388613265519828748009305183412567399661 30319527284854144491887040986350427874375000000000000000000+3658948724 8952850975626989635036036984043973038459711023487544456316770887185519 6974974526134564130489860871802571616243554346034341340988481532061923 3885431694117641997082749518637/26600439115857488285029944375998859385 2517611495000416390614534099644885360845144950159918666182284491057209 3227386611957118664513948060610103208490743561678877053125000000000000 0000*7^(1/2), a[16,13] = 166779545966699459600659373461271836409487111 3097496394713308473411640245893712505210071/14817694205771874634426898 9730526651074011467617513077874633459904607921875000000000000000+99583 2251155619761130552690028230372221382141759690163674224410995343450397 7764414401/18522117757214843293033623716315831384251433452189134734329 182488075990234375000000000000*7^(1/2), a[16,14] = 1009173346124759287 68168912044205950231837537138211695125551197160193984809290378367269/9 9638921083071752167902327460036161216912696087516535673554508574828121 4843750000000000000-14035051920781198110898020935530411673249916957009 159970154760483863951056688692206659/498194605415358760839511637300180 806084563480437582678367772542874140607421875000000000*7^(1/2), a[17,6 ] = 282346020910207495266878857047379938762003012480259/32425472010365 67051067715195934842380312990557300000-3358790251863352976542564613821 8963307158947223/2680677249534198950948838620977878951978332140625*7^( 1/2), a[17,7] = 18223033119253981180923879156484227563883974739722227/ 166545164928928328045818080519913611121586895656860000-100313230082181 54138865806552959823083322462822417/2859635386829126511775722536399615 57557669807103125*7^(1/2), a[17,8] = 130477739792862117433436067272464 64862780752381075/15028547139305830778230444498740751854340622555424-4 999044060295488160350089639911998698434613211/248487882594342440116244 1220029886219302351613*7^(1/2), a[17,10] = 154140265569929844304698928 693303965519909065225/143288135906669396836900874715366252459009313056 -2247178508319186582684151266568513851628311/6074826003369174672572449 24006945514766523*7^(1/2), c[15] = 69/400, a[17,11] = -161735651719420 7679205444849356224376563538983069/13196718082504734260923438707691958 31358057800000+3182258795374348673491299173537610086635766763/10909985 18725589803317083226495697611903156250*7^(1/2), a[16,11] = -6928011549 4548817616994817137112350435915022956794236048692461535196705336665032 10900114933332845376150551/2201686892988193948947136308373301646572872 307345365971075591622778810160246298613549949543750000000000000-333373 1846045007922723796544335071809571030575931558340030474714208942484430 219726989269407196212710461/203859897498906847124734843367898300608599 28771716351584033255766470464446724987162499532812500000000000*7^(1/2) , a[15,6] = -191031498688788616226618012353233613/16957475840682462825 521397145600000000+219561744459503945250128450660540211/11093675783624 041100808390656000000000*7^(1/2), a[17,9] = -4834722584846356480292651 5730860893559485989808658496113143557813259777019828691250781666119617 1113002225811104824321561163203/65652052283594593065389514665746547797 6134801711666046255394507802569129304793694528174865643432008312869020 528492900000000000+167753232043115470453615258085334954467158064037840 5905771396048244239186853549909489496412837700466748475273250933123162 009/586179038246380295226692095229879891050120358671130398442316524823 722436879280084400156130038778578850775911186154375000000*7^(1/2), a[1 5,7] = 14172386735756874627703502458542559611/907968808207405150684378 69813760000000+53924868594716008726045512055295307/9077835462591889896 660794982400000000*7^(1/2), a[15,1] = 30016459570250276767768375282138 1843/5685191595792547980976128000000000000-484972017420526515813879842 91872403/53061788227397114489110528000000000000*7^(1/2), a[15,8] = -87 7010740240762900922180472417/57186012045619550819463987200+51313927022 3400882486686075445021/68051354334287265475162144768000*7^(1/2), a[15, 9] = 45550848707315089359912544356558140241875548161939830581423717993 7868420320279982259692853675276040350274866593/74074089804961736793547 1008762629422134325227668913118716370672910772046081053818353815239174 47168000000000000-8088939903267911374096450504734124828801217514657511 355287393514657504893760256657562215933144919629/269503077658229281467 5576398887868995199235474954188986106083154776989963827197767483392000 000000000*7^(1/2), a[16,12] = 6876230787902827767160767205214587513807 297711754964615123643390622098477381787285420842315969989227509309/727 0805190498675407289247054348907579029951742748276627467199264704732251 4891352171003011718750000000000000+78621364964664876362030452193189774 72234187207775060875131229872919912547716368298063580928728400258823/1 0771563245183222825613699339776159376340669248515965374025480392155158 89109501513644489062500000000000000*7^(1/2), a[15,10] = -8148719013685 3529424335403395147/8815890907570223312212603699200+412817284567416435 38288815829781/9275782652527467499145003008000*7^(1/2), a[9,1] = -1472 5142644862158038813847088772642463460444333070942078290519780445318011 33057155/1246894801620032001157059621643986024803301558393487900440453 636168046069686436608, a[8,7] = 6070139212132283/92502016000000, a[8,6 ] = 1501408353528689/265697280000000, a[8,5] = -1024030607959889/16892 9280000000, c[6] = 93/200, a[9,5] = -120706792584692548079789364417331 87949484571516120469966534514296406891652614970375/2722031154761657221 710478184531100699497284085048389015085076961673446140398628096, a[15, 11] = 149526841897534768477167673223553/810982119999355452252160000000 0-10197331971622785977909572704970449/11353749679990976331530240000000 00*7^(1/2), a[9,4] = -517229431108566845837517565524698123003902533693 3699114138315270772319372469280000/12461938100480914589727863057121529 8365257079410236252921850936749076487132995191, a[9,8] = 1033284818445 2015604056836767286656859124007796970668046446015775000000/13127035500 36033648073834248740727914537972028638950165249582733679393783, a[9,7] = 6641131229599116421347821358391064699281403281605770353571553403929 50009492511875/1517846559858624813633302310729534917527976515008907830 1139943253016877823170816, a[9,6] = 7801251558438936413230905525304310 36567795592568497182701460674803126770111481625/1831104254127319721978 89874507158786859226102980861859505241443073629143100805376, a[10,7] = 318607235173649312405151265849660869927653414425413/67147167155589653 03132938072935465423910912000000, a[10,8] = 21208320243451908228184224 5535894/20022426044775672563822865371173879, a[10,6] = 211276702141728 02870128286992003940810655221489/4679473877997892906145822697976708633 673728000, a[10,5] = -180269259803172281163724663224981097/38100922558 256871086579832832000000, a[10,4] = -20462749524591049105403365239069/ 454251913499893469596231268750, a[10,1] = -290555733603374150885386184 42231036441314060511/22674759891089577691327962602370597632000000000, \+ a[11,1] = -2342659845814086836951207140065609179073838476242943917/135 8480961351056777022231400139158760857532162795520000, a[17,16] = 27281 8447134154078329357400195312500/12220217271375375959649133447445146063 -483754995004275852097660875000000000/12220217271375375959649133447445 146063*7^(1/2), a[10,9] = -2698404929400842518721166485087129798562269 848229517793703413951226714583/469545674913934315077000442080871141884 676035902717550325616728175875000000, a[15,12] = -22716443912630306217 48072222279/74102764220560004300800000000000, a[11,6] = 20980822345096 760292224086794978105312644533925634933539/377588999200755080387872783 9115494641972212962174156800, a[15,13] = -288610253631/128000000000000 -4643982156663/4096000000000000*7^(1/2), a[11,5] = -260530859592565341 52588089363841/4377552804565683061011299942400, a[11,4] = -99628603013 2538159613930889652/16353068885996164905464325675, a[5,4] = 3982992/29 07025, a[12,1] = -2866556991825663971778295329101033887534912787724034 363/868226711619262703011213925016143612030669233795338240, a[11,10] = -31155237437111730665923206875/392862141594230515010338956291, a[11,9 ] = 300760669768102517834232497565452434946672266195876496371874262392 684852243925359864884962513/465544333750134645558506533660450560376082 4779615521285751892810315680492364106674524398280000, a[11,8] = 161021 426143124178389075121929246710833125/109972077221310346506670413643464 22894371443, a[11,7] = 89072299375637918641892962209583383526432263578 2294899/13921242001395112657501941955594013822830119803764736, a[12,6] = 2346305388553404258656258473446184419154740172519949575/25672671640 7895402892744978301151486254183185289662464, a[12,5] = -45834939744845 72912949314673356033540575/451957703655250747157313034270335135744, a[ 6,1] = 5611/114400, a[15,14] = -32507875096641/1024000000000000*7^(1/2 ), a[12,4] = -16957088714171468676387054358954754000/14369041511965468 3326368228101570221, a[12,9] = -32058909627170725427914343121527275340 08102774023210240571361570757249056167015230160352087048674542196011/9 4756954968396581478301512445127360498465774712725761537244920597319265 7306017239103491074738324033259120, a[12,8] = 345685379554677052215495 825476969226377187500/74771167436930077221667203179551347546362089, a[ 12,7] = 1657121559319846802171283690913610698586256573484808662625/134 31480411255146477259155104956093505361644432088109056, a[13,1] = 44901 867737754616851973/1014046409980231013380680, a[16,15] = -881802833650 1383042724502163418043935242930059774292246092303626208484148248065395 8/11852361034459769519597506546144613989755568051785365407543288652815 101388446044921875+198068155659884577440583331314306224717883619889695 9594619834515105995312818424208/94818888275678156156780052369156911918 044544414282923260346309222520811107568359375*7^(1/2), a[12,11] = -612 2933601070769591613093993993358877250/10505170015102355131982467213020 27675953, a[12,10] = 40279545832706233433100438588458933210937500/8896 460842799482846916972126377338947215101, c[3] = 341/3200, a[13,6] = 79 1638675191615279648100000/2235604725089973126411512319, a[13,8] = -137 34512432397741476562500000/875132892924995907746928783, a[13,7] = 3847 749490868980348119500000/15517045062138271618141237517, a[13,12] = -30 6814272936976936753/1299331183183744997286, c[2] = 1/20, a[13,11] = 28 2035543183190840068750/12295407629873040425991, a[13,10] = -9798363684 577739445312500000/308722986341456031822630699, a[13,9] = 122747654703 1319687842881203774063505031923427600698639829444355496961634227421531 6330684448207141/48934514749371551765038583414351093488882928068660965 4482896526796523353052166757299452852166040, a[17,4] = 0, a[17,5] = 0, a[8,1] = -1221101821869329/690812928000000, c[10] = 909/1000, c[4] = \+ 1023/6400, a[5,3] = -3899844/2907025, a[4,3] = 3069/25600, a[5,1] = 42 02367/11628100, a[8,4] = -125/2, a[7,6] = 5611/283500, a[14,6] = 30379 13416047823635649583750/15649233075629811884880586233-3024656258143188 65951896498250/5367686944941025476514041077919*7^(1/2), a[16,8] = 5868 9925934197949396564055121796341685438385676616245216915171513431336913 58404104159327002855091833886481/2507301819316029533519626242595721981 069675551308144546055132040021979206324088409943760373235100000000000+ 5918068697688089390001547152505917181726493055771480188498447662556733 258145721786535132299414153253117/464315151725190654355486341221429996 49438436135336010112132074815221837154149785369328895800650000000000*7 ^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 1838743287949013983857 60606250/760335208044775309288920638333-114787229090554407592495836250 /37256425194193990155157111278317*7^(1/2), c[7] = 31/200, c[8] = 943/1 000, c[9] = 7067558016280/7837150160667, a[16,10] = 366684902331024995 0220210533670427622576720819211578637372330790696658489013526511398875 27426189000273823/8845076120169924130459422155107198560000775494887588 52211462407717486163006277913807113787900300000000000-5541457761682006 4135992874904494513927438264537594825743111652486874357856160822756549 1722318912765023/54599235309690889694193963920414805925930712931404867 42046064245169667672878258727204406098150000000000*7^(1/2), a[3,2] = 1 16281/1024000, a[7,4] = 8602624/76559175, a[4,1] = 1023/25600, a[2,1] \+ = 1/20, a[3,1] = -7161/1024000, c[5] = 39/100, a[14,12] = 113521280986 68146659861/254668911904014019468056-5215842639928607924801/1273344559 52007009734028*7^(1/2), c[11] = 47/50, a[6,5] = 923521/5106400, a[7,1] = 21173/343200, a[6,4] = 31744/135025\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "T he standard (simple) order conditions can be adapted to give a method \+ of stage by stage construction for an interpolation scheme that avoids dealing with the weight polynomials for a given stage (corresponding \+ to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_17 := SimpleOr derConditions(7,17,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,17,25,2 7,32,61,63,64]:\ninterp_order_eqns18 := []:\nfor ct in whch do\n tem p_eqn := convert(SO7_17[ct],'interpolation_order_condition'):\n inte rp_order_eqns18 := [op(interp_order_eqns18),temp_eqn];\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternati vely, the order conditions can be specified explicitly as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 590 "interp_order_eqns18 := [add (a[18,i],i=1..17)=c[18],seq(op(StageOrderConditions(i,18..18,'expanded ')),i=2..7),\n add(a[18,i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*add(a[l ,m]*c[m],\n m=2..l-1),l=2..k-1),k=2..j-1),j=2..i-1),i=2..17) =c[18]^6/720, ##17\n add(a[18,i]*add(a[i,j]*c[j]^2*add(a[j,k]*c[k], k=2..j-1),j=2..i-1),i=2..17)=c[18]^6/60, ##25\n add(a[18,i]*add(a[ i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..17)=c[18]^6/120, ##27 \n add(a[18,i]*c[i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..17)=c[18]^7/35 , ##61\n add(a[18,i]*add(a[i,j]*c[j]^5,j=2..i-1),i=2..17)=c[18]^7/ 42]: ##63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We specify " }{XPPEDIT 18 0 "c[18] = 1/2;" "6#/&%\"cG6#\"#=*& \"\"\"F)\"\"#!\"\"" }{TEXT -1 10 " and also " }{XPPEDIT 18 0 "a[18,17] =0" "6#/&%\"aG6$\"#=\"#<\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "e15 := `union`(e14,\{c[18]=1/2,seq(a[18,i]=0,i= 2..5),a[18,17]=0\}):\neqs_18 := expand(subs(e15,interp_order_eqns18)): \nnops(eqs_18);\nindets(eqs_18);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#= \"#6&F%6$F'\"#7&F%6$F'\"#:&F%6$F'\"#;&F%6$F'\"#8&F%6$F'\"#9&F%6$F'\"\" \"&F%6$F'\"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$F'\"#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e16 := solve(\{op(eqs_18)\} ):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e17 := `union`(e15,e16):\nseq(a[18, i]=subs(e17,a[18,i]),i=1..17):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "63/&%\"aG6$\"#=\"\"\"$\"Ii8Y$)y(*>W'**p)f8dNqG5?]!#T/&F%6 $F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F' \"\"'$\"IRBkn0q/hAK\\6)\\&zM!4Ab\"!#S/&F%6$F'\"\"($\"I+j]ZZ'[8\"48Z\" \\B*3C%oUE\"FD/&F%6$F'\"\")$!H^\\&=s0YT<\\q,Z)RNI1#\\^!#Q/&F%6$F'\"\"* $\"H^:J%\\Jj<`ugEHp.**4Mo%)FQ/&F%6$F'\"#5$!II%)\\&42i$)4UaF&\\\"3\"oI@ m5FQ/&F%6$F'\"#6$\"H.RL=!p:?'*4HOG(f\\AL=a(FQ/&F%6$F'\"#7$!HuP0d$fC\\m S%RC9KQ6opV(FD/&F%6$F'\"#8$!IkVw@6Af<#)R$>EQ=mo()e0#F+/&F%6$F'\"#9$\"H WeRiyx$*Hyhs!)H5ZE&z`xFD/&F%6$F'\"#:$\"Ixte()RLr>w8jHWDN?#fi/\"FD/&F%6 $F'\"#;$!I6^X8I1(oA]9_$z>X1L@z6FD/&F%6$F'\"# " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops (SO7_17) do\n tt := convert(SO7_17[ct],'interpolation_order_conditio n'):\n if expand(subs(e17,lhs(tt)=rhs(tt))) then recd := [op(recd),c t] end if: \nend do:\nop(recd);\nnops(recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"# 6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G \"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\" #Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"# j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#------------------------ ---------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e17" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19122 "e17 := \{a[13,5] = 0, a[1 4,10] = 30448415149825325326308593750/15127426330731345559308904251+58 87803942383224482632816406250/741243890205835932406136308299*7^(1/2), \+ a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[4,2] = 0, c[13] \+ = 1, a[11,3] = 0, a[12,2] = 0, a[11,2] = 0, c[12] = 1, a[10,3] = 0, a[ 14,13] = 3/392-3/392*7^(1/2), a[10,2] = 0, a[8,2] = 0, a[8,3] = 0, a[9 ,2] = 0, a[14,2] = 0, a[6,3] = 0, a[9,3] = 0, a[14,5] = 0, a[6,2] = 0, a[7,2] = 0, a[7,3] = 0, a[5,2] = 0, a[13,2] = 0, a[14,4] = 0, a[13,4] = 0, a[13,3] = 0, a[17,1] = 25991841383410386865330536254325266837543 6100480694679123/53371469501923522598920781313721175412940042089000000 00000+76201610563498999758072388412964731305853858164539/5883098490070 9350307452360354630925278813979375000000*7^(1/2), a[17,2] = 0, a[17,3] = 0, a[15,5] = 0, a[15,4] = 0, a[15,3] = 0, a[15,2] = 0, a[17,12] = 3 7557085142885005839199344190346713535658856538249/17432227377544307930 71508969274146027468985000000000-1546464412557185756593109560194464990 29993293/4434327273490106819982470923062032019406250000*7^(1/2), a[17, 13] = 893968763055007240865092535772471/318849293214150122419952455000 000000-5023926712398881127692249781201/6227525258088869578514696386718 75*7^(1/2), a[17,15] = 34256025405285245733900192129792/24200245252185 5518701358015079675+7819897427138839913759197790208/222642256320107077 205249373873301*7^(1/2), a[17,14] = -447281977638617672676866916305110 59789/985984743749870244050427250768515625000+513296950142361221647536 12289240491/1971969487499740488100854501537031250*7^(1/2), a[14,11] = \+ -1466584824552592368357500/602474973863778980873559-946677979546641857 718938375/59042547438650340125608782*7^(1/2), a[18,15] = 6965254389974 33496880000000000000/6011340920642891084541733094579127*7^(1/2)-121389 1006926781481467040000000000/6011340920642891084541733094579127, c[17] = 37/100, a[16,2] = 0, a[14,8] = 74200703416028798327128906250/428815 11753324799479599510367+28676647199217261041085964843750/2101194075912 915174500376007983*7^(1/2), a[16,4] = 0, a[16,3] = 0, a[16,1] = 318424 1137993004885982493411533410452435081234662965417047433163425756249010 53326413058788147679306059990010001/7717193457109162513896339989541939 1686258743566126202752882956363175641980882416112396323331250000000000 00000000-7790368985751938923820674667096775602063482288285741296170327 55363065443176248630443242600895553442815357/4763699664882199082652061 7219394686226085644176621112810421578001960272827705195131108841562500 00000000000000*7^(1/2), a[16,5] = 0, a[18,2] = 0, a[18,3] = 0, a[18,4] = 0, a[18,5] = 0, a[16,6] = 38677873603874023494821224663560520533757 129031705209294828410503456708269950565640452627167440128229776168201/ 1000800821189181857363609351052680617003947572378045335413864786223498 30693961360202657479143973437500000000000+2660388496179864594360815841 7608557665087305230552993954979509880880776475635566879209893579250983 975560997/185333485405404047659927657602348262408138439329267654706271 2567080552420258543707456619984147656250000000000*7^(1/2), a[14,1] = 8 941065567926479206438689/198753096356125278622613280+15283809417733466 6489948287/331122658529304714185273724480*7^(1/2), a[16,7] = 377308587 9033406777926055420852886774791582751698542812584349024015469545595181 6700929871689234244950950501493/13892859740661471042270156150553691881 3454339785587147820728065583231442155198771083100889637109062500000000 000-102330014535795077906358306091556548197835190067615199809734380171 8266872083464334323592817694003399143529013/69464298703307355211350780 7527684594067271698927935739103640327916157210775993855415504448185545 31250000000000*7^(1/2), c[16] = 3923/5000, a[14,9] = -1287199574154792 3519131817349480097796541904481893575052170212767231631713171072525066 4367373074947/95911648908768241459475623492128143238210539014575492278 647719252118577198224684430692759024543840-882185082068177622906787605 0054725556364447599046365045313145296081200340665919272046653909337484 91981/1597888070820078902714863887378854866348587579982827701362271002 74029549612242324261534136534890037440*7^(1/2), a[16,9] = 140826789358 8902172954752885879200378495596410425522130381311818054028954090950082 3975046055485025984309922595887619652662278786898541530825637526247745 15387959169654573113399136914397/3724061476220048359904192212639840313 9352465609300058294686034773950283950518320293022388613265519828748009 3051834125673996613031952728485414449188704098635042787437500000000000 0000000+36589487248952850975626989635036036984043973038459711023487544 4563167708871855196974974526134564130489860871802571616243554346034341 3409884815320619233885431694117641997082749518637/26600439115857488285 0299443759988593852517611495000416390614534099644885360845144950159918 6661822844910572093227386611957118664513948060610103208490743561678877 0531250000000000000000*7^(1/2), a[16,13] = 166779545966699459600659373 4612718364094871113097496394713308473411640245893712505210071/14817694 2057718746344268989730526651074011467617513077874633459904607921875000 000000000000+995832251155619761130552690028230372221382141759690163674 2244109953434503977764414401/18522117757214843293033623716315831384251 433452189134734329182488075990234375000000000000*7^(1/2), a[16,14] = 1 0091733461247592876816891204420595023183753713821169512555119716019398 4809290378367269/99638921083071752167902327460036161216912696087516535 6735545085748281214843750000000000000-14035051920781198110898020935530 411673249916957009159970154760483863951056688692206659/498194605415358 7608395116373001808060845634804375826783677725428741406074218750000000 00*7^(1/2), a[18,6] = -16530168995389981543029356724223343073092031250 /399914154794509936298351540831963893571935502067*7^(1/2)+211619505426 273318600241507183021439574335170625/799828309589019872596703081663927 787143871004134, a[17,6] = 2823460209102074952668788570473799387620030 12480259/3242547201036567051067715195934842380312990557300000-33587902 518633529765425646138218963307158947223/268067724953419895094883862097 7878951978332140625*7^(1/2), a[18,14] = -20397985870195185860619095571 9875/1363025309759820625375310631462396+292606914085123400690727956406 25/340756327439955156343827657865599*7^(1/2), a[18,1] = 19254967621004 828660163073938975933503722889627/495039717119290644395786957112776119 772197491840+2432585298063029726317162869389207640440757875/5692956746 87184241055155000679692537738027115616*7^(1/2), a[17,7] = 182230331192 53981180923879156484227563883974739722227/1665451649289283280458180805 19913611121586895656860000-1003132300821815413886580655295982308332246 2822417/285963538682912651177572253639961557557669807103125*7^(1/2), a [17,8] = 13047773979286211743343606727246464862780752381075/1502854713 9305830778230444498740751854340622555424-49990440602954881603500896399 11998698434613211/2484878825943424401162441220029886219302351613*7^(1/ 2), a[17,10] = 154140265569929844304698928693303965519909065225/143288 135906669396836900874715366252459009313056-224717850831918658268415126 6568513851628311/607482600336917467257244924006945514766523*7^(1/2), a [18,13] = 305198437376024806501272473/6121906429711682350463087136-152 74180675827991876494250/573928727785470220355914419*7^(1/2), c[15] = 6 9/400, a[17,11] = -1617356517194207679205444849356224376563538983069/1 319671808250473426092343870769195831358057800000+318225879537434867349 1299173537610086635766763/10909985187255898033170832264956976119031562 50*7^(1/2), a[16,11] = -6928011549454881761699481713711235043591502295 679423604869246153519670533666503210900114933332845376150551/220168689 2988193948947136308373301646572872307345365971075591622778810160246298 613549949543750000000000000-333373184604500792272379654433507180957103 0575931558340030474714208942484430219726989269407196212710461/20385989 7498906847124734843367898300608599287717163515840332557664704644467249 87162499532812500000000000*7^(1/2), a[18,10] = 23884714645221746779271 45048979026999230849609375/1104512714280576600617777575930948196038196 78814-3987768244192880428103766706721197663574218750/32677890955046644 9886916442583120768058638103*7^(1/2), a[15,6] = -191031498688788616226 618012353233613/16957475840682462825521397145600000000+219561744459503 945250128450660540211/11093675783624041100808390656000000000*7^(1/2), \+ a[17,9] = -48347225848463564802926515730860893559485989808658496113143 5578132597770198286912507816661196171113002225811104824321561163203/65 6520522835945930653895146657465477976134801711666046255394507802569129 304793694528174865643432008312869020528492900000000000+167753232043115 4704536152580853349544671580640378405905771396048244239186853549909489 496412837700466748475273250933123162009/586179038246380295226692095229 8798910501203586711303984423165248237224368792800844001561300387785788 50775911186154375000000*7^(1/2), a[15,7] = 141723867357568746277035024 58542559611/90796880820740515068437869813760000000+5392486859471600872 6045512055295307/9077835462591889896660794982400000000*7^(1/2), a[15,1 ] = 300164595702502767677683752821381843/56851915957925479809761280000 00000000-48497201742052651581387984291872403/5306178822739711448911052 8000000000000*7^(1/2), a[15,8] = -877010740240762900922180472417/57186 012045619550819463987200+513139270223400882486686075445021/68051354334 287265475162144768000*7^(1/2), a[15,9] = 45550848707315089359912544356 5581402418755481619398305814237179937868420320279982259692853675276040 350274866593/740740898049617367935471008762629422134325227668913118716 37067291077204608105381835381523917447168000000000000-8088939903267911 3740964505047341248288012175146575113552873935146575048937602566575622 15933144919629/2695030776582292814675576398887868995199235474954188986 106083154776989963827197767483392000000000000*7^(1/2), a[16,12] = 6876 2307879028277671607672052145875138072977117549646151236433906220984773 81787285420842315969989227509309/7270805190498675407289247054348907579 0299517427482766274671992647047322514891352171003011718750000000000000 +786213649646648763620304521931897747223418720777506087513122987291991 2547716368298063580928728400258823/10771563245183222825613699339776159 3763406692485159653740254803921551588910950151364448906250000000000000 0*7^(1/2), a[15,10] = -81487190136853529424335403395147/88158909075702 23312212603699200+41281728456741643538288815829781/9275782652527467499 145003008000*7^(1/2), a[18,17] = 0, a[9,1] = -147251426448621580388138 4708877264246346044433307094207829051978044531801133057155/12468948016 2003200115705962164398602480330155839348790044045363616804606968643660 8, a[8,7] = 6070139212132283/92502016000000, a[8,6] = 1501408353528689 /265697280000000, a[8,5] = -1024030607959889/168929280000000, c[6] = 9 3/200, a[9,5] = -12070679258469254807978936441733187949484571516120469 966534514296406891652614970375/272203115476165722171047818453110069949 7284085048389015085076961673446140398628096, a[15,11] = 14952684189753 4768477167673223553/8109821199993554522521600000000-101973319716227859 77909572704970449/1135374967999097633153024000000000*7^(1/2), a[9,4] = -51722943110856684583751756552469812300390253369336991141383152707723 19372469280000/1246193810048091458972786305712152983652570794102362529 21850936749076487132995191, a[9,8] = 103328481844520156040568367672866 56859124007796970668046446015775000000/1312703550036033648073834248740 727914537972028638950165249582733679393783, a[9,7] = 66411312295991164 2134782135839106469928140328160577035357155340392950009492511875/15178 4655985862481363330231072953491752797651500890783011399432530168778231 70816, a[9,6] = 780125155843893641323090552530431036567795592568497182 701460674803126770111481625/183110425412731972197889874507158786859226 102980861859505241443073629143100805376, a[10,7] = 3186072351736493124 05151265849660869927653414425413/6714716715558965303132938072935465423 910912000000, a[10,8] = 212083202434519082281842245535894/200224260447 75672563822865371173879, a[10,6] = 21127670214172802870128286992003940 810655221489/4679473877997892906145822697976708633673728000, a[10,5] = -180269259803172281163724663224981097/3810092255825687108657983283200 0000, a[10,4] = -20462749524591049105403365239069/45425191349989346959 6231268750, a[10,1] = -29055573360337415088538618442231036441314060511 /22674759891089577691327962602370597632000000000, a[11,1] = -234265984 5814086836951207140065609179073838476242943917/13584809613510567770222 31400139158760857532162795520000, a[17,16] = 2728184471341540783293574 00195312500/12220217271375375959649133447445146063-4837549950042758520 97660875000000000/12220217271375375959649133447445146063*7^(1/2), a[10 ,9] = -269840492940084251872116648508712979856226984822951779370341395 1226714583/46954567491393431507700044208087114188467603590271755032561 6728175875000000, a[15,12] = -2271644391263030621748072222279/74102764 220560004300800000000000, a[11,6] = 2098082234509676029222408679497810 5312644533925634933539/37758899920075508038787278391154946419722129621 74156800, a[18,7] = -814586114736897681347580051914111367091406250/703 9102490656311413601778551137515262958026021*7^(1/2)+720476286898479329 5755345584089235457889776021875/16654516492892832804581808051991361112 158689565686, a[15,13] = -288610253631/128000000000000-4643982156663/4 096000000000000*7^(1/2), a[11,5] = -26053085959256534152588089363841/4 377552804565683061011299942400, a[11,4] = -996286030132538159613930889 652/16353068885996164905464325675, a[5,4] = 3982992/2907025, a[12,1] = -2866556991825663971778295329101033887534912787724034363/868226711619 262703011213925016143612030669233795338240, a[11,10] = -31155237437111 730665923206875/392862141594230515010338956291, a[11,9] = 300760669768 1025178342324975654524349466722661958764963718742623926848522439253598 64884962513/4655443337501346455585065336604505603760824779615521285751 892810315680492364106674524398280000, a[11,8] = 1610214261431241783890 75121929246710833125/10997207722131034650667041364346422894371443, a[1 1,7] = 890722993756379186418929622095833835264322635782294899/13921242 001395112657501941955594013822830119803764736, a[12,6] = 2346305388553 404258656258473446184419154740172519949575/256726716407895402892744978 301151486254183185289662464, a[12,5] = -458349397448457291294931467335 6033540575/451957703655250747157313034270335135744, a[6,1] = 5611/1144 00, a[15,14] = -32507875096641/1024000000000000*7^(1/2), a[12,4] = -16 957088714171468676387054358954754000/143690415119654683326368228101570 221, a[12,9] = -320589096271707254279143431215272753400810277402321024 0571361570757249056167015230160352087048674542196011/94756954968396581 4783015124451273604984657747127257615372449205973192657306017239103491 074738324033259120, a[12,8] = 3456853795546770522154958254769692263771 87500/74771167436930077221667203179551347546362089, a[12,7] = 16571215 59319846802171283690913610698586256573484808662625/1343148041125514647 7259155104956093505361644432088109056, a[13,1] = 449018677377546168519 73/1014046409980231013380680, a[16,15] = -8818028336501383042724502163 4180439352429300597742922460923036262084841482480653958/11852361034459 7695195975065461446139897555680517853654075432886528151013884460449218 75+1980681556598845774405833313143062247178836198896959594619834515105 995312818424208/948188882756781561567800523691569119180445444142829232 60346309222520811107568359375*7^(1/2), a[12,11] = -6122933601070769591 613093993993358877250/1050517001510235513198246721302027675953, a[12,1 0] = 40279545832706233433100438588458933210937500/88964608427994828469 16972126377338947215101, c[3] = 341/3200, a[13,6] = 791638675191615279 648100000/2235604725089973126411512319, a[18,11] = -472881944207955026 637075845907843049717091875/263934361650094685218468774153839166271611 56+21164021072452475373967469896889463184140625/2199453013750789043487 239784615326385596763*7^(1/2), a[13,8] = -1373451243239774147656250000 0/875132892924995907746928783, a[13,7] = 3847749490868980348119500000/ 15517045062138271618141237517, a[13,12] = -306814272936976936753/12993 31183183744997286, c[2] = 1/20, a[13,11] = 282035543183190840068750/12 295407629873040425991, a[13,10] = -9798363684577739445312500000/308722 986341456031822630699, a[13,9] = 1227476547031319687842881203774063505 0319234276006986398294443554969616342274215316330684448207141/48934514 7493715517650385834143510934888829280686609654482896526796523353052166 757299452852166040, a[17,4] = 0, a[17,5] = 0, a[8,1] = -12211018218693 29/690812928000000, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = -3899 844/2907025, a[18,16] = -14362833758457519226074218750000000000/109981 955442378383636842201027006314567*7^(1/2)+2503126771431561481994628906 2500000000/109981955442378383636842201027006314567, a[4,3] = 3069/2560 0, a[5,1] = 4202367/11628100, a[8,4] = -125/2, a[7,6] = 5611/283500, a [14,6] = 3037913416047823635649583750/15649233075629811884880586233-30 2465625814318865951896498250/5367686944941025476514041077919*7^(1/2), \+ a[16,8] = 586899259341979493965640551217963416854383856766162452169151 7151343133691358404104159327002855091833886481/25073018193160295335196 2624259572198106967555130814454605513204002197920632408840994376037323 5100000000000+59180686976880893900015471525059171817264930557714801884 98447662556733258145721786535132299414153253117/4643151517251906543554 8634122142999649438436135336010112132074815221837154149785369328895800 650000000000*7^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 18387432 8794901398385760606250/760335208044775309288920638333-1147872290905544 07592495836250/37256425194193990155157111278317*7^(1/2), c[7] = 31/200 , c[8] = 943/1000, c[9] = 7067558016280/7837150160667, a[16,10] = 3666 8490233102499502202105336704276225767208192115786373723307906966584890 1352651139887527426189000273823/88450761201699241304594221551071985600 0077549488758852211462407717486163006277913807113787900300000000000-55 4145776168200641359928749044945139274382645375948257431116524868743578 561608227565491722318912765023/545992353096908896941939639204148059259 3071293140486742046064245169667672878258727204406098150000000000*7^(1/ 2), a[18,9] = 18030940246661589735836582988057718311368779168321462409 1405760862055998989936441237796383108094390439124899145193625/19098778 8461366088917496769936717229956693760497939213456114765906201928525030 89295365087000536203878192553324465248*7^(1/2)-69370150528589696110404 3504903948243548896400091226792059437337315219769049232950742425054420 70216860013335461409030103/4201731346150053956184928938607779059047262 7309546626960345248499364424275506796449803191401179648532023617313823 54560, a[18,12] = 2566273931876403497786160206125629704906909/11156625 521628357075657657403354534575801504-575382395115950859775015293456281 18125/500117694173765334214526510819191974888*7^(1/2), a[3,2] = 116281 /1024000, a[7,4] = 8602624/76559175, a[4,1] = 1023/25600, a[2,1] = 1/2 0, a[3,1] = -7161/1024000, c[18] = 1/2, c[5] = 39/100, a[14,12] = 1135 2128098668146659861/254668911904014019468056-5215842639928607924801/12 7334455952007009734028*7^(1/2), c[11] = 47/50, a[6,5] = 923521/5106400 , a[7,1] = 21173/343200, a[6,4] = 31744/135025, a[18,8] = -10389620297 77403408608910724198710303293457031250/1565473660344357372732337968618 82831816048151619*7^(1/2)+11656442810656902213111547056893375403759951 171875/939284196206614423639402781171296990896288909714\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 " calculation for stage 19" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 253 "The standard (simple) order conditions can be ada pted to give a method of stage by stage construction for an interpolat ion scheme that avoids dealing with the weight polynomials for a given stage (corresponding to an \"approximate\" interpolation scheme)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_18 := SimpleOrderConditions(7,18,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch : = [1,2,4,8,16,17,25,27,32,61,63,64]:\ninterp_order_eqns19 := []:\nfor \+ ct in whch do\n temp_eqn := convert(SO7_18[ct],'interpolation_order_ condition'):\n interp_order_eqns19 := [op(interp_order_eqns19),temp_ eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified expl icitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 584 "inter p_order_eqns19 := [add(a[19,i],i=1..18)=c[19],seq(op(StageOrderConditi ons(i,19..19,'expanded')),i=2..7),\nadd(a[19,i]*add(a[i,j]*add(a[j,k]* add(a[k,l]*add(a[l,m]*c[m],\n m=2..l-1),l=2..k-1),k=2..j-1),j= 2..i-1),i=2..18)=c[19]^6/720, ##17\n add(a[19,i]*add(a[i,j]*c[j]^2* add(a[j,k]*c[k],k=2..j-1),j=2..i-1),i=2..18)=c[19]^6/60, ##25\n ad d(a[19,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..18)=c[ 19]^6/120, ##27\n add(a[19,i]*c[i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2. .18)=c[19]^7/35, ##61\n add(a[19,i]*add(a[i,j]*c[j]^5,j=2..i-1),i= 2..18)=c[19]^7/42]: ##63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We specify " }{XPPEDIT 18 0 "c[19] = 7/10;" "6#/ &%\"cG6#\"#>*&\"\"(\"\"\"\"#5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[19,17]=0" "6#/&%\"aG6$\"#>\"#<\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[19,18] = 0;" "6#/&%\"aG6$\"#>\"#=\"\"!" }{TEXT -1 2 " . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "e18 := `union`(e17,\{ c[19]=7/10,seq(a[19,i]=0,i=2..5),a[19,17]=0,a[19,18]=0\}):\neqs_19 := \+ expand(subs(e18,interp_order_eqns19)):\nnops(eqs_19);\nindets(eqs_19); \nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<.&%\"aG6$\"#>\"#7&F%6$F'\"#8&F%6$F'\"#9&F%6$F'\"#:&F %6$F'\"#;&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\"\"\"&F%6$F'\"\"'&F%6$F'\"\"(&F %6$F'\"\")&F%6$F'\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e19 := solve(\{op(eqs_19)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e20 := `u nion`(e18,e19):\nseq(a[19,i]=subs(e20,a[19,i]),i=1..18):\nevalf[40](%) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "64/&%\"aG6$\"#>\"\"\"$\"Ix\\8%4+)[l ]dFpDyXY9MPP!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F% 6$F'\"\"&F0/&F%6$F'\"\"'$\"IC#*3r!R$ou3nnS;$Q`qI\\]$!#S/&F%6$F'\"\"($ \"It`![[tT#)*)*)HLa-t$>GlA\\FD/&F%6$F'\"\")$\"Iz`&eJDn+uD`[8fI!*H<`.\"!#Q/&F%6$F'\"#5$\"I$\\tZaga(G 3^.*\\\"HDF/K$Q\"FX/&F%6$F'\"#6$!I7![!>Xy$eB&HP'=YyIV#4G7FX/&F%6$F'\"# 7$\"I7rIW'yL65oui(4ll&f^\">'HF+/&F%6$F'\"#:$!IZ+.D%QPi:wk- q1F'QRz^EFD/&F%6$F'\"#;$\"HIcZR(e.(z+<`l1Q<'R]H%*FD/&F%6$F'\"# " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_18) do\n tt := convert(SO7_18[c t],'interpolation_order_condition'):\n if expand(subs(e20,lhs(tt)=rh s(tt))) then recd := [op(recd),ct] end if: \nend do:\nop(recd);\nnops( recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"& \"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\" #A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R \"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c \"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#- --------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21826 "e20 := \{ a[13,5] = 0, a[19,12] = 1435378532952865366640508161506288132628507179 /4980636393584087980204311340783274364197100000-2026770206693789575425 07423185706165490917/4611700364429711092781769759984513300182500*7^(1/ 2), a[19,13] = 403746694578349438308454175287/637698586428300244839904 9100000-2025930373551673081586588852/199280808258843826512470284375*7^ (1/2), a[14,10] = 30448415149825325326308593750/1512742633073134555930 8904251+5887803942383224482632816406250/741243890205835932406136308299 *7^(1/2), a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[4,2] = 0, c[13] = 1, a[11,3] = 0, a[12,2] = 0, a[11,2] = 0, c[12] = 1, a[19, 14] = -13527697495128038175393365162508971/236636338499968858572102540 184443750+31048538567801978197319138691898/946545353999875434288410160 737775*7^(1/2), a[10,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[10,2] = 0, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[14,2] = 0, a[6,3] = 0, a[9,3 ] = 0, a[14,5] = 0, a[6,2] = 0, a[7,2] = 0, a[7,3] = 0, a[19,15] = 295 63343742281752194031206400000/667926768960321231615748121619903*7^(1/2 )-255337671203169229981390407680000/667926768960321231615748121619903, a[5,2] = 0, a[13,2] = 0, a[14,4] = 0, a[13,4] = 0, a[13,3] = 0, a[19, 16] = 2765199492553514035802070312500000000/12220217271375375959649133 447445146063-609616487411729384267187500000000000/12220217271375375959 649133447445146063*7^(1/2), a[17,1] = 25991841383410386865330536254325 2668375436100480694679123/53371469501923522598920781313721175412940042 08900000000000+76201610563498999758072388412964731305853858164539/5883 0984900709350307452360354630925278813979375000000*7^(1/2), a[17,2] = 0 , a[17,3] = 0, a[19,10] = 12386307405357854564905965716278384136478564 21875/47336259183453282883619038968469208401637005206-4086346085795561 461894163610670055194160612500/876597392286171905252204425342022377808 092689*7^(1/2), a[15,5] = 0, a[15,4] = 0, a[15,3] = 0, a[15,2] = 0, a[ 17,12] = 37557085142885005839199344190346713535658856538249/1743222737 754430793071508969274146027468985000000000-154646441255718575659310956 019446499029993293/4434327273490106819982470923062032019406250000*7^(1 /2), a[17,13] = 893968763055007240865092535772471/31884929321415012241 9952455000000000-5023926712398881127692249781201/622752525808886957851 469638671875*7^(1/2), a[17,15] = 34256025405285245733900192129792/2420 02452521855518701358015079675+7819897427138839913759197790208/22264225 6320107077205249373873301*7^(1/2), a[17,14] = -44728197763861767267686 691630511059789/985984743749870244050427250768515625000+51329695014236 122164753612289240491/1971969487499740488100854501537031250*7^(1/2), a [19,7] = 1449472813240462521572915144813245178698300209565/23792166418 41833257797401150284480158879812795098-5258753177038055077747585848756 8858362192928292/1189608320920916628898700575142240079439906397549*7^( 1/2), a[14,11] = -1466584824552592368357500/602474973863778980873559-9 46677979546641857718938375/59042547438650340125608782*7^(1/2), a[18,15 ] = 696525438997433496880000000000000/60113409206428910845417330945791 27*7^(1/2)-1213891006926781481467040000000000/601134092064289108454173 3094579127, c[17] = 37/100, a[16,2] = 0, a[14,8] = 7420070341602879832 7128906250/42881511753324799479599510367+28676647199217261041085964843 750/2101194075912915174500376007983*7^(1/2), a[16,4] = 0, a[16,3] = 0, a[16,1] = 31842411379930048859824934115334104524350812346629654170474 3316342575624901053326413058788147679306059990010001/77171934571091625 1389633998954193916862587435661262027528829563631756419808824161123963 2333125000000000000000000-77903689857519389238206746670967756020634822 8828574129617032755363065443176248630443242600895553442815357/47636996 6488219908265206172193946862260856441766211128104215780019602728277051 9513110884156250000000000000000*7^(1/2), a[16,5] = 0, a[18,2] = 0, a[1 8,3] = 0, a[18,4] = 0, a[18,5] = 0, a[16,6] = 386778736038740234948212 2466356052053375712903170520929482841050345670826995056564045262716744 0128229776168201/10008008211891818573636093510526806170039475723780453 3541386478622349830693961360202657479143973437500000000000+26603884961 7986459436081584176085576650873052305529939549795098808807764756355668 79209893579250983975560997/1853334854054040476599276576023482624081384 393292676547062712567080552420258543707456619984147656250000000000*7^( 1/2), a[14,1] = 8941065567926479206438689/198753096356125278622613280+ 152838094177334666489948287/331122658529304714185273724480*7^(1/2), a[ 16,7] = 37730858790334067779260554208528867747915827516985428125843490 240154695455951816700929871689234244950950501493/138928597406614710422 7015615055369188134543397855871478207280655832314421551987710831008896 37109062500000000000-1023300145357950779063583060915565481978351900676 151998097343801718266872083464334323592817694003399143529013/694642987 0330735521135078075276845940672716989279357391036403279161572107759938 5541550444818554531250000000000*7^(1/2), c[16] = 3923/5000, a[14,9] = \+ -128719957415479235191318173494800977965419044818935750521702127672316 317131710725250664367373074947/959116489087682414594756234921281432382 10539014575492278647719252118577198224684430692759024543840-8821850820 6817762290678760500547255563644475990463650453131452960812003406659192 7204665390933748491981/15978880708200789027148638873788548663485875799 8282770136227100274029549612242324261534136534890037440*7^(1/2), a[16, 9] = 14082678935889021729547528858792003784955964104255221303813118180 5402895409095008239750460554850259843099225958876196526622787868985415 3082563752624774515387959169654573113399136914397/37240614762200483599 0419221263984031393524656093000582946860347739502839505183202930223886 1326551982874800930518341256739966130319527284854144491887040986350427 874375000000000000000000+365894872489528509756269896350360369840439730 3845971102348754445631677088718551969749745261345641304898608718025716 162435543460343413409884815320619233885431694117641997082749518637/266 0043911585748828502994437599885938525176114950004163906145340996448853 6084514495015991866618228449105720932273866119571186645139480606101032 084907435616788770531250000000000000000*7^(1/2), a[19,17] = 0, a[19,18 ] = 0, a[19,2] = 0, a[19,3] = 0, a[16,13] = 16677954596669945960065937 34612718364094871113097496394713308473411640245893712505210071/1481769 4205771874634426898973052665107401146761751307787463345990460792187500 0000000000000+99583225115561976113055269002823037222138214175969016367 42244109953434503977764414401/1852211775721484329303362371631583138425 1433452189134734329182488075990234375000000000000*7^(1/2), a[19,4] = 0 , a[19,5] = 0, a[16,14] = 10091733461247592876816891204420595023183753 7138211695125551197160193984809290378367269/99638921083071752167902327 4600361612169126960875165356735545085748281214843750000000000000-14035 0519207811981108980209355304116732499169570091599701547604838639510566 88692206659/4981946054153587608395116373001808060845634804375826783677 72542874140607421875000000000*7^(1/2), c[19] = 7/10, a[19,6] = 1344631 17364912049288884412301220752886554060845/3427835612524370882557298921 41683337347373287486-501147800176472936538357321490008588109143452/317 39218634484915579234249272378086791423452545*7^(1/2), a[18,6] = -16530 168995389981543029356724223343073092031250/399914154794509936298351540 831963893571935502067*7^(1/2)+2116195054262733186002415071830214395743 35170625/799828309589019872596703081663927787143871004134, a[17,6] = 2 82346020910207495266878857047379938762003012480259/3242547201036567051 067715195934842380312990557300000-335879025186335297654256461382189633 07158947223/2680677249534198950948838620977878951978332140625*7^(1/2), a[18,14] = -203979858701951858606190955719875/13630253097598206253753 10631462396+29260691408512340069072795640625/3407563274399551563438276 57865599*7^(1/2), a[18,1] = 192549676210048286601630739389759335037228 89627/495039717119290644395786957112776119772197491840+243258529806302 9726317162869389207640440757875/56929567468718424105515500067969253773 8027115616*7^(1/2), a[17,7] = 1822303311925398118092387915648422756388 3974739722227/166545164928928328045818080519913611121586895656860000-1 0031323008218154138865806552959823083322462822417/28596353868291265117 7572253639961557557669807103125*7^(1/2), a[17,8] = 1304777397928621174 3343606727246464862780752381075/15028547139305830778230444498740751854 340622555424-4999044060295488160350089639911998698434613211/2484878825 943424401162441220029886219302351613*7^(1/2), a[19,8] = 20478027115277 33090468449247225042610312438078125/1341834566009449176627718258816138 55842326987102-6299675893634686965347686169110786579741787500/24848788 25943424401162441220029886219302351613*7^(1/2), a[17,10] = 15414026556 9929844304698928693303965519909065225/14328813590666939683690087471536 6252459009313056-2247178508319186582684151266568513851628311/607482600 336917467257244924006945514766523*7^(1/2), a[18,13] = 3051984373760248 06501272473/6121906429711682350463087136-15274180675827991876494250/57 3928727785470220355914419*7^(1/2), a[19,1] = 5040532884653950230124928 27525699510597293259085433/1524899128626386359969165180392033583226858 3454000000+23046587787568553137495508163715092754674894221/14119436376 170244073788566485111422066915355050000*7^(1/2), c[15] = 69/400, a[17, 11] = -1617356517194207679205444849356224376563538983069/1319671808250 473426092343870769195831358057800000+318225879537434867349129917353761 0086635766763/1090998518725589803317083226495697611903156250*7^(1/2), \+ a[16,11] = -6928011549454881761699481713711235043591502295679423604869 246153519670533666503210900114933332845376150551/220168689298819394894 7136308373301646572872307345365971075591622778810160246298613549949543 750000000000000-333373184604500792272379654433507180957103057593155834 0030474714208942484430219726989269407196212710461/20385989749890684712 4734843367898300608599287717163515840332557664704644467249871624995328 12500000000000*7^(1/2), a[18,10] = 23884714645221746779271450489790269 99230849609375/110451271428057660061777757593094819603819678814-398776 8244192880428103766706721197663574218750/32677890955046644988691644258 3120768058638103*7^(1/2), a[19,9] = -373180627876116715225887951001890 9600819678915637197319873999019312706024588858115735861737031895257321 8254826689380705113/18757729223884169447254147047356156513603851477476 1727501541287943591179801369627008049961612409145232248291579569400000 0+16911889220978055035271100221021035910099550046093461456725580685439 4098091886931880046671159021110688326728803940450517/46894323059710423 6181353676183903912840096286936904318753853219858977949503424067520124 90403102286308062072894892350000*7^(1/2), a[15,6] = -19103149868878861 6226618012353233613/16957475840682462825521397145600000000+21956174445 9503945250128450660540211/11093675783624041100808390656000000000*7^(1/ 2), a[17,9] = -4834722584846356480292651573086089355948598980865849611 3143557813259777019828691250781666119617111300222581110482432156116320 3/65652052283594593065389514665746547797613480171166604625539450780256 9129304793694528174865643432008312869020528492900000000000+16775323204 3115470453615258085334954467158064037840590577139604824423918685354990 9489496412837700466748475273250933123162009/58617903824638029522669209 5229879891050120358671130398442316524823722436879280084400156130038778 578850775911186154375000000*7^(1/2), a[15,7] = 14172386735756874627703 502458542559611/90796880820740515068437869813760000000+539248685947160 08726045512055295307/9077835462591889896660794982400000000*7^(1/2), a[ 15,1] = 300164595702502767677683752821381843/5685191595792547980976128 000000000000-48497201742052651581387984291872403/530617882273971144891 10528000000000000*7^(1/2), a[15,8] = -877010740240762900922180472417/5 7186012045619550819463987200+513139270223400882486686075445021/6805135 4334287265475162144768000*7^(1/2), a[15,9] = 4555084870731508935991254 4356558140241875548161939830581423717993786842032027998225969285367527 6040350274866593/74074089804961736793547100876262942213432522766891311 871637067291077204608105381835381523917447168000000000000-808893990326 7911374096450504734124828801217514657511355287393514657504893760256657 562215933144919629/269503077658229281467557639888786899519923547495418 8986106083154776989963827197767483392000000000000*7^(1/2), a[16,12] = \+ 6876230787902827767160767205214587513807297711754964615123643390622098 477381787285420842315969989227509309/727080519049867540728924705434890 7579029951742748276627467199264704732251489135217100301171875000000000 0000+78621364964664876362030452193189774722341872077750608751312298729 19912547716368298063580928728400258823/1077156324518322282561369933977 6159376340669248515965374025480392155158891095015136444890625000000000 00000*7^(1/2), a[15,10] = -81487190136853529424335403395147/8815890907 570223312212603699200+41281728456741643538288815829781/927578265252746 7499145003008000*7^(1/2), a[18,17] = 0, a[9,1] = -14725142644862158038 81384708877264246346044433307094207829051978044531801133057155/1246894 8016200320011570596216439860248033015583934879004404536361680460696864 36608, a[8,7] = 6070139212132283/92502016000000, a[8,6] = 150140835352 8689/265697280000000, a[8,5] = -1024030607959889/168929280000000, c[6] = 93/200, a[9,5] = -1207067925846925480797893644173318794948457151612 0469966534514296406891652614970375/27220311547616572217104781845311006 99497284085048389015085076961673446140398628096, a[15,11] = 1495268418 97534768477167673223553/8109821199993554522521600000000-10197331971622 785977909572704970449/1135374967999097633153024000000000*7^(1/2), a[9, 4] = -5172294311085668458375175655246981230039025336933699114138315270 772319372469280000/124619381004809145897278630571215298365257079410236 252921850936749076487132995191, a[9,8] = 10332848184452015604056836767 286656859124007796970668046446015775000000/131270355003603364807383424 8740727914537972028638950165249582733679393783, a[9,7] = 6641131229599 11642134782135839106469928140328160577035357155340392950009492511875/1 5178465598586248136333023107295349175279765150089078301139943253016877 823170816, a[9,6] = 78012515584389364132309055253043103656779559256849 7182701460674803126770111481625/18311042541273197219788987450715878685 9226102980861859505241443073629143100805376, a[10,7] = 318607235173649 312405151265849660869927653414425413/671471671555896530313293807293546 5423910912000000, a[10,8] = 212083202434519082281842245535894/20022426 044775672563822865371173879, a[10,6] = 2112767021417280287012828699200 3940810655221489/4679473877997892906145822697976708633673728000, a[10, 5] = -180269259803172281163724663224981097/381009225582568710865798328 32000000, a[10,4] = -20462749524591049105403365239069/4542519134998934 69596231268750, a[10,1] = -2905557336033741508853861844223103644131406 0511/22674759891089577691327962602370597632000000000, a[11,1] = -23426 59845814086836951207140065609179073838476242943917/1358480961351056777 022231400139158760857532162795520000, a[17,16] = 272818447134154078329 357400195312500/12220217271375375959649133447445146063-483754995004275 852097660875000000000/12220217271375375959649133447445146063*7^(1/2), \+ a[10,9] = -26984049294008425187211664850871297985622698482295177937034 13951226714583/4695456749139343150770004420808711418846760359027175503 25616728175875000000, a[15,12] = -2271644391263030621748072222279/7410 2764220560004300800000000000, a[11,6] = 209808223450967602922240867949 78105312644533925634933539/3775889992007550803878727839115494641972212 962174156800, a[18,7] = -814586114736897681347580051914111367091406250 /7039102490656311413601778551137515262958026021*7^(1/2)+72047628689847 93295755345584089235457889776021875/1665451649289283280458180805199136 1112158689565686, a[15,13] = -288610253631/128000000000000-46439821566 63/4096000000000000*7^(1/2), a[11,5] = -260530859592565341525880893638 41/4377552804565683061011299942400, a[11,4] = -99628603013253815961393 0889652/16353068885996164905464325675, a[5,4] = 3982992/2907025, a[12, 1] = -2866556991825663971778295329101033887534912787724034363/86822671 1619262703011213925016143612030669233795338240, a[11,10] = -3115523743 7111730665923206875/392862141594230515010338956291, a[11,9] = 30076066 9768102517834232497565452434946672266195876496371874262392684852243925 359864884962513/465544333750134645558506533660450560376082477961552128 5751892810315680492364106674524398280000, a[11,8] = 161021426143124178 389075121929246710833125/10997207722131034650667041364346422894371443, a[11,7] = 890722993756379186418929622095833835264322635782294899/1392 1242001395112657501941955594013822830119803764736, a[12,6] = 234630538 8553404258656258473446184419154740172519949575/25672671640789540289274 4978301151486254183185289662464, a[12,5] = -45834939744845729129493146 73356033540575/451957703655250747157313034270335135744, a[6,1] = 5611/ 114400, a[15,14] = -32507875096641/1024000000000000*7^(1/2), a[12,4] = -16957088714171468676387054358954754000/14369041511965468332636822810 1570221, a[12,9] = -32058909627170725427914343121527275340081027740232 10240571361570757249056167015230160352087048674542196011/9475695496839 6581478301512445127360498465774712725761537244920597319265730601723910 3491074738324033259120, a[12,8] = 345685379554677052215495825476969226 377187500/74771167436930077221667203179551347546362089, a[12,7] = 1657 121559319846802171283690913610698586256573484808662625/134314804112551 46477259155104956093505361644432088109056, a[13,1] = 44901867737754616 851973/1014046409980231013380680, a[16,15] = -881802833650138304272450 21634180439352429300597742922460923036262084841482480653958/1185236103 4459769519597506546144613989755568051785365407543288652815101388446044 921875+198068155659884577440583331314306224717883619889695959461983451 5105995312818424208/94818888275678156156780052369156911918044544414282 923260346309222520811107568359375*7^(1/2), a[12,11] = -612293360107076 9591613093993993358877250/1050517001510235513198246721302027675953, a[ 12,10] = 40279545832706233433100438588458933210937500/8896460842799482 846916972126377338947215101, c[3] = 341/3200, a[13,6] = 79163867519161 5279648100000/2235604725089973126411512319, a[18,11] = -47288194420795 5026637075845907843049717091875/26393436165009468521846877415383916627 161156+21164021072452475373967469896889463184140625/219945301375078904 3487239784615326385596763*7^(1/2), a[13,8] = -137345124323977414765625 00000/875132892924995907746928783, a[13,7] = 3847749490868980348119500 000/15517045062138271618141237517, a[13,12] = -306814272936976936753/1 299331183183744997286, c[2] = 1/20, a[13,11] = 28203554318319084006875 0/12295407629873040425991, a[13,10] = -9798363684577739445312500000/30 8722986341456031822630699, a[13,9] = 122747654703131968784288120377406 35050319234276006986398294443554969616342274215316330684448207141/4893 4514749371551765038583414351093488882928068660965448289652679652335305 2166757299452852166040, a[17,4] = 0, a[17,5] = 0, a[8,1] = -1221101821 869329/690812928000000, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = - 3899844/2907025, a[18,16] = -14362833758457519226074218750000000000/10 9981955442378383636842201027006314567*7^(1/2)+250312677143156148199462 89062500000000/109981955442378383636842201027006314567, a[4,3] = 3069/ 25600, a[5,1] = 4202367/11628100, a[8,4] = -125/2, a[7,6] = 5611/28350 0, a[14,6] = 3037913416047823635649583750/1564923307562981188488058623 3-302465625814318865951896498250/5367686944941025476514041077919*7^(1/ 2), a[16,8] = 58689925934197949396564055121796341685438385676616245216 91517151343133691358404104159327002855091833886481/2507301819316029533 5196262425957219810696755513081445460551320400219792063240884099437603 73235100000000000+5918068697688089390001547152505917181726493055771480 188498447662556733258145721786535132299414153253117/464315151725190654 3554863412214299964943843613533601011213207481522183715414978536932889 5800650000000000*7^(1/2), a[19,11] = -82973304729022771245245622953720 437681491775/3770490880715638360263839630769130946737308+6416330411568 09452351541809340157183397438/1745597629960943685307333162393116179045 05*7^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 183874328794901398 385760606250/760335208044775309288920638333-11478722909055440759249583 6250/37256425194193990155157111278317*7^(1/2), c[7] = 31/200, c[8] = 9 43/1000, c[9] = 7067558016280/7837150160667, a[16,10] = 36668490233102 4995022021053367042762257672081921157863737233079069665848901352651139 887527426189000273823/884507612016992413045942215510719856000077549488 758852211462407717486163006277913807113787900300000000000-554145776168 2006413599287490449451392743826453759482574311165248687435785616082275 65491722318912765023/5459923530969088969419396392041480592593071293140 486742046064245169667672878258727204406098150000000000*7^(1/2), a[18,9 ] = 180309402466615897358365829880577183113687791683214624091405760862 055998989936441237796383108094390439124899145193625/190987788461366088 9174967699367172299566937604979392134561147659062019285250308929536508 7000536203878192553324465248*7^(1/2)-693701505285896961104043504903948 2435488964000912267920594373373152197690492329507424250544207021686001 3335461409030103/42017313461500539561849289386077790590472627309546626 96034524849936442427550679644980319140117964853202361731382354560, a[1 8,12] = 2566273931876403497786160206125629704906909/111566255216283570 75657657403354534575801504-57538239511595085977501529345628118125/5001 17694173765334214526510819191974888*7^(1/2), a[3,2] = 116281/1024000, \+ a[7,4] = 8602624/76559175, a[4,1] = 1023/25600, a[2,1] = 1/20, a[3,1] \+ = -7161/1024000, c[18] = 1/2, c[5] = 39/100, a[14,12] = 11352128098668 146659861/254668911904014019468056-5215842639928607924801/127334455952 007009734028*7^(1/2), c[11] = 47/50, a[6,5] = 923521/5106400, a[7,1] = 21173/343200, a[6,4] = 31744/135025, a[18,8] = -103896202977740340860 8910724198710303293457031250/15654736603443573727323379686188283181604 8151619*7^(1/2)+11656442810656902213111547056893375403759951171875/939 284196206614423639402781171296990896288909714\}: " }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculatio n for stage 20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order conditions can be adapted to give a method of stage by stage construction for an interpolation sch eme that avoids dealing with the weight polynomials for a given stage \+ (corresponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO 7_19 := SimpleOrderConditions(7,19,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1, 2,4,8,16,17,25,27,32,61,63,64]:\ninterp_order_eqns20 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_19[ct],'interpolation_order_condi tion'):\n interp_order_eqns20 := [op(interp_order_eqns20),temp_eqn]; \nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 590 "interp_orde r_eqns20 := [add(a[20,i],i=1..19)=c[20],seq(op(StageOrderConditions(i, 20..20,'expanded')),i=2..7),\n add(a[20,i]*add(a[i,j]*add(a[j,k]*ad d(a[k,l]*add(a[l,m]*c[m],\n m=2..l-1),l=2..k-1),k=2..j-1),j= 2..i-1),i=2..19)=c[20]^6/720, ##17\n add(a[20,i]*add(a[i,j]*c[j]^2* add(a[j,k]*c[k],k=2..j-1),j=2..i-1),i=2..19)=c[20]^6/60, ##25\n ad d(a[20,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..19)=c[ 20]^6/120, ##27\n add(a[20,i]*c[i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2. .19)=c[20]^7/35, ##61\n add(a[20,i]*add(a[i,j]*c[j]^5,j=2..i-1),i= 2..19)=c[20]^7/42]: ##63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We specify " }{XPPEDIT 18 0 "c[20] = 9/10;" "6#/ &%\"cG6#\"#?*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[20,17] = 0;" "6#/&%\"aG6$\"#?\"#<\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[20,18] = 0;" "6#/&%\"aG6$\"#?\"#=\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[20,19] = 0;" "6#/&%\"aG6$\"#?\"#>\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "e21 := `union`(e20, \{c[20]=9/10,seq(a[20,i]=0,i=2..5),seq(a[20,i]=0,i=17..19)\}):\neqs_20 := expand(subs(e21,interp_order_eqns20)):\nnops(eqs_20);\nindets(eqs_ 20);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#?\"\"\"&F%6$F'\"\"'&F%6$F'\"\"(&F%6$F '\"\")&F%6$F'\"\"*&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\"#7&F%6$F'\"#8&F%6$F' \"#9&F%6$F'\"#:&F%6$F'\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e22 := solve(\{op(eqs_20)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e23 : = `union`(e21,e22):\nseq(a[20,i]=subs(e23,a[20,i]),i=1..19):\nevalf[40 ](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "65/&%\"aG6$\"#?\"\"\"$\"IKr)>_B \\jq1TV4DGbMe!RR!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0 /&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"Ij]rB`vAtpOJ=CWBTh^eN!#S/&F%6$F'\"\"($ \"IeiOl+sEbAs$H+h_fA#Q(>%FD/&F%6$F'\"\")$\"Hg52OS?DD<$HmT>2y(\\/s)!#R/ &F%6$F'\"\"*$\"H6O/<9H=.6v'p<(>+C^>#)H%F+/&F%6$F'\"#8$\"IkTc!>*)=*e\\8qq:R(ocdDL\"F+/&F%6$ F'\"#9$\"I+Qx(4G>H,hWM![T'R0Fi(=F+/&F%6$F'\"#:$!I*p8lf#fo$z`^qb5AH86%f =FD/&F%6$F'\"#;$\"IU++hj$)HZ1E_u-Y#>F9Ox\"FD/&F%6$F'\"#F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "These linking coefficients agree to 40 digits with those \+ of the published scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 70 "#------------------------------------------------- --------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check whic h of the (adapted) simple order conditions are satisfied at this stage ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_19) do\n tt := con vert(SO7_19[ct],'interpolation_order_condition'):\n if expand(subs(e 23,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop( recd);\nnops(recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\" \"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#= \"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\" #O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"# `\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 70 "#------------------------------------------------------ ---------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e23" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24544 "e23 := \{a[13,5] = 0, a[19,12] = 1435378532952865366640508161 506288132628507179/4980636393584087980204311340783274364197100000-2026 77020669378957542507423185706165490917/4611700364429711092781769759984 513300182500*7^(1/2), a[19,13] = 403746694578349438308454175287/637698 5864283002448399049100000-2025930373551673081586588852/199280808258843 826512470284375*7^(1/2), a[14,10] = 30448415149825325326308593750/1512 7426330731345559308904251+5887803942383224482632816406250/741243890205 835932406136308299*7^(1/2), a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), a[1 2,3] = 0, a[4,2] = 0, c[13] = 1, a[11,3] = 0, a[12,2] = 0, a[11,2] = 0 , c[12] = 1, a[19,14] = -13527697495128038175393365162508971/236636338 499968858572102540184443750+31048538567801978197319138691898/946545353 999875434288410160737775*7^(1/2), a[10,3] = 0, a[14,13] = 3/392-3/392* 7^(1/2), a[10,2] = 0, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[14,2] = 0, a[6,3] = 0, a[9,3] = 0, a[14,5] = 0, a[6,2] = 0, a[7,2] = 0, a[7,3] = 0, a[19,15] = 29563343742281752194031206400000/6679267689603212316157 48121619903*7^(1/2)-255337671203169229981390407680000/6679267689603212 31615748121619903, a[5,2] = 0, a[13,2] = 0, a[14,4] = 0, a[13,4] = 0, \+ a[13,3] = 0, a[19,16] = 2765199492553514035802070312500000000/12220217 271375375959649133447445146063-609616487411729384267187500000000000/12 220217271375375959649133447445146063*7^(1/2), a[17,1] = 25991841383410 3868653305362543252668375436100480694679123/53371469501923522598920781 31372117541294004208900000000000+7620161056349899975807238841296473130 5853858164539/58830984900709350307452360354630925278813979375000000*7^ (1/2), a[17,2] = 0, a[17,3] = 0, a[19,10] = 12386307405357854564905965 71627838413647856421875/4733625918345328288361903896846920840163700520 6-4086346085795561461894163610670055194160612500/876597392286171905252 204425342022377808092689*7^(1/2), a[20,1] = 48304471866039810753010257 751589919886361140232907/131781406177588944688693287194373272624543313 8000000+68127962326089052629919488875228931247075214511/65890703088794 472344346643597186636312271656900000*7^(1/2), a[20,12] = 1506113044966 46214218585292683337607910401323/1291276102040319105978895532795663724 051100000-1797399149384124340751361777823955407632741/6456380510201595 5298944776639783186202555000*7^(1/2), a[20,13] = -12833245434064828305 08432214/199280808258843826512470284375*7^(1/2)+1936284506566333313130 82794159/6376985864283002448399049100000, a[20,14] = -5712741970443114 906032237142293823/157757558999979239048068360122962500+65558935458559 83544500111964137/315515117999958478096136720245925*7^(1/2), a[15,5] = 0, a[15,4] = 0, a[15,3] = 0, a[15,2] = 0, a[20,15] = -579139089626543 58753739683840000/222642256320107077205249373873301+624229491545010644 4957081600000/222642256320107077205249373873301*7^(1/2), a[17,12] = 37 557085142885005839199344190346713535658856538249/174322273775443079307 1508969274146027468985000000000-15464644125571857565931095601944649902 9993293/4434327273490106819982470923062032019406250000*7^(1/2), a[20,1 6] = 3189081802658751841399101562500000000/122202172713753759596491334 47445146063-386161247481177249147656250000000000/122202172713753759596 49133447445146063*7^(1/2), a[17,13] = 89396876305500724086509253577247 1/318849293214150122419952455000000000-5023926712398881127692249781201 /622752525808886957851469638671875*7^(1/2), a[17,15] = 342560254052852 45733900192129792/242002452521855518701358015079675+781989742713883991 3759197790208/222642256320107077205249373873301*7^(1/2), a[17,14] = -4 4728197763861767267686691630511059789/98598474374987024405042725076851 5625000+51329695014236122164753612289240491/19719694874997404881008545 01537031250*7^(1/2), a[19,7] = 144947281324046252157291514481324517869 8300209565/2379216641841833257797401150284480158879812795098-525875317 70380550777475858487568858362192928292/1189608320920916628898700575142 240079439906397549*7^(1/2), a[14,11] = -1466584824552592368357500/6024 74973863778980873559-946677979546641857718938375/590425474386503401256 08782*7^(1/2), a[18,15] = 696525438997433496880000000000000/6011340920 642891084541733094579127*7^(1/2)-1213891006926781481467040000000000/60 11340920642891084541733094579127, c[17] = 37/100, a[16,2] = 0, a[14,8] = 74200703416028798327128906250/42881511753324799479599510367+2867664 7199217261041085964843750/2101194075912915174500376007983*7^(1/2), a[1 6,4] = 0, a[16,3] = 0, a[16,1] = 3184241137993004885982493411533410452 4350812346629654170474331634257562490105332641305878814767930605999001 0001/77171934571091625138963399895419391686258743566126202752882956363 17564198088241611239632333125000000000000000000-7790368985751938923820 6746670967756020634822882857412961703275536306544317624863044324260089 5553442815357/47636996648821990826520617219394686226085644176621112810 42157800196027282770519513110884156250000000000000000*7^(1/2), a[16,5] = 0, a[18,2] = 0, a[18,3] = 0, a[18,4] = 0, a[18,5] = 0, a[16,6] = 38 6778736038740234948212246635605205337571290317052092948284105034567082 69950565640452627167440128229776168201/1000800821189181857363609351052 6806170039475723780453354138647862234983069396136020265747914397343750 0000000000+26603884961798645943608158417608557665087305230552993954979 509880880776475635566879209893579250983975560997/185333485405404047659 9276576023482624081384393292676547062712567080552420258543707456619984 147656250000000000*7^(1/2), a[14,1] = 8941065567926479206438689/198753 096356125278622613280+152838094177334666489948287/33112265852930471418 5273724480*7^(1/2), a[16,7] = 3773085879033406777926055420852886774791 5827516985428125843490240154695455951816700929871689234244950950501493 /138928597406614710422701561505536918813454339785587147820728065583231 442155198771083100889637109062500000000000-102330014535795077906358306 0915565481978351900676151998097343801718266872083464334323592817694003 399143529013/694642987033073552113507807527684594067271698927935739103 64032791615721077599385541550444818554531250000000000*7^(1/2), c[16] = 3923/5000, a[14,9] = -12871995741547923519131817349480097796541904481 8935750521702127672316317131710725250664367373074947/95911648908768241 4594756234921281432382105390145754922786477192521185771982246844306927 59024543840-8821850820681776229067876050054725556364447599046365045313 14529608120034066591927204665390933748491981/1597888070820078902714863 8873788548663485875799828277013622710027402954961224232426153413653489 0037440*7^(1/2), a[16,9] = 1408267893588902172954752885879200378495596 4104255221303813118180540289540909500823975046055485025984309922595887 6196526622787868985415308256375262477451538795916965457311339913691439 7/37240614762200483599041922126398403139352465609300058294686034773950 2839505183202930223886132655198287480093051834125673996613031952728485 4144491887040986350427874375000000000000000000+36589487248952850975626 9896350360369840439730384597110234875444563167708871855196974974526134 5641304898608718025716162435543460343413409884815320619233885431694117 641997082749518637/266004391158574882850299443759988593852517611495000 4163906145340996448853608451449501599186661822844910572093227386611957 1186645139480606101032084907435616788770531250000000000000000*7^(1/2), a[19,17] = 0, a[19,18] = 0, a[19,2] = 0, a[19,3] = 0, a[16,13] = 1667 7954596669945960065937346127183640948711130974963947133084734116402458 93712505210071/1481769420577187463442689897305266510740114676175130778 74633459904607921875000000000000000+9958322511556197611305526900282303 722213821417596901636742244109953434503977764414401/185221177572148432 9303362371631583138425143345218913473432918248807599023437500000000000 0*7^(1/2), a[19,4] = 0, a[19,5] = 0, a[16,14] = 1009173346124759287681 68912044205950231837537138211695125551197160193984809290378367269/9963 8921083071752167902327460036161216912696087516535673554508574828121484 3750000000000000-14035051920781198110898020935530411673249916957009159 970154760483863951056688692206659/498194605415358760839511637300180806 084563480437582678367772542874140607421875000000000*7^(1/2), c[19] = 7 /10, a[19,6] = 134463117364912049288884412301220752886554060845/342783 561252437088255729892141683337347373287486-501147800176472936538357321 490008588109143452/31739218634484915579234249272378086791423452545*7^( 1/2), a[18,6] = -16530168995389981543029356724223343073092031250/39991 4154794509936298351540831963893571935502067*7^(1/2)+211619505426273318 600241507183021439574335170625/799828309589019872596703081663927787143 871004134, a[17,6] = 2823460209102074952668788570473799387620030124802 59/3242547201036567051067715195934842380312990557300000-33587902518633 529765425646138218963307158947223/268067724953419895094883862097787895 1978332140625*7^(1/2), a[18,14] = -203979858701951858606190955719875/1 363025309759820625375310631462396+29260691408512340069072795640625/340 756327439955156343827657865599*7^(1/2), a[18,1] = 19254967621004828660 163073938975933503722889627/495039717119290644395786957112776119772197 491840+2432585298063029726317162869389207640440757875/5692956746871842 41055155000679692537738027115616*7^(1/2), a[17,7] = 182230331192539811 80923879156484227563883974739722227/1665451649289283280458180805199136 11121586895656860000-1003132300821815413886580655295982308332246282241 7/285963538682912651177572253639961557557669807103125*7^(1/2), c[20] = 9/10, a[17,8] = 13047773979286211743343606727246464862780752381075/15 028547139305830778230444498740751854340622555424-499904406029548816035 0089639911998698434613211/24848788259434244011624412200298862193023516 13*7^(1/2), a[19,8] = 204780271152773309046844924722504261031243807812 5/134183456600944917662771825881613855842326987102-6299675893634686965 347686169110786579741787500/248487882594342440116244122002988621930235 1613*7^(1/2), a[17,10] = 154140265569929844304698928693303965519909065 225/143288135906669396836900874715366252459009313056-22471785083191865 82684151266568513851628311/607482600336917467257244924006945514766523* 7^(1/2), a[18,13] = 305198437376024806501272473/6121906429711682350463 087136-15274180675827991876494250/573928727785470220355914419*7^(1/2), a[19,1] = 504053288465395023012492827525699510597293259085433/1524899 1286263863599691651803920335832268583454000000+23046587787568553137495 508163715092754674894221/141194363761702440737885664851114220669153550 50000*7^(1/2), c[15] = 69/400, a[17,11] = -161735651719420767920544484 9356224376563538983069/13196718082504734260923438707691958313580578000 00+3182258795374348673491299173537610086635766763/10909985187255898033 17083226495697611903156250*7^(1/2), a[16,11] = -6928011549454881761699 4817137112350435915022956794236048692461535196705336665032109001149333 32845376150551/2201686892988193948947136308373301646572872307345365971 075591622778810160246298613549949543750000000000000-333373184604500792 2723796544335071809571030575931558340030474714208942484430219726989269 407196212710461/203859897498906847124734843367898300608599287717163515 84033255766470464446724987162499532812500000000000*7^(1/2), a[18,10] = 2388471464522174677927145048979026999230849609375/1104512714280576600 61777757593094819603819678814-3987768244192880428103766706721197663574 218750/326778909550466449886916442583120768058638103*7^(1/2), a[19,9] \+ = -3731806278761167152258879510018909600819678915637197319873999019312 7060245888581157358617370318952573218254826689380705113/18757729223884 1694472541470473561565136038514774761727501541287943591179801369627008 0499616124091452322482915795694000000+16911889220978055035271100221021 0359100995500460934614567255806854394098091886931880046671159021110688 326728803940450517/468943230597104236181353676183903912840096286936904 31875385321985897794950342406752012490403102286308062072894892350000*7 ^(1/2), a[15,6] = -191031498688788616226618012353233613/16957475840682 462825521397145600000000+219561744459503945250128450660540211/11093675 783624041100808390656000000000*7^(1/2), a[17,9] = -4834722584846356480 2926515730860893559485989808658496113143557813259777019828691250781666 1196171113002225811104824321561163203/65652052283594593065389514665746 5477976134801711666046255394507802569129304793694528174865643432008312 869020528492900000000000+167753232043115470453615258085334954467158064 0378405905771396048244239186853549909489496412837700466748475273250933 123162009/586179038246380295226692095229879891050120358671130398442316 524823722436879280084400156130038778578850775911186154375000000*7^(1/2 ), a[15,7] = 14172386735756874627703502458542559611/907968808207405150 68437869813760000000+53924868594716008726045512055295307/9077835462591 889896660794982400000000*7^(1/2), a[15,1] = 30016459570250276767768375 2821381843/5685191595792547980976128000000000000-484972017420526515813 87984291872403/53061788227397114489110528000000000000*7^(1/2), a[20,7] = 8224414033808965946363932975223105619473504454675/16654516492892832 804581808051991361112158689565686-333115447002188004162420367084790378 87482563294/1189608320920916628898700575142240079439906397549*7^(1/2), a[15,8] = -877010740240762900922180472417/571860120456195508194639872 00+513139270223400882486686075445021/68051354334287265475162144768000* 7^(1/2), a[15,9] = 455508487073150893599125443565581402418755481619398 305814237179937868420320279982259692853675276040350274866593/740740898 0496173679354710087626294221343252276689131187163706729107720460810538 1835381523917447168000000000000-80889399032679113740964505047341248288 01217514657511355287393514657504893760256657562215933144919629/2695030 7765822928146755763988878689951992354749541889861060831547769899638271 97767483392000000000000*7^(1/2), a[16,12] = 68762307879028277671607672 0521458751380729771175496461512364339062209847738178728542084231596998 9227509309/72708051904986754072892470543489075790299517427482766274671 992647047322514891352171003011718750000000000000+786213649646648763620 3045219318977472234187207775060875131229872919912547716368298063580928 728400258823/107715632451832228256136993397761593763406692485159653740 2548039215515889109501513644489062500000000000000*7^(1/2), a[15,10] = \+ -81487190136853529424335403395147/8815890907570223312212603699200+4128 1728456741643538288815829781/9275782652527467499145003008000*7^(1/2), \+ a[18,17] = 0, a[9,1] = -1472514264486215803881384708877264246346044433 307094207829051978044531801133057155/124689480162003200115705962164398 6024803301558393487900440453636168046069686436608, a[8,7] = 6070139212 132283/92502016000000, a[8,6] = 1501408353528689/265697280000000, a[8, 5] = -1024030607959889/168929280000000, c[6] = 93/200, a[9,5] = -12070 6792584692548079789364417331879494845715161204699665345142964068916526 14970375/2722031154761657221710478184531100699497284085048389015085076 961673446140398628096, a[15,11] = 149526841897534768477167673223553/81 09821199993554522521600000000-10197331971622785977909572704970449/1135 374967999097633153024000000000*7^(1/2), a[9,4] = -51722943110856684583 75175655246981230039025336933699114138315270772319372469280000/1246193 8100480914589727863057121529836525707941023625292185093674907648713299 5191, a[9,8] = 1033284818445201560405683676728665685912400779697066804 6446015775000000/13127035500360336480738342487407279145379720286389501 65249582733679393783, a[9,7] = 664113122959911642134782135839106469928 140328160577035357155340392950009492511875/151784655985862481363330231 07295349175279765150089078301139943253016877823170816, a[9,6] = 780125 1558438936413230905525304310365677955925684971827014606748031267701114 81625/1831104254127319721978898745071587868592261029808618595052414430 73629143100805376, a[10,7] = 31860723517364931240515126584966086992765 3414425413/6714716715558965303132938072935465423910912000000, a[10,8] \+ = 212083202434519082281842245535894/2002242604477567256382286537117387 9, a[10,6] = 21127670214172802870128286992003940810655221489/467947387 7997892906145822697976708633673728000, a[10,5] = -18026925980317228116 3724663224981097/38100922558256871086579832832000000, a[10,4] = -20462 749524591049105403365239069/454251913499893469596231268750, a[10,1] = \+ -29055573360337415088538618442231036441314060511/226747598910895776913 27962602370597632000000000, a[11,1] = -2342659845814086836951207140065 609179073838476242943917/135848096135105677702223140013915876085753216 2795520000, a[17,16] = 272818447134154078329357400195312500/1222021727 1375375959649133447445146063-483754995004275852097660875000000000/1222 0217271375375959649133447445146063*7^(1/2), a[10,9] = -269840492940084 2518721166485087129798562269848229517793703413951226714583/46954567491 3934315077000442080871141884676035902717550325616728175875000000, a[15 ,12] = -2271644391263030621748072222279/741027642205600043008000000000 00, a[11,6] = 20980822345096760292224086794978105312644533925634933539 /3775889992007550803878727839115494641972212962174156800, a[18,7] = -8 14586114736897681347580051914111367091406250/7039102490656311413601778 551137515262958026021*7^(1/2)+7204762868984793295755345584089235457889 776021875/16654516492892832804581808051991361112158689565686, a[15,13] = -288610253631/128000000000000-4643982156663/4096000000000000*7^(1/2 ), a[11,5] = -26053085959256534152588089363841/43775528045656830610112 99942400, a[11,4] = -996286030132538159613930889652/163530688859961649 05464325675, a[5,4] = 3982992/2907025, a[12,1] = -28665569918256639717 78295329101033887534912787724034363/8682267116192627030112139250161436 12030669233795338240, a[11,10] = -31155237437111730665923206875/392862 141594230515010338956291, a[11,9] = 3007606697681025178342324975654524 34946672266195876496371874262392684852243925359864884962513/4655443337 5013464555850653366045056037608247796155212857518928103156804923641066 74524398280000, a[11,8] = 161021426143124178389075121929246710833125/1 0997207722131034650667041364346422894371443, a[11,7] = 890722993756379 186418929622095833835264322635782294899/139212420013951126575019419555 94013822830119803764736, a[12,6] = 23463053885534042586562584734461844 19154740172519949575/2567267164078954028927449783011514862541831852896 62464, a[12,5] = -4583493974484572912949314673356033540575/45195770365 5250747157313034270335135744, a[6,1] = 5611/114400, a[20,2] = 0, a[15, 14] = -32507875096641/1024000000000000*7^(1/2), a[12,4] = -16957088714 171468676387054358954754000/143690415119654683326368228101570221, a[12 ,9] = -320589096271707254279143431215272753400810277402321024057136157 0757249056167015230160352087048674542196011/94756954968396581478301512 4451273604984657747127257615372449205973192657306017239103491074738324 033259120, a[12,8] = 345685379554677052215495825476969226377187500/747 71167436930077221667203179551347546362089, a[12,7] = 16571215593198468 02171283690913610698586256573484808662625/1343148041125514647725915510 4956093505361644432088109056, a[20,9] = 149979584265997790954827772549 8597697014972400647371959801228580080259919057575126361473563579004627 500002497186668723541/656520522835945930653895146657465477976134801711 6660462553945078025691293047936945281748656434320083128690205284929000 00*7^(1/2)-67558126503821557161864339921937166639594246662806827122510 503022547617419025194068959010701808469924021246390403506719787/131304 1045671891861307790293314930955952269603423332092510789015605138258609 5873890563497312868640166257380410569858000000, a[13,1] = 449018677377 54616851973/1014046409980231013380680, a[16,15] = -8818028336501383042 7245021634180439352429300597742922460923036262084841482480653958/11852 3610344597695195975065461446139897555680517853654075432886528151013884 46044921875+1980681556598845774405833313143062247178836198896959594619 834515105995312818424208/948188882756781561567800523691569119180445444 14282923260346309222520811107568359375*7^(1/2), a[12,11] = -6122933601 070769591613093993993358877250/105051700151023551319824672130202767595 3, a[20,11] = 2845094817811868725022379378375854841597087/122191834097 2660579715133213675181325331535*7^(1/2)-711861433254182938486103089425 3928312032669/977534672778128463772106570940145060265228, a[12,10] = 4 0279545832706233433100438588458933210937500/88964608427994828469169721 26377338947215101, c[3] = 341/3200, a[13,6] = 791638675191615279648100 000/2235604725089973126411512319, a[18,11] = -472881944207955026637075 845907843049717091875/26393436165009468521846877415383916627161156+211 64021072452475373967469896889463184140625/2199453013750789043487239784 615326385596763*7^(1/2), a[13,8] = -13734512432397741476562500000/8751 32892924995907746928783, a[13,7] = 3847749490868980348119500000/155170 45062138271618141237517, a[13,12] = -306814272936976936753/12993311831 83744997286, c[2] = 1/20, a[13,11] = 282035543183190840068750/12295407 629873040425991, a[13,10] = -9798363684577739445312500000/308722986341 456031822630699, a[13,9] = 1227476547031319687842881203774063505031923 4276006986398294443554969616342274215316330684448207141/48934514749371 5517650385834143510934888829280686609654482896526796523353052166757299 452852166040, a[20,3] = 0, a[20,4] = 0, a[20,5] = 0, a[20,17] = 0, a[2 0,18] = 0, a[20,19] = 0, a[17,4] = 0, a[17,5] = 0, a[8,1] = -122110182 1869329/690812928000000, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = \+ -3899844/2907025, a[18,16] = -14362833758457519226074218750000000000/1 09981955442378383636842201027006314567*7^(1/2)+25031267714315614819946 289062500000000/109981955442378383636842201027006314567, a[4,3] = 3069 /25600, a[5,1] = 4202367/11628100, a[8,4] = -125/2, a[7,6] = 5611/2835 00, a[14,6] = 3037913416047823635649583750/156492330756298118848805862 33-302465625814318865951896498250/5367686944941025476514041077919*7^(1 /2), a[16,8] = 5868992593419794939656405512179634168543838567661624521 691517151343133691358404104159327002855091833886481/250730181931602953 3519626242595721981069675551308144546055132040021979206324088409943760 373235100000000000+591806869768808939000154715250591718172649305577148 0188498447662556733258145721786535132299414153253117/46431515172519065 4355486341221429996494384361353360101121320748152218371541497853693288 95800650000000000*7^(1/2), a[19,11] = -8297330472902277124524562295372 0437681491775/3770490880715638360263839630769130946737308+641633041156 809452351541809340157183397438/174559762996094368530733316239311617904 505*7^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 18387432879490139 8385760606250/760335208044775309288920638333-1147872290905544075924958 36250/37256425194193990155157111278317*7^(1/2), c[7] = 31/200, c[8] = \+ 943/1000, c[9] = 7067558016280/7837150160667, a[16,10] = 3666849023310 2499502202105336704276225767208192115786373723307906966584890135265113 9887527426189000273823/88450761201699241304594221551071985600007754948 8758852211462407717486163006277913807113787900300000000000-55414577616 8200641359928749044945139274382645375948257431116524868743578561608227 565491722318912765023/545992353096908896941939639204148059259307129314 0486742046064245169667672878258727204406098150000000000*7^(1/2), a[18, 9] = 18030940246661589735836582988057718311368779168321462409140576086 2055998989936441237796383108094390439124899145193625/19098778846136608 8917496769936717229956693760497939213456114765906201928525030892953650 87000536203878192553324465248*7^(1/2)-69370150528589696110404350490394 8243548896400091226792059437337315219769049232950742425054420702168600 13335461409030103/4201731346150053956184928938607779059047262730954662 696034524849936442427550679644980319140117964853202361731382354560, a[ 18,12] = 2566273931876403497786160206125629704906909/11156625521628357 075657657403354534575801504-57538239511595085977501529345628118125/500 117694173765334214526510819191974888*7^(1/2), a[3,2] = 116281/1024000, a[7,4] = 8602624/76559175, a[4,1] = 1023/25600, a[2,1] = 1/20, a[3,1] = -7161/1024000, a[20,8] = -27933684971335718931332160504293597603876 493750/17394151781603970808137088540209203535116461291*7^(1/2)+1781481 32680281061166207127585231501733817765625/3478830356320794161627417708 0418407070232922582, a[20,10] = -6039818882312132338065105140914067839 480368750/2045393915334401112255143659131385548218882941*7^(1/2)+29380 145942317975737545613572395823054173078125/409078783066880222451028731 8262771096437765882, c[18] = 1/2, c[5] = 39/100, a[14,12] = 1135212809 8668146659861/254668911904014019468056-5215842639928607924801/12733445 5952007009734028*7^(1/2), c[11] = 47/50, a[6,5] = 923521/5106400, a[7, 1] = 21173/343200, a[6,4] = 31744/135025, a[18,8] = -10389620297774034 08608910724198710303293457031250/1565473660344357372732337968618828318 16048151619*7^(1/2)+11656442810656902213111547056893375403759951171875 /939284196206614423639402781171296990896288909714, a[20,6] = -22221626 97028947028783013162206764717253288398/2221745304413944090546397449066 46607539964167815*7^(1/2)+33976182057500502962081400240417788844155222 553/88869812176557763621855897962658643015985667126\}: " }{TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 " calculation of the interpolation coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) o rder condition gives rise to a group \{list) of equations to be satis fied by the \"d\" coefficients of the weight polynomials for a given s tage (corresponding to an \"approximate\" interpolation scheme)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO8_20 := SimpleOrderConditions(8,20,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "whch : = [1,2,4,8,16,17,25,27,32,58,61,63,64,102,117,121,123,125,127,128]:\no rdeqns := []:\nfor ct in whch do\n eqn_group := convert(SO8_20[ct],' polynom_order_conditions',8):\n ordeqns := [op(ordeqns),op(eqn_group )];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "eqns := []:\nfor ct to nops(ordeqns) do\n \+ eqns := [op(eqns),expand(subs(e23,ordeqns[ct]))];\nend do:\nnops(eqn s);\nnops(indets(eqns));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$g\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$g\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Solve the system of equations. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=0:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dd := solve(\{op(eqns)\}):\n infolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " dd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67899 "dd := \{d[5,8] = 0, d[8,1] = 0, d[4,2] = 0, d[4,8] = 0, d[4,5] = 0, d[16,8] = 195398509993934353753639090112131443251797 7633691173970644071010498687479218135509177272659891654094883984375000 00000000000000000000000/5220608815327855202818977789052154649860940120 3765965426640093021818622721672730278217121262940173467542306964578221 8707205760970841-13802411549352279737458613894097751985459357574958902 0903424845317948740855371739683606579784329294980468750000000000000000 000000/370687887753449956425748428065622185984160274729816129196220084 977165752161367680767142998990598076109553948072670650060464613*7^(1/2 ), d[15,4] = -10396957121472031619939811038933607422018457591549803217 6538811922969285115091229170642176190595575066153716787995888194560000 00000000/9900702794006894886175314765204702965450936777758658994701842 249655120074477969385609622360039884014810076399072960392464949348617* 7^(1/2)-12964436448449927496558576626329464761432978622131891630081108 9508201759440217768731050109343726862778830568500494464205209600000000 000000/753740503707744907684526713075034036759779816890766709266651250 4662442912700078093264605502698363700474911162614244746783565939102122 1, d[19,7] = 251044937580126631977814391869579609064028672619364480937 6491171482120279406971516939799443734595709919241034670377816139597500 00/8571461668773467460895684372688561057291393739998751335768452415387 9167917193127384594627394222377959266228233019940104071397-10031589997 8668348505071735336390981477458047030310233568521737103646475494090268 57654577454529570721372068389891274644437500000/8571461668773467460895 6843726885610572913937399987513357684524153879167917193127384594627394 222377959266228233019940104071397*7^(1/2), d[13,4] = -8292137072387462 1610645227273153492018716137776797773310317459250199092262061384809648 9949680394079015283756098965180953355466590649251/55501073477456928138 8652275478483716102728444201953442284846219619670527461147817737273326 3808776127038795448036449965180403636770302-77366430492096668950553374 3937862056552715463883347205286416663709038320330572020238467654249610 1885260527793588270202185564220513/46706673856934694509644301936268395 4340041946159568322787237307071228847723323277766600178728153575898037 97457156501907618541238*7^(1/2), d[9,3] = -269654356531086231156493911 6196605041407266964362138241618491887343083698468206300258440234394032 3794242978822242172052227106940509325049668582383136599524653996870158 4009374112846597108713915489411773326678361337020015/54771346353513864 3401171986418999347288456504333584487483417299555990513416591102508098 1711546089266959823603463009222657389547091787963620635086799342037835 867854162987278578515620772967204789205199659174368380116-878238477199 9897316056079141210076290936297477840113392323998410382164111558369150 0011849870786931920776782454424207398179467079796728671640700579014256 6868618319637661848839887491464534593842398909956877219157019181693/38 6376128080766282138435455636881061293704642730974274322541116751997655 6601822016606040359558056450279266868095101071200701973981054769988904 5340573619375603024319041073302146267803061475172915154071508740924768 427*7^(1/2), d[20,2] = -2437141551304952649192026089700912510694702448 9849889322832049800283061956029496422934255313523940133798545082538914 1391065625/57095199218055355101902547769479732220751219298692981247892 07934841553288158235129514262989393111895904202048198647036676729-1425 5685113169849281386173633668670203573842919167746349749061948416806056 9866556480676214298851265779243484230275765005515625/22838079687222142 0407610191077918928883004877194771924991568317393662131526329405180570 51957572447583616808192794588146706916*7^(1/2), d[4,4] = 0, d[11,1] = \+ 0, d[6,1] = 0, d[18,5] = 509683320977153495000573988388300650034765501 3185254244325253818444946699080714613777117723819561154421890984991795 10427035427650/1784793533627722012420270209204847562145956878328744325 7595781869270943622584369814714292543991759220089634536832290558466814 7*7^(1/2)+210221531134936005976622357117343791600977923262508960515712 339046048807145517490958841699737374132033313672994501363608349455702/ 1784793533627722012420270209204847562145956878328744325759578186927094 36225843698147142925439917592200896345368322905584668147, d[7,6] = 215 1108717168087381768841370743436454189397967056557248989028953781565619 0624898695166209700388882623950070035147676605604597632332214585727306 6182000000000/91301280289827564029925424805955788333683823712575001680 6173347018965981574473290509116165584161930092993104142279250150166507 38820542603882059615975967*7^(1/2)+39674173609127147425139899962160773 9009527678979899145101428623942067574210220564194300163715418447164902 636685982760910401701671728387693823008570800000000/913012802898275640 2992542480595578833368382371257500168061733470189659815744732905091161 6558416193009299310414227925015016650738820542603882059615975967, d[11 ,4] = 1258632247730767236788396287851163004654506840380503669237093609 0048135139110405215484003940900350765717147689941898522628034743842956 172432537119125000/723453759267506313768387418477880084695350877527903 0261134908470879115484663267347544671383036611279023646476651324723759 0257011405769724458745341+27383348963066076394429690337535078201201600 4564806341468892522832700795757838719442534439114936865432630026761127 196671509392723886382548649517875000/314545112725002745116690181946904 3846501525554469143591797786291686571949853594498932465818711570121314 628902891880314677837261365468248889510667*7^(1/2), d[15,2] = -9543024 9245870271570731557368989297685825376108074335319790134588894998979620 572519687281691486433379280042558557798400000000000000/243385483453693 9867882484785027072352222479953794977911029259099313004266427743257213 537893602752333131489929482464641022818282509-765312255141896926276752 6977508422657561360358942188828894733746456476768648997309695962490058 672865586009714983424000000000000/319697206690784167592602756472753494 3153132738467066742452724417854990498394513670318583861293514164102837 15944104116776936593*7^(1/2), d[8,8] = -486968665809770805857465922450 4690631223273456682215833115280693442078888759245765773849551522974860 699046710660090397390347887140689713093750000000000000/223879339946123 4110196227003541701413226720497576105950612544237988452811053877809710 66347527298171646882701113377244406139245417581055730034990201971*7^(1 /2)-199077812151469265281948249769407489368375394538793444974693848191 5337935958690783391584532670393089563277680246642487569393412983078711 73002734375000000000/5149224818760838453451322108145913250421457144425 0436864088517473734414654239189623345259931278579478783021256076766213 41202644604364281790804774645333, d[16,6] = 13395570428355373016129648 4560259292245148558507231637208547566144503292217312266498599390192675 9972148363242187500000000000000000000000/52206088153278552028189777890 5215464986094012037659654266400930218186227216727302782171212629401734 675423069645782218707205760970841-946226130364215871469503738679229253 5610701144881628939162680771406179849332268170520078868551839363105273 43750000000000000000000/3706878877534499564257484280656221859841602747 2981612919622008497716575216136768076714299899059807610955394807267065 0060464613*7^(1/2), d[20,1] = 0, d[4,3] = 0, d[3,3] = 0, d[2,3] = 0, d [5,7] = 0, d[9,5] = -1730113207784114464204040659129627488052889826218 3650702397457263684416905243306269960656955007636865429733343050613503 6332580170365810930609014412836123080967356296966837493587645929404868 9064577348441477257628446984568608/44433254729288122445920077398241322 0487760339140620415470922284264797304009209531909694641349176491782115 6898309366231880807270078212985487240214165966228194347796689723429746 820797352069644885242718223505206348369105-790335040321686466256331677 1035114134505837900455617836367307403237887424748381649041102133954687 4890653888560482019573540179952253418072708504351364109159343630836512 45392760317300579788728713068809901451603734258873900/3863761280807662 8213843545563688106129370464273097427432254111675199765566018220166060 4035955805645027926686809510107120070197398105476998890453405736193756 03024319041073302146267803061475172915154071508740924768427*7^(1/2), d [11,5] = -588365078178231646007854060948090885162693751699381569916559 4145109751953197427677226648812495404740363751722260622443716631102410 72491075264725000000/3145451127250027451166901819469043846501525554469 1435917977862916865719498535944989324658187115701213146289028918803146 77837261365468248889510667*7^(1/2)-25759662442418468490091305673420172 3274658543249307791037853096354403458772729102416681481282720027374703 15453312749062733246568740635068153622590400000/7234537592675063137683 8741847788008469535087752790302611349084708791154846632673475446713830 366112790236464766513247237590257011405769724458745341, d[17,4] = -921 1470773400427862668590377375063625382975296268805132574456397652717409 01023751317610603499501800859796628390590953930820156250000000/6548812 8014915989192389920409370961263948791333952313890482723680417603582558 74176057166095974250510702775147164098867220057094791837*7^(1/2)-11896 5031906802054699885506874167396691856137438594967592822068564208288608 98811841868862425876575221122654143703544566787793625000000000/6548812 8014915989192389920409370961263948791333952313890482723680417603582558 74176057166095974250510702775147164098867220057094791837, d[16,7] = -8 0479633425145694979535497072990347527394902872888625090381035735031804 5105404629507631861823856776630017187500000000000000000000000000/52206 0881532785520281897778905215464986094012037659654266400930218186227216 727302782171212629401734675423069645782218707205760970841+568485922389 7618546525624281784722268021553621501920992895615566215911803984494249 96333110914849056414843750000000000000000000000/3706878877534499564257 4842806562218598416027472981612919622008497716575216136768076714299899 0598076109553948072670650060464613*7^(1/2), d[16,3] = 9024257309869448 9902118330581195780256541215932972642868960778553350848914210883132331 230082422146548033007812500000000000000000/370687887753449956425748428 0656221859841602747298161291962200849771657521613676807671429989905980 76109553948072670650060464613*7^(1/2)-12775495252009857263804546548269 4120832088354060111253472605825889684271490394681993803573581180972581 177706640625000000000000000000000/522060881532785520281897778905215464 9860940120376596542664009302181862272167273027821712126294017346754230 69645782218707205760970841, d[10,5] = 47013700420612497405110315890399 8482181422171497184844177550470067529511603029052907544286267915626682 907561440628289229353576346909204303127281250000000000/181650590012462 7262640201696649812174169629188359339232374432176372532726113332766271 153950526393073851541029433016351564436515773989435348498410427849*7^( 1/2)+68840898058460162718873118871189732943062401046853431598675105231 6230282776579725984682930576876991949726118943041279172185963700501823 06877634000000000000/1397312230865097894338616689730624749361253221814 8763325957170587481020970102559740547338080972254414242623303330895012 0341270444153033488346031571373, d[16,1] = 0, d[19,6] = 16697251789770 7163697547218010724005084059591136967141360302878080955415100661361601 83907920238703782067359620085428246873437500/8571461668773467460895684 3726885610572913937399987513357684524153879167917193127384594627394222 377959266228233019940104071397*7^(1/2)-4387528225629889178911861294629 4739832067939381712320693575518740978851571917550273760613995284312058 9990930475622519083225800000/85714616687734674608956843726885610572913 9373999875133576845241538791679171931273845946273942223779592662282330 19940104071397, d[7,4] = 859245651934450811084899343286497857291421143 6940293475190227730050731704992713782501329367518959847997311436874600 0085394361194627085479383144130610000000/91301280289827564029925424805 9557883336838237125750016806173347018965981574473290509116165584161930 09299310414227925015016650738820542603882059615975967*7^(1/2)+17171245 6371207265467167296597685364852350680823702449070473874112889386285358 0658389126779723973576089959195536801230776807590702383056194421333299 70000000/9130128028982756402992542480595578833368382371257500168061733 4701896598157447329050911616558416193009299310414227925015016650738820 542603882059615975967, d[11,7] = -119687217542847681847754199572804762 7805488585352545398550253429984677800410242965033977884811101224203635 17558871177390848167316265090008864939525000000/5064176314872544196378 7119293451605928674561426953211827944359296153808392642871432812699681 2562789531655253365592730663131799079840388071211217387-41186639221182 0148693368327135585756474056232997955589408745656658792156474721227294 263483755892735368990024449922669803318863572976045786737000000000/314 5451127250027451166901819469043846501525554469143591797786291686571949 8535944989324658187115701213146289028918803146778372613654682488895106 67*7^(1/2), d[3,2] = 0, d[5,5] = 0, d[5,4] = 0, d[2,4] = 0, d[3,4] = 0 , d[3,5] = 0, d[3,7] = 0, d[10,1] = 0, d[2,7] = 0, d[15,1] = 0, d[14,2 ] = 214382158733035474438957175414276691559614007389155767482988343183 4929066858157350198700881817074241476862904182773793087019667/31969720 6690784167592602756472753494315313273846706674245272441785499049839451 367031858386129351416410283715944104116776936593*7^(1/2)-1084464323273 4393713292817403956749717060683816218118985209091913994321995084278132 44465808250363966178459545992615406748444000/3196972066907841675926027 5647275349431531327384670667424527244178549904983945136703185838612935 1416410283715944104116776936593, d[1,3] = 4213874270803753212124697980 3882440252551164988949607528569720573249805893290031999397574017645557 8712815549474220415029872005415543040719763484960111027/10275234418818 2942069322599560446133917511688778235816746261671588403589348818222223 09944652680835028047947315566255509291453742438857799900604189917364+8 1150418524196608412620003413618046134786447563429998688294570113373712 7907347999014386498467656739192804734051570418037753008238285078783683 1585606/16797227906501217962323735645042951548651735134812292023710333 2180225087598784069178503234688450849132437711716412981150238773233631 2632840150193411*7^(1/2), d[9,2] = 14726305284415201050094344243141225 0374747127089993578556840439460283525127524638988697181412673707769544 9770745137409375833975843507419189582352150574984397586932401859688284 240929974840607290798552784868588628002433/269486401451275523723407466 8783826059589919042587440448631498634713148426574941249594448376326456 1117902471617053887157459124491585386364351557343033038797281969882504 6718759520612401474979410044666584193485089928+13291818432014865836546 0982766992724804133422719375241575532733096573383966122423286438972033 0268967522980972391958143707112571818972611271402388007803586861702691 3265498360754013571907254497706173480056258753618529/61246909420744437 2098653333814505922634072509678963738325340598798442824221577556726010 9946196491163159652640239519808513437384451224173716263032507508817564 084064205607244534559372760794986591924240549851933862*7^(1/2), d[10,7 ] = 415812444291202033577507639917262766158892950733998117684563506883 0689842971132282796661424601112119781849514762631806204372862855277778 102052468750000000000/127155413008723908384814118765486852191874043185 1537462662102523460772908279332936389807765368475151696078720603111446 0951055610417926047439488872994943+32910456270994581289583086748306194 0758517487445179585688895793458371217795950072336162373450451428768765 340095616264626457107935303851348136250000000000000/181650590012462726 2640201696649812174169629188359339232374432176372532726113332766271153 950526393073851541029433016351564436515773989435348498410427849*7^(1/2 ), d[1,8] = -315558082540469430895909422645449273814073607712836006617 0349405940374172024304797821426047585565847556641377835276745462174813 233606611295033155888146250/151559707677569839552250834351658047528329 7409478978297007359655928952942895068777790716836270423166637072229046 02268762048942700973152548533911801281119-1241178565075009107351278043 9774882169457948245327405587678471888500808055781896287373864728823978 609222029670117813108179951651930346571735684200000/167972279065012179 6232373564504295154865173513481229202371033321802250875987840691785032 346884508491324377117164129811502387732336312632840150193411*7^(1/2), \+ d[16,4] = 535076890995029437268096097204550734635696854122150939340428 1888103482680864429323637776808226044594308252791015625000000000000000 00000/5220608815327855202818977789052154649860940120376596542664009302 18186227216727302782171212629401734675423069645782218707205760970841-3 7796355050458493631102809827579259334097684638765166756254692365943618 5831073022771514115561178040804338769531250000000000000000/37068788775 3449956425748428065622185984160274729816129196220084977165752161367680 767142998990598076109553948072670650060464613*7^(1/2), d[19,1] = 0, d[ 2,1] = 0, d[7,1] = 0, d[13,1] = 0, d[16,5] = 8121013767932221265275525 0448027013526215919284718529896464888870924333520694191457956749086413 3389370449218750000000000000000000/37068788775344995642574842806562218 5984160274729816129196220084977165752161367680767142998990598076109553 948072670650060464613*7^(1/2)-1149678796506143627298355266120016773795 4297103479097471391813390195887384596133335486438941955921362180768359 37500000000000000000000000/5220608815327855202818977789052154649860940 1203765965426640093021818622721672730278217121262940173467542306964578 2218707205760970841, d[14,3] = -69537292130138629369023922527252630704 5271178664190744876827898620675962288203956612776865588439092002702739 230177018876641637163246/990070279400689488617531476520470296545093677 7758658994701842249655120074477969385609622360039884014810076399072960 392464949348617*7^(1/2)+3517583407959113267515523305597344057907232083 5698664652192058865185604043791920256044511025727863449276740998377350 2387900819672000/99007027940068948861753147652047029654509367777586589 9470184224965512007447796938560962236003988401481007639907296039246494 9348617, d[6,6] = 8162592265890581274904267140876836238713749426686139 2419450427131265108865578644029154881712938474354859943950545138008226 945582205110546522070202640000000/131541522761148302511191260257722195 8464883069039860624042748458072382214029609403382137397916420475807565 2253810560465309795576182203094819198747826469+44257061409356319069484 1211655127009120980446804634250385047831348465218998718873844049440827 23631860368040867745762630095578091751944396417645035600000000/1315415 2276114830251119126025772219584648830690398606240427484580723822140296 0940338213739791642047580756522538105604653097955761822030948191987478 26469*7^(1/2), d[9,7] = -160772631502208585650983829936857435139789463 7214925990923378503481718582681762547589463919745422824546646464955212 0059858620164446919952403977750765962269947068297157127663669740244655 53037284694221254682820428633803893100/6220655662100337142428810835753 7850868286447479686858165929119797071622561289334467357249788884708849 4961965763311272463313017810949817968213629983235271947208691536561280 1645549116292897502839339805512907288887716747-55324908593108225694677 6962556284850523363391454381561072169790859324601579202052650665664121 3841224185072335245003346515932899463224200185115505809412305356424725 993493254199982749602115856219837329783416738781282748000/386376128080 7662821384354556368810612937046427309742743225411167519976556601822016 6060403595580564502792668680951010712007019739810547699889045340573619 375603024319041073302146267803061475172915154071508740924768427*7^(1/2 ), d[14,7] = 221591271125389424581121911567394697854994772585518697228 1633357636637201842311667852354187102328116657845599068755034236015392 000000/990070279400689488617531476520470296545093677775865899470184224 9655120074477969385609622360039884014810076399072960392464949348617-43 8052354888582719562698261701047240821083499565976777048096804715818700 8612922660400815609630794712428619505097414253365445796456000/99007027 9400689488617531476520470296545093677775865899470184224965512007447796 9385609622360039884014810076399072960392464949348617*7^(1/2), d[6,5] = -37983753935780826845491560567435699912337813884541364724875868454675 4409945250829352818740717874747201967186260699114771684352059165462128 54066374800000000/1315415227611483025111912602577221958464883069039860 6240427484580723822140296094033821373979164204758075652253810560465309 795576182203094819198747826469*7^(1/2)-7230416712426172469193006691281 7750380192743571742784794551465291383971580846818515186344973622679103 810295735104343879946877647708372919532508368614400000/131541522761148 3025111912602577221958464883069039860624042748458072382214029609403382 1373979164204758075652253810560465309795576182203094819198747826469, d [2,8] = 0, d[5,2] = 0, d[5,3] = 0, d[14,4] = -147327172576762221828222 3438041412461091536412132115133311588519623701982172428530038148835093 719226576389233401247041408970391836000/990070279400689488617531476520 4702965450936777758658994701842249655120074477969385609622360039884014 810076399072960392464949348617+291243488782590462114448020528561996020 2415191887672718786681701459505685629467794725596578010663098124551310 380813803704380834095123/990070279400689488617531476520470296545093677 7758658994701842249655120074477969385609622360039884014810076399072960 392464949348617*7^(1/2), d[1,2] = -34526622656681398212098528289998126 9245058458792437036355181222634328552609814391281742925584254890968273 060306956735227745352027544581331833732294791/391513339604300660795636 4993423308405911194832197302585184822644396533159985969912598319186981 6220520832373768504573931879994239652078542680464154808-35740744020484 5554472881112140114408021040394498956705404353902548452587993719526599 7709738666590698990636816002511628009933721760047442122841/77484064278 5698969122289831077296261637293290286244402176846580129553920735408538 3932468629387491137793909656957094474670005177233974217720717*7^(1/2), d[1,5] = 730280225263209507765580583530815330217522135086423721048559 1115091790362450072914059927648370501747154506660339648300083519102873 1264643821938053800/16797227906501217962323735645042951548651735134812 2920237103332180225087598784069178503234688450849132437711716412981150 2387732336312632840150193411*7^(1/2)+138676489300620688342130927286764 2023659816748829651588360531520324793611684581929110473352804183890434 53765374932033685619411161630006418467158243649434172/7577985383878491 9776125417175829023764164870473948914850367982796447647144753438889535 8418135211583318536114523011343810244713504865762742669559006405595, d [2,2] = 0, d[18,1] = 0, d[15,3] = 309538870169024301470563385656739544 7874515922155612719497055529881588346474260993859858599921494599274662 7898897849140019200000000000000/75374050370774490768452671307503403675 9779816890766709266651250466244291270007809326460550269836370047491116 26142447467835659391021221+2482377366245665316964026392734497114671560 0084186708084366057450366893551535014530056424942108070556077606061783 84294912000000000000/9900702794006894886175314765204702965450936777758 6589947018422496551200744779693856096223600398840148100763990729603924 64949348617*7^(1/2), d[12,1] = 0, d[14,1] = 0, d[2,6] = 0, d[5,6] = 0, d[5,1] = 0, d[19,3] = -1592434320753434931592281346468546690744785112 7071014387214356429033522763090853817696362854518576180383590619185986 32466078125/8571461668773467460895684372688561057291393739998751335768 4524153879167917193127384594627394222377959266228233019940104071397*7^ (1/2)+5091796679412412114278737399280824710898381458635953147862013790 3211951855418414175730080645271866282268124971252604810903662500/85714 6166877346746089568437268856105729139373999875133576845241538791679171 93127384594627394222377959266228233019940104071397, d[20,6] = -6801184 0009188320025394891017762120454841491283830969423875193642423226002451 5482561518943089541903554601499199787917725791816587500/17681812245839 5629215082000187301782714444451046122293626596980534108063780972383725 927210518515282304257233230663900078841620401-121210628923635975055811 0384054977777309753288304350946337692919639143578015357004256278422282 84898678661529858166089302283880468750/1768181224583956292150820001873 0178271444445104612229362659698053410806378097238372592721051851528230 4257233230663900078841620401*7^(1/2), d[15,6] = -324562739359195509624 7548400353940087213319609143918412746471729898776061010704325168729498 92668598249686819652735732203520000000000000000/7537405037077449076845 2671307503403675977981689076670926665125046624429127000780932646055026 983637004749111626142447467835659391021221-260286276057029937911054237 0631288611189303360540254254317208065405545467610208577742291525290986 0847316669815138806067200000000000000/99007027940068948861753147652047 0296545093677775865899470184224965512007447796938560962236003988401481 0076399072960392464949348617*7^(1/2), d[14,5] = -625772611953315670490 9756747360746277870199010127944476405579664448471299756182784405064551 389990170123265077552459755577661337025800/990070279400689488617531476 5204702965450936777758658994701842249655120074477969385609622360039884 014810076399072960392464949348617*7^(1/2)+3165506290987968550806429845 6491406169566440031122352980412966971051701464295368002501751450544175 96954251063186896627242843165600000/9900702794006894886175314765204702 9654509367777586589947018422496551200744779693856096223600398840148100 76399072960392464949348617, d[12,3] = 81793288064001405035051802077326 1181499853542347766137189202482139155110545615505001277478763564907623 796961751224239072887644092271365112874456940/382258993481407451700934 0409360962220010170665236967566985909215747458647765894333746038996390 701657590889787189289978886713973532215119457016193*7^(1/2)+1610559715 7527356608797264349063792957397470513682855346349967627106213388482965 2530090255062307578402569190331617510431175371811190260279157345525/34 7508175892188592455394582669178383637288242294269778816900837795223513 4332631212496399087627910597809899806535718162624285430483831926779105 63, d[20,7] = 41005093718975289778836814119900709888082369388928172863 3739794265900726272290516959224636707468292399586615990636550375134586 625000/176818122458395629215082000187301782714444451046122293626596980 534108063780972383725927210518515282304257233230663900078841620401+728 2248288849647423818750048139265893342513472500573180792592242584761681 1105599124817119652350853092996924788489181048401968750000/17681812245 8395629215082000187301782714444451046122293626596980534108063780972383 725927210518515282304257233230663900078841620401*7^(1/2), d[1,1] = 1, \+ d[3,6] = 0, d[4,7] = 0, d[4,6] = 0, d[10,6] = -10103100080444946873952 0738617393533920252360035964385638772689683529625552202236334039118681 7130641794372274265723171823266527662376078799434731250000000000/18165 0590012462726264020169664981217416962918835933923237443217637253272611 3332766271153950526393073851541029433016351564436515773989435348498410 427849-547783726198824397777037201117257022442448102930264510397871076 7297651493563716028972238016514230083186399077353183298808623792416365 85227543156250000000000/1816505900124627262640201696649812174169629188 3593392323744321763725327261133327662711539505263930738515410294330163 51564436515773989435348498410427849*7^(1/2), d[8,7] = 5828505013191212 6406709353348490172763725290684361195686227019945490541408145527777044 2517203693434228794599954407666266920827889552060277962934375000000000 0/36044573731325869174159254757021392752950200010975305804861962231614 0902579674327363416819518950056351481148792537363493884185122305499725 35633422517331+2005699005336285327497127182582905772654829718797259451 0969261863439867121513735387533884634513580436726314483466386983345753 276732134992128750000000000000/223879339946123411019622700354170141322 6720497576105950612544237988452811053877809710663475272981716468827011 13377244406139245417581055730034990201971*7^(1/2), d[12,4] = -68460580 1141385513385347864687884460954942747756571403232221753013246612442837 7211969348888663442360106913008869448126554216730415321581869463218190 /382258993481407451700934040936096222001017066523696756698590921574745 8647765894333746038996390701657590889787189289978886713973532215119457 016193-342575355539828849211109505827556367940910699371237668673013553 1344332033895307841084703163416505227062430432094368610994856466429304 421286052243470/382258993481407451700934040936096222001017066523696756 6985909215747458647765894333746038996390701657590889787189289978886713 973532215119457016193*7^(1/2), d[17,2] = -1749083901770424910015952571 4812151887627475867159173341855360067337533408349166015757817747028313 61806842464954124362375000000000/3020906990627308838441663802483172631 7999470130938456378259699183246658447645222070259965476878955041229133 037019041471056573139-880583752871865309316034990835128208189585402977 9995546328939123123089412319527210225730649032978512128091309187620781 250000000/274627908238846258040151254771197511981813364826713239802360 9016658787131604111097296360497898086821929921185183549224641506649*7^ (1/2), d[12,7] = 65101274498595179624055006508397379942285664436299505 1736727005635479601247294413932254794020420091445812054382264704518474 78382395009443841361326000/2675812954369852161906538286552673554007119 4656658772968901364510232210534361260336222272974734911603136228510325 029852206997814725505836199113351+515259386049443372267043753893792426 5346057301121017297317828917043468691398032685480792242816884146530237 629541886497718708096817362817205693840000/382258993481407451700934040 9360962220010170665236967566985909215747458647765894333746038996390701 657590889787189289978886713973532215119457016193*7^(1/2), d[17,8] = -2 8504487591710923397038399865217159843956508784729598662931765363312962 66156008727418895735371274515801601052625612694954262500000000000/6548 8128014915989192389920409370961263948791333952313890482723680417603582 55874176057166095974250510702775147164098867220057094791837-3363829935 9705254815872536649901897552842162393759582986976547450330201555060593 9430622910793059779163293088010967113843750000000000000/65488128014915 9891923899204093709612639487913339523138904827236804176035825587417605 7166095974250510702775147164098867220057094791837*7^(1/2), d[12,8] = - 2223592373365157587973339004172678190485217792037090356741641558923136 5933667448165989441705124500081466937606377029491283817957303411319128 74475000/3822589934814074517009340409360962220010170665236967566985909 2157474586477658943337460389963907016575908897871892899788867139735322 15119457016193-1251011119329888698111117061495326606347049555745810626 6686649249444532974742261693786944098973214928398782061861243721262435 12599934278342258000000/3822589934814074517009340409360962220010170665 2369675669859092157474586477658943337460389963907016575908897871892899 78886713973532215119457016193*7^(1/2), d[17,6] = -23060780155233581032 2868833020897876237613539083120310166995037492694852915188517753024047 7807145832543211828904839642766218750000000000/65488128014915989192389 9204093709612639487913339523138904827236804176035825587417605716609597 4250510702775147164098867220057094791837*7^(1/2)-232638016706155219771 7657181403505134473073459426969897643089432387120344376201061699776905 9520870257224345221992259076887455500000000000/65488128014915989192389 9204093709612639487913339523138904827236804176035825587417605716609597 4250510702775147164098867220057094791837, d[20,8] = -10058805201314826 2644607260455844694689171805732730313407816433798589755428967916485597 741040090477881490177601592195261933660937500/176818122458395629215082 0001873017827144444510461222936265969805341080637809723837259272105185 15282304257233230663900078841620401-1768075231568468319964129327292672 7330406519464682075018195265143815649070678254732343608396805579067425 652914274461763671093750000/176818122458395629215082000187301782714444 4510461222936265969805341080637809723837259272105185152823042572332306 63900078841620401*7^(1/2), d[12,5] = 736065468529743240339568189838422 2748095329548154901761463973685799223409676638193306845021718324588125 457006459478446445551482807819591972519362000/382258993481407451700934 0409360962220010170665236967566985909215747458647765894333746038996390 701657590889787189289978886713973532215119457016193*7^(1/2)+1077800884 2118839789806251301506325555086046569852970034319930283981328096294871 96841854241406818125328359833747820729973180946561431535883818323072/2 9404537960108265515456464687392017077001312809515135130660840121134297 2905068794903541461260823204430068445168406921452824151810170393804385 861, d[8,2] = -2471639358420968741360873596016663862254554479049199103 4086084400006220361764728156448171638703388027372722233003504210331991 014276048306679687500000/722914333514557819172794408454164297596538634 6269191612943731596074954990648318672577943993260943900251306180805878 278476516691452131348446349259-981586054119171519660337068813772718859 1415082841115684551743806152881529697956124727393075059399588320067976 984998968561696002875212248906250000000/722914333514557819172794408454 1642975965386346269191612943731596074954990648318672577943993260943900 251306180805878278476516691452131348446349259*7^(1/2), d[8,3] = 158612 4234059706409096223420780209907415744625136804971910262208518226493127 2400092946616400914240077881162345729492871888143053658400510081978242 1875000000/51492248187608384534513221081459132504214571444250436864088 5174737344146542391896233452599312785794787830212560767662134120264460 4364281790804774645333+31838860378840274206743061989665991986717969896 4695183095110702985781946083571408736137229709465589327732013118635504 9986706358866322088853911562500000000/22387933994612341101962270035417 0141322672049757610595061254423798845281105387780971066347527298171646 882701113377244406139245417581055730034990201971*7^(1/2), d[10,4] = -4 3726887644062781540811485738789865262417943341027981084779797585175544 7481935186225863616156585927386131448133129778114765656047710054763967 097968750000000/181650590012462726264020169664981217416962918835933923 2374432176372532726113332766271153950526393073851541029433016351564436 515773989435348498410427849-218808459647003806575748777788182284280565 2460937914437874454853591617806228460301150180255772242665011538914560 39904834446426497851553075035257968750000000/1816505900124627262640201 6966498121741696291883593392323744321763725327261133327662711539505263 93073851541029433016351564436515773989435348498410427849*7^(1/2), d[17 ,5] = 1979198281787298725488101613331052490470495772562545842993783091 5253209516950544718648931755305224437902073770347478092695937500000000 00/6548812801491598919238992040937096126394879133395231389048272368041 760358255874176057166095974250510702775147164098867220057094791837*7^( 1/2)+22359181545887551831462319487707094827367763626020747452484410958 1989396546655893511694249202876085011278230549677582946387065000000000 00/6548812801491598919238992040937096126394879133395231389048272368041 760358255874176057166095974250510702775147164098867220057094791837, d[ 18,7] = 20297584237676749329406254044151416959910735998312029305564972 7308681729566027986900258842686774658206717662387827601542077891000/17 8479353362772201242027020920484756214595687832874432575957818692709436 225843698147142925439917592200896345368322905584668147+356787712879533 7168364196312555077512698884093839357223957122825251019683442338507478 81335628023549492445997911845923748588698000/1784793533627722012420270 2092048475621459568783287443257595781869270943622584369814714292543991 7592200896345368322905584668147*7^(1/2), d[13,7] = 4376089915149419517 4066878704552080964209969973173515341184163005591874780786902908897351 8182295147401302720403960493031415536805158000/27750536738728464069432 6137739241858051364222100976721142423109809835263730573908868636663190 4388063519397724018224982590201818385151+58182497414878959218323764306 9777493134087487664596222656748837192827045735383734916135336984629559 0975624032400403981594646268000/23353336928467347254822150968134197717 0020973079784161393618653535614423861661638883300089364076787949018987 28578250953809270619*7^(1/2), d[17,7] = 127106543127824078534612097101 0321792458874748922521535190380188528631949078687322549459893506373441 9610369058198179685323830000000000000/65488128014915989192389920409370 9612639487913339523138904827236804176035825587417605716609597425051070 2775147164098867220057094791837+13854752534184780030654768131803573176 3716608962946537927720996587569439577670513314807555739625270718418537 4222343503238750000000000000/65488128014915989192389920409370961263948 7913339523138904827236804176035825587417605716609597425051070277514716 4098867220057094791837*7^(1/2), d[3,1] = 0, d[9,1] = 0, d[18,2] = -591 2034700498324467872734005922020507441411750635564776981464290569752878 6124985128848509627815563673602984974002137944229/57631616572305273416 0053669542073545205191281064530441977325127361908476947410953363501971 1321566476182516237796600005963-69844546427325205156872709632059024883 7682603410867204567800014095511556478009012751992241911776409999449378 126103018473619/230526466289221093664021467816829418082076512425812176 79093005094476339077896438134540078845286265904730064951186400023852*7 ^(1/2), d[7,3] = -4443556874769042450831206410459729378618591296079586 4005682142311283817881786724825154276381470942933737584010302517957481 751494388002036040172988825000000/913012802898275640299254248059557883 3368382371257500168061733470189659815744732905091161655841619300929931 0414227925015016650738820542603882059615975967-20515348226280741375975 9807703547187236822333040165273830091511393399988045265781840311307732 00644291719217914028845787196571844769767813567988371220000000/9130128 0289827564029925424805955788333683823712575001680617334701896598157447 3290509116165584161930092993104142279250150166507388205426038820596159 75967*7^(1/2), d[8,4] = -612926302095430866779818876513633778074275351 7391949508477793069152586718714921094775099309267735864733180785488520 57191067389178215649262438344843750000000/5149224818760838453451322108 1459132504214571444250436864088517473734414654239189623345259931278579 47878302125607676621341202644604364281790804774645333-1333509041197797 4961484005083057523261962629278053337151675291330605458005657872255053 418083415630556725181908942432427401802442301475843788908906250000000/ 2238793399461234110196227003541701413226720497576105950612544237988452 8110538778097106634752729817164688270111337724440613924541758105573003 4990201971*7^(1/2), d[16,2] = 3938661065322760533777541868752387730468 1748113670260664815043928900244999946710525301045762213969975410156250 0000000000000000000/16857531128960751728563976198947833801094449676697 9771470309319066868877657246699209587397923536999798321892746224359426 26683489-2782161490314464763833280811113944771938142757256113772137097 66740367880781770260618788132026004076269531250000000000000000/1196964 3441940326017170345444335373631184741991340247641067521875978099136600 073646780425554283253450532918340039092320077*7^(1/2), d[17,3] = 33200 8941727435812521342153646660651862104281144722950507480792528583267585 0584335260906065882780964516981035235628300315714750000000000/65488128 0149159891923899204093709612639487913339523138904827236804176035825587 4176057166095974250510702775147164098867220057094791837+41496804887084 3552092706961151127487083676604470158466124314193025877090084027864160 1193762511598860012266004724192792197812500000000/12356250568852073432 5264000772398040120658096856513799793363629585693591665205173133154077 282533028503825946172907525796604850845129*7^(1/2), d[6,8] = 114745751 8039621551255824818977190128144087495374027109532121924022107863952585 0680273720863149232774083290557059491215099556682274497692598064415000 000000/131541522761148302511191260257722195846488306903986062404274845 8072382214029609403382137397916420475807565225381056046530979557618220 3094819198747826469+64556891416829325595492504161155319441079075990879 1577715524829159218795349906063329726930501932077850004605590443939898 4312251861818996684573200000000000/13154152276114830251119126025772219 5846488306903986062404274845807238221402960940338213739791642047580756 52253810560465309795576182203094819198747826469*7^(1/2), d[9,6] = 3906 3332725217265717071490143611295954614729115106886733209602458505958486 4020056016326828734054000201163377320437416095541808172011318758401632 1081644077505836717591541152616139475501436583990476139792192245844169 58221460/8886650945857624489184015479648264409755206782812408309418445 6852959460801841906381938928269835298356423137966187324637616145401564 2597097448042833193245638869559337944685949364159470413928977048543644 701041269673821+920864916948971501109575170465865475326490561423376187 5382396154224998872242994678229261867564101464981448963983847456320234 5125792497043206777834104869843781697837622134127335384302095633656401 88716034976391938816258300/3863761280807662821384354556368810612937046 4273097427432254111675199765566018220166060403595580564502792668680951 0107120070197398105476998890453405736193756030243190410733021462678030 61475172915154071508740924768427*7^(1/2), d[13,5] = 831156663603289259 8082197081966910014320872331189118452552901897702519972166300222644759 947729568743126576641058043415508315809900/233533369284673472548221509 6813419771700209730797841613936186535356144238616616388833000893640767 8794901898728578250953809270619*7^(1/2)+792933684811743045881583033613 9833751577210010662997799890138299729019110773540943233164665874442762 04111497462281081616679858486737604/2775053673872846406943261377392418 5805136422210097672114242310980983526373057390886863666319043880635193 97724018224982590201818385151, d[15,5] = 27855693161625913070367935958 3669620129955511285396960929986580417832755735840570849485837604183798 299224759863025182556160000000000000000/753740503707744907684526713075 0340367597798168907667092666512504662442912700078093264605502698363700 4749111626142447467835659391021221+22339146675745678541633465064407760 7120644135325165528927872023104780315224181156562052284425076532666823 17225165342617600000000000000/9900702794006894886175314765204702965450 9367777586589947018422496551200744779693856096223600398840148100763990 72960392464949348617*7^(1/2), d[4,1] = 0, d[3,8] = 0, d[19,8] = -58298 6430347992097911710859056207410310312161050982104244163220727421776322 37400380577658856590526947451093362815756807028125000/8571461668773467 4608956843726885610572913937399987513357684524153879167917193127384594 627394222377959266228233019940104071397+243559476482504379982568534210 2338620810810873899076449329797603446951342270204059861728140813479442 444278566214005004687500000/857146166877346746089568437268856105729139 3739998751335768452415387916791719312738459462739422237795926622823301 9940104071397*7^(1/2), d[18,3] = 1648600132090977282526501986405242270 6442598727037013033733537159019665916899003914719364978689253867887112 758933844599050717104/178479353362772201242027020920484756214595687832 8744325759578186927094362258436981471429254399175922008963453683229055 84668147+1132743661440127252865165909038201446251082009481612065609855 73732773013088207311226789216800898928357473427038219188619947361911/3 5695870672554440248405404184096951242919137566574886515191563738541887 2451687396294285850879835184401792690736645811169336294*7^(1/2), d[8,6 ] = -14161665980926452803750393133517619102224065541199277512545986152 5068870789025594846342064581049267165852603512722425468205796744576363 8566385856250000000000/51492248187608384534513221081459132504214571444 2504368640885174737344146542391896233452599312785794787830212560767662 1341202644604364281790804774645333-33384200623943001178124788023791180 2439562719057594945472704077792415624250158255677569420121090986990780 83037059970401121014107150159855466468750000000000/2238793399461234110 1962270035417014132267204975761059506125442379884528110538778097106634 7527298171646882701113377244406139245417581055730034990201971*7^(1/2), d[17,1] = 0, d[2,5] = 0, d[7,5] = -1846195422833748992131866326562642 0160802986139333665646532211620339724753470146722383476511881617732594 5218969476347783698687223240747265935403006000000000/91301280289827564 0299254248059557883336838237125750016806173347018965981574473290509116 16558416193009299310414227925015016650738820542603882059615975967*7^(1 /2)-270333434216489366239362995789451067435259467150343402600591678349 4109613653516913231466470912308995921359054252418624940312682016399753 0206000292736000000/70231754069098126176865711389196760256679864394288 4628312441036168435370441902531160858588910893792379225464724830192423 2050056832349431067850739690459, d[12,6] = -15817821247389300266475983 9735869048307364155339506409214276794582464561922773640658114489511987 87801960879566494334415203137390380235046630632171600/3822589934814074 5170093404093609622200101706652369675669859092157474586477658943337460 38996390701657590889787189289978886713973532215119457016193-8576323103 0571426752502147312731159389039043426294851936745018780996962359824302 03471050242946504998371806506883402153063977142036082428608312314000/3 8225899348140745170093404093609622200101706652369675669859092157474586 4776589433374603899639070165759088978718928997888671397353221511945701 6193*7^(1/2), d[1,6] = -2696059269776327206084263156125380162079875826 7391811759935890785632971468151803651018578207043144014525781306920514 921820002789333771676890104946885356240/151559707677569839552250834351 6580475283297409478978297007359655928952942895068777790716836270423166 63707222904602268762048942700973152548533911801281119-8508915898664461 5193355263069018586582904428305112255804832220556118448013952787143330 893180713633836917033068802486059110749910432207288326806798600/167972 2790650121796232373564504295154865173513481229202371033321802250875987 8406917850323468845084913243771171641298115023877323363126328401501934 11*7^(1/2), d[9,8] = 5491333311176146435762383876397485099732836276004 1893298923587697605989569412874021195949651548316788146956257417285069 6928656695333537754072313391477759945159947269551224168530971970458433 87736173492959471872595661778750/8886650945857624489184015479648264409 7552067828124083094184456852959460801841906381938928269835298356423137 9661873246376161454015642597097448042833193245638869559337944685949364 159470413928977048543644701041269673821+134324725953166104485730411156 0614308867168451820141330207764657219339488758168404640732468693552237 0205786546394920923177698477112368717081368048121189687332551007585768 18467733646684807200561840812005685408434465100000/3863761280807662821 3843545563688106129370464273097427432254111675199765566018220166060403 5955805645027926686809510107120070197398105476998890453405736193756030 24319041073302146267803061475172915154071508740924768427*7^(1/2), d[11 ,3] = -325707528469545807806170992670687460523990132573907179269082635 4163800163591657325343015058638708151782063177712234125474120434700299 898965310311562500/723453759267506313768387418477880084695350877527903 0261134908470879115484663267347544671383036611279023646476651324723759 0257011405769724458745341-65380480926998113445490508799317818310967623 3638353406518551024493169666845312628477142902351718833149434372705357 59867183264854301545874673160750000/3145451127250027451166901819469043 8465015255544691435917977862916865719498535944989324658187115701213146 28902891880314677837261365468248889510667*7^(1/2), d[6,2] = 3276624655 4655149990288207623619736902717015147048170067486812143127841073937005 814249190700185465942984014249816424438069522534530724434490928500000/ 4247522450229206707068076471882275690092941551357359372413537595893901 0430740721475738234941923228900111893357262296055118975673034980447606 3119501+13012776542135754112180426231437067007756777097637945414684662 7867172374978907766692117732064525575901702499032628752283191634396167 55569233302000000/4247522450229206707068076471882275690092941551357359 3724135375958939010430740721475738234941923228900111893357262296055118 9756730349804476063119501*7^(1/2), d[10,8] = -142024466801520635138450 2571342190610698550324609220580352800877902795431425211987069023890612 15442986677986030364231870842465747320669367073046875000000000/1816505 9001246272626402016966498121741696291883593392323744321763725327261133 3276627115395052639307385154102943301635156443651577398943534849841042 7849-79904117910204467207891004842203730468394823946241547855009783584 8742850956252400165340587393047019052653937538296467987660899166662882 08031250000000000000/1816505900124627262640201696649812174169629188359 3392323744321763725327261133327662711539505263930738515410294330163515 64436515773989435348498410427849*7^(1/2), d[19,4] = -19682654899165043 7242542262461615294664842541145911546791968576707071459227370613802318 648911504395238812735248641634201808175000/857146166877346746089568437 2688561057291393739998751335768452415387916791719312738459462739422237 7959266228233019940104071397+13339205857064209657992406748900707959915 5014209412414333594964038001055340534724335168930257412034828971692766 71730570911390625/1714292333754693492179136874537712211458278747999750 2671536904830775833583438625476918925478844475591853245646603988020814 2794*7^(1/2), d[13,3] = 1141292339734172012166041107011526750493096637 8426873005008617355445586670369074294845276232112092790772913248577032 2236685645140655590/27750536738728464069432613773924185805136422210097 6721142423109809835263730573908868636663190438806351939772401822498259 0201818385151+92360040402671143063122892980378734219969770177760344850 0801894253723034930402845279880050908849376015497763224251615485426652 013/233533369284673472548221509681341977170020973079784161393618653535 61442386166163888330008936407678794901898728578250953809270619*7^(1/2) , d[7,7] = -1292370802283018277405743021465848627032102105099845087138 7107692290929845159049815460445597804439261716248622654732142890328543 1628815347910135920000000000/91301280289827564029925424805955788333683 8237125750016806173347018965981574473290509116165584161930092993104142 27925015016650738820542603882059615975967*7^(1/2)-16328666421483296842 7608815220808436506686649492041163460709373907162938380456727702116493 7398034410400515952686591879890431037132214876681277077138000000000/63 9108962028792948209477973641690518335786765988025011764321342913276187 1021313033563813159089133510650951728995954751051165551717437982271744 17311831769, d[20,5] = 58785477955290480228019526274117504728305475024 6131963742599896431075402868587223539394837561497330299593054067533345 877754525106250/176818122458395629215082000187301782714444451046122293 6265969805341080637809723837259272105185152823042572332306639000788416 20401+1040293810032153470946266269048260902993901370046610011694912082 37460118376854198900277564255087502358083148269155632751580164843750/1 7681812245839562921508200018730178271444445104612229362659698053410806 3780972383725927210518515282304257233230663900078841620401*7^(1/2), d[ 7,2] = 122507627289048641947100243988443731883035716908511222910709886 9298704502345467558092421182268692989283529716893887294609944252376975 4760499227500000/22678082621039790169803904352480467647221371175784966 5249908307071082492312280839278264906490649937802068347290784394853952 341271347135234197124211+632484312400165298216823947811501145837391710 0516605971276804369670015750224022701673198193871344276911521482333441 6471917126266305351488464690000000/29481507407351727220745075658224607 9413877825285204564824880799192407240005965091061744378437844919142688 8514780197133101380436527512758044562614743*7^(1/2), d[13,8] = -914100 6833869224588315095647107193862305946586668790904224523017500927836344 3137132577478243566789260376443130423677085792324595475000/27750536738 7284640694326137739241858051364222100976721142423109809835263730573908 8686366631904388063519397724018224982590201818385151-14126273715159761 5430668619802623050272805600935498396460363842863464135017361943063230 0927493571029994844396944726140850889100000/23353336928467347254822150 9681341977170020973079784161393618653535614423861661638883300089364076 78794901898728578250953809270619*7^(1/2), d[6,7] = -265893274030320910 2589635768245951136016130235628524389304968966507718544114471737697537 0768966488324766577127009970609387323362799244860271309136000000000/13 1541522761148302511191260257722195846488306903986062404274845807238221 4029609403382137397916420475807565225381056046530979557618220309481919 8747826469*7^(1/2)-335947126605415679480598702942334468326814016765404 0959848386091331707957182441379742566325045498351382522220081596331182 52068332576383788851733020400000000/9207906593280381175783388218040553 7092541814832790243682992392065066754982072658236749617854149433306529 565776673923257168569033275421663734391234785283, d[19,5] = -143304657 6500324291727908566371343476297679816375344615015904302071246320152834 0587793456664492394680311192714307500577292187500/85714616687734674608 9568437268856105729139373999875133576845241538791679171931273845946273 94222377959266228233019940104071397*7^(1/2)+39743786391643150792166540 8289346112123154258835907749235530037281825985188654489397632669420451 571093469036756742124232661962500/857146166877346746089568437268856105 7291393739998751335768452415387916791719312738459462739422237795926622 8233019940104071397, d[1,7] = 2037065375027368807025271943201192952668 7727691365952933666962265575023386848673317598678819769110364129363463 21230035983714578513768849836145638690060100/2165138681108140565032154 7764522578218332820135413975671533709370413613469929553968438811946720 330951958174700657466966006991814424736078361987400183017+511209609200 6325468958656605425185500036219228080713836499947782851212398598168676 1001423585713337656035530371142833591324630186076256759439937416000/16 7972279065012179623237356450429515486517351348122920237103332180225087 5987840691785032346884508491324377117164129811502387732336312632840150 193411*7^(1/2), d[14,6] = -3688313865681068397757598697972254532407379 3690928451048285040466129629099556724777811513317414320218336973114321 26211539306023200000/9900702794006894886175314765204702965450936777758 6589947018422496551200744779693856096223600398840148100763990729603924 64949348617+7291237449130203878877762139194289422291772990061889075572 1398838829912335877189839538429370231752286039706943523027350377215922 82600/9900702794006894886175314765204702965450936777758658994701842249 655120074477969385609622360039884014810076399072960392464949348617*7^( 1/2), d[13,6] = -82513151875964588993809896845883696955673483761318222 0627066402275652598384129940345657141805307572721378853571722640948275 822161193500/277505367387284640694326137739241858051364222100976721142 4231098098352637305739088686366631904388063519397724018224982590201818 385151-968428543531483625328606192541434711499279959590284852462773805 9526100983462763368210611158967876373760938418285943680673176650300/23 3533369284673472548221509681341977170020973079784161393618653535614423 86166163888330008936407678794901898728578250953809270619*7^(1/2), d[15 ,7] = 1563781415259811620684144966926318425000096355109776911146729647 7045145829508553775562648757608718882855070448267948032000000000000000 /990070279400689488617531476520470296545093677775865899470184224965512 0074477969385609622360039884014810076399072960392464949348617*7^(1/2)+ 1949949830564707149071768871455493524765846931862335535308376703334855 00772176394778669066911840382885226036051860386611200000000000000000/7 5374050370774490768452671307503403675977981689076670926665125046624429 127000780932646055026983637004749111626142447467835659391021221, d[10, 2] = -3119678890638239310012160520858034913209835904825256245872247358 1631680672936448446057852147071959414840371338255688715643136806608601 639882812500000/451197078002227354560565949733806306100052704903250454 5176521872673002347541916025879859886006088157267791437673446030557695 451714715796065292117-161063274321616334424282915519625320577654488059 1779891318260035611340301665744366825424744221586923221318146346827959 03271961350348277277968750000000/5865562014028955609287357346539481979 3006851637422559087294784344749030518044908336438178518079146044481288 689754798397250040872291305348848797521*7^(1/2), d[13,2] = -1247650376 8645101289658202040732429734578239270094672764970228183592982411936235 6220396914903741991057842871446404823016889335239/25602133690121886005 2980296184887060379609307096014651649274260260108277614549027247188813 87418645036920770761713095423443736994-8135550307160072092865732030517 5578691144586926203264977172893795712329067590647837462357080781577643 6821380439371075882711/21545358195492586830906621799803672536132535584 4124457539215393767605784458801325642047659977098562109592530120703661 761386*7^(1/2), d[9,4] = 338137392973937475185729389431818654071466012 0187334941000847271459375486493183596150892865907430694671353347975337 4209171175888188130458283600379234256093257777443219406092601740448504 9563918113164951290655851522853174503/17773301891715248978368030959296 5288195104135656248166188368913705918921603683812763877856539670596712 8462759323746492752322908031285194194896085666386491277739118675889371 898728318940827857954097087289402082539347642+735666374825587025310757 4867042287692516679854014047419166639033502076342840095353895265824223 4186252800282851415222994359560172056936619230038777317750915416214609 58549736380197540267383395148992252404297705904153910193/7727522561615 3256427687091127376212258740928546194854864508223350399531132036440332 1208071911611290055853373619020214240140394796210953997780906811472387 51206048638082146604292535606122950345830308143017481849536854*7^(1/2) , d[12,2] = -976859036304749167634648894456031167117954082111213320664 7277177374614578196954917730159057202141616626512200358501277214722138 31479625871785/1898965931099374569114693059988505736510789034809979988 4181497704888305912691323252513252688871013234529267665627364232156295 121584190242138-229242875269874892324026686661447283885584997126195664 5727920019531720702177901357500225358450589123005012419712793352063797 01261768061476330/1122116232013266790840500444538662480665466247842260 9022470885007433998948408509194666922043423780547676385438779806137183 265299117930597627*7^(1/2), d[6,3] = -91422050365506498669218542017384 7439212009889923028518119879144639881739178180652685314693815322842391 0546087772207539535091311174943330104078559435000000/13154152276114830 2511191260257722195846488306903986062404274845807238221402960940338213 73979164204758075652253810560465309795576182203094819198747826469-4220 8421128995358842156208592494534485204404559920822814891166575288939603 6357668954665980202365137178014898649883641509363244848609862054158317 0876000000/13154152276114830251119126025772219584648830690398606240427 4845807238221402960940338213739791642047580756522538105604653097955761 82203094819198747826469*7^(1/2), d[11,2] = 507546338056293667330690493 5589653169772101650739549311989045477829289708150119540454204453304843 00575640078033630308292882163258363565838406250/1015677331282904663104 0401109073731300660420273399669320280881822747172817506521033719092701 4484488401776902802540615282309317748893030091043+20156678827674627198 7271579228149331675537257467093229677082267012806961207565557162113491 185766389277801850070194015195858242837743145405875000/101567733128290 4663104040110907373130066042027339966932028088182274717281750652103371 90927014484488401776902802540615282309317748893030091043*7^(1/2), d[18 ,6] = -593861419646616270181977900067188203635589684385714123547201975 231626700381152230730192790607795952326304554588606279674293597050/178 4793533627722012420270209204847562145956878328744325759578186927094362 25843698147142925439917592200896345368322905584668147*7^(1/2)-29316930 9397321889731944270635417780999693557702329121031086847433460867749942 907967397050459644916254775493491143831425988008700/178479353362772201 2420270209204847562145956878328744325759578186927094362258436981471429 25439917592200896345368322905584668147, d[20,4] = -2801654311936665895 8826284306348615895267093288507180448920884916275781506511325955036119 7011834959146313257916899640088743553603125/17681812245839562921508200 0187301782714444451046122293626596980534108063780972383725927210518515 282304257233230663900078841620401-193667023967038570529407180728994236 4254336997921541416806614052647627416283178748325382154834044277100819 57648717062052700191515625/7072724898335825168603280007492071308577778 0418448917450638792213643225512388953490370884207406112921702893292265 5600315366481604*7^(1/2), d[19,2] = 4718167869032807360918866893077828 7418848168761395721793299744150543036473297146709363320005287298099705 91986906711265625/2659920764875628003815632320963416362485498220297832 809126115972439577585911127477062316789840722989007315186675353848906* 7^(1/2)-65982720444966500804870923793913193304821923219818369171555628 95636573961752136615101847930997569404308848396598089086575000/1024069 4944771167814690184435709152995569168148146656315135546493892373705757 8407866899196408867835076781634687001123182881, d[8,5] = 2865208898787 1969730461484093398674485225567996576007701775493863924161325176710116 9156353515809823323493612271889996256380919645069107795028437500000000 00/2238793399461234110196227003541701413226720497576105950612544237988 4528110538778097106634752729817164688270111337724440613924541758105573 0034990201971*7^(1/2)+125443906848281132789256985493180093380607312578 6780629631174239438193931241474520151950311661207508633275253528493006 418641801407161213398088322000000000000/514922481876083845345132210814 5913250421457144425043686408851747373441465423918962334525993127857947 878302125607676621341202644604364281790804774645333, d[10,3] = 1131559 7965879973974644533121777215900726160873880109633888855374931244222296 9467857998443623249332012729005478902442853841149710994275042928758984 375000000/181650590012462726264020169664981217416962918835933923237443 2176372532726113332766271153950526393073851541029433016351564436515773 989435348498410427849+522427053824280208745285749713098908044378958686 3775531930360165218524545609239151469088413996475381591039157636481916 4592640745000306658367547187500000000/18165059001246272626402016966498 1217416962918835933923237443217637253272611333276627115395052639307385 1541029433016351564436515773989435348498410427849*7^(1/2), d[20,3] = 4 3622581585067442305577813217385009636834707044899007307457450978544367 0123799063695239094666358884339884792179503974368371390625/66723819795 6209921566347170518119934771488494513669032553196152958898353890461825 3808573981830765369971971065308071701088363034*7^(1/2)+705137502262766 6027753839535396466798074963133495973198382538619125336283488169284607 6916400546946903261336086721333143319830737500/17681812245839562921508 2000187301782714444451046122293626596980534108063780972383725927210518 515282304257233230663900078841620401, d[18,8] = -549965585506198964446 4185782389657418859233767257081017458026707461142199119291028475384522 2261665164711045525020601300886512500/17847935336277220124202702092048 4756214595687832874432575957818692709436225843698147142925439917592200 896345368322905584668147-866253790105137284737836853228784009922060891 2433482861847389785210565408266215677638345337996707422720443585849719 2550948850000/17847935336277220124202702092048475621459568783287443257 5957818692709436225843698147142925439917592200896345368322905584668147 *7^(1/2), d[11,8] = 40880241772120074452319139480276259105767389256023 7852548270444566162545051583806471623274014663760902270923134798699340 4683889077141021865585312500000/72345375926750631376838741847788008469 5350877527903026113490847087911548466326734754467138303661127902364647 66513247237590257011405769724458745341+9999806898924295571316590034435 4603506565236821337160696942111686873063871805250436269290432419169745 610787956901446499763439953269976032525000000000/314545112725002745116 6901819469043846501525554469143591797786291686571949853594498932465818 711570121314628902891880314677837261365468248889510667*7^(1/2), d[1,4] = -339882820903098897021856353785710196660904219969461756314974330255 8653084599078745969102563427995054187029435525124912699177227282350996 6390500216303/16797227906501217962323735645042951548651735134812292023 7103332180225087598784069178503234688450849132437711716412981150238773 2336312632840150193411*7^(1/2)-169023264501229588249421376038752908235 7569218631323265960376566769786443613461061130235197379353580129662766 6611080451180220999384173194579769504617980019/15155970767756983955225 0834351658047528329740947897829700735965592895294289506877779071683627 042316663707222904602268762048942700973152548533911801281119, d[14,8] \+ = 10635582420262279381257355975097882620230201405539935477727473648843 41433168592347760913943966392675641768349867298167479534432200000/9900 7027940068948861753147652047029654509367777586589947018422496551200744 77969385609622360039884014810076399072960392464949348617*7^(1/2)-53800 6975961628363438682629650841349599633924389002786019464040408549480795 090161895306413963167578026196813726206604386890400000000/990070279400 6894886175314765204702965450936777758658994701842249655120074477969385 609622360039884014810076399072960392464949348617, d[18,4] = -796866088 5827415981490828196930203818729245579271200734364509932953136611833115 8857213702226267642497024098283438504812608714705/17847935336277220124 2027020920484756214595687832874432575957818692709436225843698147142925 439917592200896345368322905584668147-948855515503175960512929409376655 1165413947581209400453139581139724792639378108598137294419508986065981 78330140938940662311931411/7139174134510888049681080836819390248583827 5133149773030383127477083774490337479258857170175967036880358538147329 1622338672588*7^(1/2), d[6,4] = 35328241040167355316548528784265374891 3763949016351206660699987623516928169268675618764436576775300544082017 28161521917397113366618167386721518719126000000/1315415227611483025111 9126025772219584648830690398606240427484580723822140296094033821373979 164204758075652253810560465309795576182203094819198747826469+176781802 2393979995747005190755505733665038478733060055220727402900096441459024 9521432871850604032521865375346674368489623684501026634997341154316438 000000/131541522761148302511191260257722195846488306903986062404274845 8072382214029609403382137397916420475807565225381056046530979557618220 3094819198747826469*7^(1/2), d[15,8] = -473433184570421347272979933960 4404118959129048478492998981796287397128520807150480846823585214002279 3357112022557245440000000000000000000/75374050370774490768452671307503 4036759779816890766709266651250466244291270007809326460550269836370047 49111626142447467835659391021221-3796743915119540595295058838192490201 5434281261245664060231016768134728904206713926024125600031338112387736 50809958400000000000000000/9900702794006894886175314765204702965450936 7777586589947018422496551200744779693856096223600398840148100763990729 60392464949348617*7^(1/2), d[11,6] = 290807058275182405006934572570979 0855088935948723106858442020701090367458365427824477989930468758788100 0462976582575696730187843539765360572245115000000/72345375926750631376 8387418477880084695350877527903026113490847087911548466326734754467138 30366112790236464766513247237590257011405769724458745341+6855380708310 0363319207627962617251777647665334853058758145597837239084006771419024 6085432984717058232523573733668192414291872068751656685425825000000/31 4545112725002745116690181946904384650152555446914359179778629168657194 9853594498932465818711570121314628902891880314677837261365468248889510 667*7^(1/2), d[7,8] = 557720235151732634699281612058896493828823833752 2293725943527768787851064891777906574032357987392416213512092278848941 7248086761403863712181951925000000000/91301280289827564029925424805955 7883336838237125750016806173347018965981574473290509116165584161930092 99310414227925015016650738820542603882059615975967+3137779316063157713 2315162086232913988296575745420045208152458041739558656955541195313788 442322772906236379301965905314639405498348888660511054000000000000/913 0128028982756402992542480595578833368382371257500168061733470189659815 7447329050911616558416193009299310414227925015016650738820542603882059 615975967*7^(1/2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "subs(dd,mat rix([seq([seq(d[j,i],j=1..11)],i=1..8)])):\nevalf[8](%);\nsubs(dd,matr ix([seq([seq(d[j,i],j=12..20)],i=1..8)])):\nevalf[8](%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7-$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F +F+7-$!)b\"R+\"!\"'F+F+F+F+$\")n(>e\"!\"&$\"):\"y5\"F2$!)>W6q!\"%$\")) R17\"!\"$$!)K#zT\"F:$\")ixC5F:7-$\")0@z`F/F+F+F+F+$!)9'Ra\"F7$!)0>\"3 \"F7$\")]&H%oF:$!)0r$4\"!\"#$\"),&QQ\"FJ$!)0:+5FJ7-$!)\"z0l\"F2F+F+F+F +$\")')RTiF7$\")pmqVF7$!)dBmFFJ$\")%y7U%FJ$!)a:%f&FJ$\")i1VSFJ7-$\")YE !)HF2F+F+F+F+$!);l88F:$!)P6*>*F7$\")X?AeFJ$!)vj0$*FJ$\")QUx6!\"\"$!)'* f4&)FJ7-$!)b7>JF2F+F+F+F+$\")[p5:F:$\")\\*y0\"F:$!)^^&p'FJ$\")^9q5Feo$ !)L.a8Feo$\")0,'y*FJ7-$\"))fgu\"F2F+F+F+F+$!)'*['**)F7$!)c(**H'F7$\")I J()RFJ$!)!GHP'FJ$\")!RN1)FJ$!)YwFeFJ7-$!)i1PSF/F+F+F+F+$\")kxq@F7$\")0 8?:F7$!)r/@'*F:$\")LtP:FJ$!)MmX>FJ$\")+>19FJQ)pprint156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+$\"\"!F)F(F(F(F(F(F(F(F(7+$!)/$ \\0\"!\"&$!)8O'[\"!\"'$\")'o\\V\"F0$!)CXD5F-$!)i?8QF0$!)'z#QmF0$!)w)=/ *F0$!)W)Q(fF0$!)Q0?fF07+$\"),eH5!\"%$\")gjd9F-$!)%\\H]\"F-$\")F.u5FB$ \")Y&Q*RF-$\")xHefF-$\")Q]>$*F-$\"),()[aF-$\"),mW>%FB$\")TB\")[FB7+$!)mR25!\"$$!)J\\%R\"FB$\")p *ed\"FB$!)A;E6Ffp$!):q(=%FB$!)d.%[%FB$!)3fW5Ffp$!)&pLg%FB$!))=,m&FB7+$ \")yC**fFB$\")Gco\")F-$!)i(yY*F-$\")D!fw'FB$\")7%f^#FB$\")$[1]#FB$\")) =iU'FB$\")9?>EFB$\")nq3MFB7+$!)ucZ9FB$!)izH>F-$\")%H()H#F-$!).rU;FB$!) m^3hF-$!)Bi6dF-$!)8E#f\"FB$!)wo\\gF-$!)!zVL)F-Q)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check wh ich of the groups of order conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "nm := NULL:\nfor ct to nops(SO8_20) do\n eqn_group := convert(SO8_20[ct],'polynom_order_conditions',8):\n \+ tt := expand(subs(\{op(e23),op(dd)\},eqn_group));\n tt := map(_Z->`i f`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op(i,tt),i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\s\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"# 6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G \"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\" #Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"# j\"#k\"#l\"#m\"#n\"#o\"#p\"#q\"#r\"#s\"#t\"#u\"#v\"#w\"#x\"#y\"#z\"#!) \"#\")\"##)\"#$)\"#%)\"#&)\"#')\"#()\"#))\"#*)\"#!*\"#\"*\"##*\"#$*\"# %*\"#&*\"#'*\"#(*\"#)*\"#**\"$+\"\"$,\"\"$-\"\"$.\"\"$/\"\"$0\"\"$1\" \"$2\"\"$3\"\"$4\"\"$5\"\"$6\"\"$7\"\"$8\"\"$9\"\"$:\"\"$;\"\"$<\"\"$= \"\"$>\"\"$?\"\"$@\"\"$A\"\"$B\"\"$C\"\"$D\"\"$E\"\"$F\"\"$G\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$G\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "principle error g raph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "T he interpolation scheme amounts to having a Runge-Kutta method for eac h value of the parameter " }{TEXT 267 1 "u" }{TEXT -1 8 " where " } {XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "The nodes and linki ng coefficients are ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "e _u := map(_U->lhs(_U)=rhs(_U)/u,e23):" }}}{PARA 0 "" 0 "" {TEXT -1 43 " ... and the weight polynomials (of degree " }{XPPEDIT 18 0 "`` <= 7; " "6#1%!G\"\"(" }{TEXT -1 33 " and re-using the weight symbol " } {XPPEDIT 18 0 "b[j]" "6#&%\"bG6#%\"jG" }{TEXT -1 10 ") are ... " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "pols := [seq(b[j]=add(simpli fy(subs(dd,d[j,i]))*u^(i-1),i=1..8),j=1..20)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "The whole interpolation \+ scheme (Runge-Kutta scheme with a parameter), including the weights, i s given by the set of equations:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eu := `union`(e_u,\{op(pols)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can calculate the principal er ror norm, that is, the root mean square of the residues of the princip al error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "errterms 8_20 := PrincipalErrorTerms(8,20,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "sm := 0:\nfo r ct to nops(errterms8_20) do\n sm := sm+expand(subs(eu,errterms8_20 [ct]))^2;\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 28 "Because the step has width " }{XPPEDIT 18 0 "u*h" "6#* &%\"uG\"\"\"%\"hGF%" }{TEXT -1 17 " we multiply by " }{XPPEDIT 18 0 " u^9;" "6#*$%\"uG\"\"*" }{TEXT -1 45 " in order to provide appropriate \+ weighting. 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"" {XPPMATH 20 "6#/&%\"aG6$\"#=\"#<\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"\",&#\"TLa3fKH(f5&*pD v#G\\7I-&Rl%)G`S]\"V+++aMeoA$eL?R!=l\"p*fjQE'G\"**[_\"F(*(\"P@U*[naF4: P;3b\\PJbovy(eYI#F(\"S++0b`\"p1A96&[m&)ytSCqhPO%>T\"!\"\"\"\"(#F(\"\"# F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"',&#\"QX31al)Gv?7I7W)))G\\?\"\\O\"\"(,&#\"Rl&4-I)py^C8[9:Hd@DYSK\"GZ\\9\"R )4&z7)z)e,[%G]6S(zdK$=%=k;#zB\"\"\"*(\"P#HGH>i$e)ov[eeZx2b!Qq<`(e_F-\" R\\vR1*R%z+CU^d+()*)Gm\"4#4K3'*=\"!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"),&#\"RD\"y!QCJ5E/DsC\\%o/4Lx_6F!y/# \"Q-r)pKUe&Qh\")e#=xiw\"\\%4gcM=M\"\"\"\"*(\"O+vyT(zly5\"phoZ`'poMO*en *H'F-\"O8;N-$>i))H+ATC;,WUVf#)y[[#!\"\"\"\"(#F-\"\"#F1" }}{PARA 12 "" 1 "" {XPPMATH 20 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9F[_lFgp*(\"ft++++++]7.3#)Gmm\"**3m()zY'HQv$Rl_!>q/$R(eS`;+CDc4&Gu[e$y 4]&yaTi%R#[Ro/t.A%[+\"*y?nW?5z6/*zF(F[_lFgpFAF[qFgp/&F&6$FQFE,&#\"bt++ ]7`el=-Tr2*)Qo/M*p)zMJ#4F-4wjY,uKirk!Qe^]aihcW/F[D&yBgD*Qnd5fiF![R\">B Xu+7sCV!HpiV]_!=(Q 1too6@%ppgrL@oBll].YNW.!f;8d&HC*)*o!)****F(Fg_lFgpFAF[qF(/&F&6$FUFE,&# \"_t+]ZuG\">8T.t&z\"QG\"\\HqP1w$pY\"3+X70yE6,^7F(F^`lFgpFAF[qFgp/ &F&6$FYFE,&#\"as+]Z&fCBz&3xOUIJWw.E*ymNCyudKr8VMOy#4] 2rk&4:$)eC#pQ$o+T\"*Fh`lFgp*(\"]s++5*)3&39EZ%pRW[**H5d$\\F4IK1V>O<]8kM 'G%QOgkR)\\N4g0GF]Ii-)>'o1V:wf^rti79F(F[alFgpFAF[qFgp/&F&6$FgnFE,&*(\" `s++?KW`zu;)Hn)\\$oE!yv;jRT1`*=;!4&z![\\&3//k %>gy-!*QCRj*f\\8%3lHEoQMOG;'f(p+Q&FbalFgp/&F&6$F[oFE,&#\"ds+++++++++Sa CdD-7rN$zA+9_eBo%3[]r!3_GrR(G'z\")*)*H\\y%[!H\"f*=TS/'R$*zHFZ8UqX=LMZF [blFgp*(\"`s++++++++Se*43lt(Q7\"Q8.+c7Cg#Rr1U!*GZ8on,J-1kcCh7GMa,-\\#> Q)e]H&fS&>^\"Ru'z$F(FbalFgpFAF[qFgp/&F&6$F_oFE,&#\"bs+++++++++++++vV)R )[4a;*)fEFx\"4b8=#zuo)\\552W1(R<\"pLw(z^KWJ@6!4ROv`V$R**4&)R&>FgblF(*( \"\\s+++++++++++vo/)\\HHVyzlg$oRv(4%*QheutzAN \\:T-Q\"F(FdblFgpFAF[qFgp/&F&6$FcoFE,&#\"cs+++++]iU&\\p7ci_5g,e^u7PNd* )=us3g:miHJj`wJHm)fHZy3l&R%)fr@l)*RQqRB4r\"f([/&GF^clFgp*(\"bs++++++]P %Q6n4,)3$Hj\"z(fIz5HiI%Rfg]b,-L]uawp)HefPRi@%Gb(*=!*\\m`se\"[D0(f$*HQO $F(F^clFgpFAF[qFgp/&F&6$FgoFE,&#\"[s+D^')3I,1-Db/6Z;l;EA_%Qv%G5H>\"*>U 6Y2n-eu,\"3dswL#f)=ul*Q#y&=kWk*)>1bel*\\&FhclFgp*(\"[s++&)[4b#>(\\eeV/ sAuq'*zLX$Qwn:iE3ac5_y*QZ='G[LC\"*31A*4SyGK&o$yt%GP^5!z`i')F(FhclFgpFA F[qFgp/&F&6$F[pFE,&#\"[s+]7Gq!ov:GO$4^u%p_!fc)ewd!Q+uBKwE&>#f,w= ]2#ok%>lSItsEHF$HT'*>$o%o:Bv!o " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Checking the interpolation scheme .. A" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "nodes and linking coef ficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24543 "ee := \{a[13,5] = 0, a[19,12] = 143537853295286536 6640508161506288132628507179/49806363935840879802043113407832743641971 00000-202677020669378957542507423185706165490917/461170036442971109278 1769759984513300182500*7^(1/2), a[19,13] = 403746694578349438308454175 287/6376985864283002448399049100000-2025930373551673081586588852/19928 0808258843826512470284375*7^(1/2), a[14,10] = 304484151498253253263085 93750/15127426330731345559308904251+5887803942383224482632816406250/74 1243890205835932406136308299*7^(1/2), a[14,3] = 0, c[14] = 1/2-1/14*7^ (1/2), a[12,3] = 0, a[4,2] = 0, c[13] = 1, a[11,3] = 0, a[12,2] = 0, a [11,2] = 0, c[12] = 1, a[19,14] = -13527697495128038175393365162508971 /236636338499968858572102540184443750+31048538567801978197319138691898 /946545353999875434288410160737775*7^(1/2), a[10,3] = 0, a[14,13] = 3/ 392-3/392*7^(1/2), a[10,2] = 0, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[ 14,2] = 0, a[6,3] = 0, a[9,3] = 0, a[14,5] = 0, a[6,2] = 0, a[7,2] = 0 , a[7,3] = 0, a[19,15] = 29563343742281752194031206400000/667926768960 321231615748121619903*7^(1/2)-255337671203169229981390407680000/667926 768960321231615748121619903, a[5,2] = 0, a[13,2] = 0, a[14,4] = 0, a[1 3,4] = 0, a[13,3] = 0, a[19,16] = 276519949255351403580207031250000000 0/12220217271375375959649133447445146063-60961648741172938426718750000 0000000/12220217271375375959649133447445146063*7^(1/2), a[17,1] = 2599 18413834103868653305362543252668375436100480694679123/5337146950192352 259892078131372117541294004208900000000000+762016105634989997580723884 12964731305853858164539/5883098490070935030745236035463092527881397937 5000000*7^(1/2), a[17,2] = 0, a[17,3] = 0, a[19,10] = 1238630740535785 456490596571627838413647856421875/473362591834532828836190389684692084 01637005206-4086346085795561461894163610670055194160612500/87659739228 6171905252204425342022377808092689*7^(1/2), a[20,1] = 4830447186603981 0753010257751589919886361140232907/13178140617758894468869328719437327 26245433138000000+68127962326089052629919488875228931247075214511/6589 0703088794472344346643597186636312271656900000*7^(1/2), a[20,12] = 150 611304496646214218585292683337607910401323/129127610204031910597889553 2795663724051100000-1797399149384124340751361777823955407632741/645638 05102015955298944776639783186202555000*7^(1/2), a[20,13] = -1283324543 406482830508432214/199280808258843826512470284375*7^(1/2)+193628450656 633331313082794159/6376985864283002448399049100000, a[20,14] = -571274 1970443114906032237142293823/157757558999979239048068360122962500+6555 893545855983544500111964137/315515117999958478096136720245925*7^(1/2), a[15,5] = 0, a[15,4] = 0, a[15,3] = 0, a[15,2] = 0, a[20,15] = -57913 908962654358753739683840000/222642256320107077205249373873301+62422949 15450106444957081600000/222642256320107077205249373873301*7^(1/2), a[1 7,12] = 37557085142885005839199344190346713535658856538249/17432227377 54430793071508969274146027468985000000000-1546464412557185756593109560 19446499029993293/4434327273490106819982470923062032019406250000*7^(1/ 2), a[20,16] = 3189081802658751841399101562500000000/12220217271375375 959649133447445146063-386161247481177249147656250000000000/12220217271 375375959649133447445146063*7^(1/2), a[17,13] = 8939687630550072408650 92535772471/318849293214150122419952455000000000-502392671239888112769 2249781201/622752525808886957851469638671875*7^(1/2), a[17,15] = 34256 025405285245733900192129792/242002452521855518701358015079675+78198974 27138839913759197790208/222642256320107077205249373873301*7^(1/2), a[1 7,14] = -44728197763861767267686691630511059789/9859847437498702440504 27250768515625000+51329695014236122164753612289240491/1971969487499740 488100854501537031250*7^(1/2), a[19,7] = 14494728132404625215729151448 13245178698300209565/2379216641841833257797401150284480158879812795098 -52587531770380550777475858487568858362192928292/118960832092091662889 8700575142240079439906397549*7^(1/2), a[14,11] = -14665848245525923683 57500/602474973863778980873559-946677979546641857718938375/59042547438 650340125608782*7^(1/2), a[18,15] = 696525438997433496880000000000000/ 6011340920642891084541733094579127*7^(1/2)-121389100692678148146704000 0000000/6011340920642891084541733094579127, c[17] = 37/100, a[16,2] = \+ 0, a[14,8] = 74200703416028798327128906250/428815117533247994795995103 67+28676647199217261041085964843750/2101194075912915174500376007983*7^ (1/2), a[16,4] = 0, a[16,3] = 0, a[16,1] = 318424113799300488598249341 1533410452435081234662965417047433163425756249010533264130587881476793 06059990010001/7717193457109162513896339989541939168625874356612620275 288295636317564198088241611239632333125000000000000000000-779036898575 1938923820674667096775602063482288285741296170327553630654431762486304 43242600895553442815357/4763699664882199082652061721939468622608564417 662111281042157800196027282770519513110884156250000000000000000*7^(1/2 ), a[16,5] = 0, a[18,2] = 0, a[18,3] = 0, a[18,4] = 0, a[18,5] = 0, a[ 16,6] = 38677873603874023494821224663560520533757129031705209294828410 503456708269950565640452627167440128229776168201/100080082118918185736 3609351052680617003947572378045335413864786223498306939613602026574791 43973437500000000000+2660388496179864594360815841760855766508730523055 2993954979509880880776475635566879209893579250983975560997/18533348540 5404047659927657602348262408138439329267654706271256708055242025854370 7456619984147656250000000000*7^(1/2), a[14,1] = 8941065567926479206438 689/198753096356125278622613280+152838094177334666489948287/3311226585 29304714185273724480*7^(1/2), a[16,7] = 377308587903340677792605542085 2886774791582751698542812584349024015469545595181670092987168923424495 0950501493/13892859740661471042270156150553691881345433978558714782072 8065583231442155198771083100889637109062500000000000-10233001453579507 7906358306091556548197835190067615199809734380171826687208346433432359 2817694003399143529013/69464298703307355211350780752768459406727169892 793573910364032791615721077599385541550444818554531250000000000*7^(1/2 ), c[16] = 3923/5000, a[14,9] = -1287199574154792351913181734948009779 65419044818935750521702127672316317131710725250664367373074947/9591164 8908768241459475623492128143238210539014575492278647719252118577198224 684430692759024543840-882185082068177622906787605005472555636444759904 636504531314529608120034066591927204665390933748491981/159788807082007 8902714863887378854866348587579982827701362271002740295496122423242615 34136534890037440*7^(1/2), a[16,9] = 140826789358890217295475288587920 0378495596410425522130381311818054028954090950082397504605548502598430 9922595887619652662278786898541530825637526247745153879591696545731133 99136914397/3724061476220048359904192212639840313935246560930005829468 6034773950283950518320293022388613265519828748009305183412567399661303 19527284854144491887040986350427874375000000000000000000+3658948724895 2850975626989635036036984043973038459711023487544456316770887185519697 4974526134564130489860871802571616243554346034341340988481532061923388 5431694117641997082749518637/26600439115857488285029944375998859385251 7611495000416390614534099644885360845144950159918666182284491057209322 7386611957118664513948060610103208490743561678877053125000000000000000 0*7^(1/2), a[19,17] = 0, a[19,18] = 0, a[19,2] = 0, a[19,3] = 0, a[16, 13] = 1667795459666994596006593734612718364094871113097496394713308473 411640245893712505210071/148176942057718746344268989730526651074011467 617513077874633459904607921875000000000000000+995832251155619761130552 6900282303722213821417596901636742244109953434503977764414401/18522117 7572148432930336237163158313842514334521891347343291824880759902343750 00000000000*7^(1/2), a[19,4] = 0, a[19,5] = 0, a[16,14] = 100917334612 4759287681689120442059502318375371382116951255511971601939848092903783 67269/9963892108307175216790232746003616121691269608751653567355450857 48281214843750000000000000-1403505192078119811089802093553041167324991 6957009159970154760483863951056688692206659/49819460541535876083951163 7300180806084563480437582678367772542874140607421875000000000*7^(1/2), c[19] = 7/10, a[19,6] = 134463117364912049288884412301220752886554060 845/342783561252437088255729892141683337347373287486-50114780017647293 6538357321490008588109143452/31739218634484915579234249272378086791423 452545*7^(1/2), a[18,6] = -1653016899538998154302935672422334307309203 1250/399914154794509936298351540831963893571935502067*7^(1/2)+21161950 5426273318600241507183021439574335170625/79982830958901987259670308166 3927787143871004134, a[17,6] = 282346020910207495266878857047379938762 003012480259/3242547201036567051067715195934842380312990557300000-3358 7902518633529765425646138218963307158947223/26806772495341989509488386 20977878951978332140625*7^(1/2), a[18,14] = -2039798587019518586061909 55719875/1363025309759820625375310631462396+29260691408512340069072795 640625/340756327439955156343827657865599*7^(1/2), a[18,1] = 1925496762 1004828660163073938975933503722889627/49503971711929064439578695711277 6119772197491840+2432585298063029726317162869389207640440757875/569295 674687184241055155000679692537738027115616*7^(1/2), a[17,7] = 18223033 119253981180923879156484227563883974739722227/166545164928928328045818 080519913611121586895656860000-100313230082181541388658065529598230833 22462822417/285963538682912651177572253639961557557669807103125*7^(1/2 ), c[20] = 9/10, a[17,8] = 1304777397928621174334360672724646486278075 2381075/15028547139305830778230444498740751854340622555424-49990440602 95488160350089639911998698434613211/2484878825943424401162441220029886 219302351613*7^(1/2), a[19,8] = 20478027115277330904684492472250426103 12438078125/134183456600944917662771825881613855842326987102-629967589 3634686965347686169110786579741787500/24848788259434244011624412200298 86219302351613*7^(1/2), a[17,10] = 15414026556992984430469892869330396 5519909065225/143288135906669396836900874715366252459009313056-2247178 508319186582684151266568513851628311/607482600336917467257244924006945 514766523*7^(1/2), a[18,13] = 305198437376024806501272473/612190642971 1682350463087136-15274180675827991876494250/57392872778547022035591441 9*7^(1/2), a[19,1] = 5040532884653950230124928275256995105972932590854 33/15248991286263863599691651803920335832268583454000000+2304658778756 8553137495508163715092754674894221/14119436376170244073788566485111422 066915355050000*7^(1/2), c[15] = 69/400, a[17,11] = -16173565171942076 79205444849356224376563538983069/1319671808250473426092343870769195831 358057800000+3182258795374348673491299173537610086635766763/1090998518 725589803317083226495697611903156250*7^(1/2), a[16,11] = -692801154945 4881761699481713711235043591502295679423604869246153519670533666503210 900114933332845376150551/220168689298819394894713630837330164657287230 7345365971075591622778810160246298613549949543750000000000000-33337318 4604500792272379654433507180957103057593155834003047471420894248443021 9726989269407196212710461/20385989749890684712473484336789830060859928 771716351584033255766470464446724987162499532812500000000000*7^(1/2), \+ a[18,10] = 2388471464522174677927145048979026999230849609375/110451271 428057660061777757593094819603819678814-398776824419288042810376670672 1197663574218750/326778909550466449886916442583120768058638103*7^(1/2) , a[19,9] = -373180627876116715225887951001890960081967891563719731987 39990193127060245888581157358617370318952573218254826689380705113/1875 7729223884169447254147047356156513603851477476172750154128794359117980 13696270080499616124091452322482915795694000000+1691188922097805503527 1100221021035910099550046093461456725580685439409809188693188004667115 9021110688326728803940450517/46894323059710423618135367618390391284009 6286936904318753853219858977949503424067520124904031022863080620728948 92350000*7^(1/2), a[15,6] = -191031498688788616226618012353233613/1695 7475840682462825521397145600000000+21956174445950394525012845066054021 1/11093675783624041100808390656000000000*7^(1/2), a[17,9] = -483472258 4846356480292651573086089355948598980865849611314355781325977701982869 12507816661196171113002225811104824321561163203/6565205228359459306538 9514665746547797613480171166604625539450780256912930479369452817486564 3432008312869020528492900000000000+16775323204311547045361525808533495 4467158064037840590577139604824423918685354990948949641283770046674847 5273250933123162009/58617903824638029522669209522987989105012035867113 0398442316524823722436879280084400156130038778578850775911186154375000 000*7^(1/2), a[15,7] = 14172386735756874627703502458542559611/90796880 820740515068437869813760000000+53924868594716008726045512055295307/907 7835462591889896660794982400000000*7^(1/2), a[15,1] = 3001645957025027 67677683752821381843/5685191595792547980976128000000000000-48497201742 052651581387984291872403/53061788227397114489110528000000000000*7^(1/2 ), a[20,7] = 8224414033808965946363932975223105619473504454675/1665451 6492892832804581808051991361112158689565686-33311544700218800416242036 708479037887482563294/118960832092091662889870057514224007943990639754 9*7^(1/2), a[15,8] = -877010740240762900922180472417/57186012045619550 819463987200+513139270223400882486686075445021/68051354334287265475162 144768000*7^(1/2), a[15,9] = 45550848707315089359912544356558140241875 5481619398305814237179937868420320279982259692853675276040350274866593 /740740898049617367935471008762629422134325227668913118716370672910772 04608105381835381523917447168000000000000-8088939903267911374096450504 7341248288012175146575113552873935146575048937602566575622159331449196 29/2695030776582292814675576398887868995199235474954188986106083154776 989963827197767483392000000000000*7^(1/2), a[16,12] = 6876230787902827 7671607672052145875138072977117549646151236433906220984773817872854208 42315969989227509309/7270805190498675407289247054348907579029951742748 2766274671992647047322514891352171003011718750000000000000+78621364964 6648763620304521931897747223418720777506087513122987291991254771636829 8063580928728400258823/10771563245183222825613699339776159376340669248 51596537402548039215515889109501513644489062500000000000000*7^(1/2), a [15,10] = -81487190136853529424335403395147/88158909075702233122126036 99200+41281728456741643538288815829781/9275782652527467499145003008000 *7^(1/2), a[18,17] = 0, a[9,1] = -147251426448621580388138470887726424 6346044433307094207829051978044531801133057155/12468948016200320011570 59621643986024803301558393487900440453636168046069686436608, a[8,7] = \+ 6070139212132283/92502016000000, a[8,6] = 1501408353528689/26569728000 0000, a[8,5] = -1024030607959889/168929280000000, c[6] = 93/200, a[9,5 ] = -12070679258469254807978936441733187949484571516120469966534514296 406891652614970375/272203115476165722171047818453110069949728408504838 9015085076961673446140398628096, a[15,11] = 14952684189753476847716767 3223553/8109821199993554522521600000000-101973319716227859779095727049 70449/1135374967999097633153024000000000*7^(1/2), a[9,4] = -5172294311 0856684583751756552469812300390253369336991141383152707723193724692800 00/1246193810048091458972786305712152983652570794102362529218509367490 76487132995191, a[9,8] = 103328481844520156040568367672866568591240077 96970668046446015775000000/1312703550036033648073834248740727914537972 028638950165249582733679393783, a[9,7] = 66411312295991164213478213583 9106469928140328160577035357155340392950009492511875/15178465598586248 136333023107295349175279765150089078301139943253016877823170816, a[9,6 ] = 780125155843893641323090552530431036567795592568497182701460674803 126770111481625/183110425412731972197889874507158786859226102980861859 505241443073629143100805376, a[10,7] = 3186072351736493124051512658496 60869927653414425413/6714716715558965303132938072935465423910912000000 , a[10,8] = 212083202434519082281842245535894/200224260447756725638228 65371173879, a[10,6] = 21127670214172802870128286992003940810655221489 /4679473877997892906145822697976708633673728000, a[10,5] = -1802692598 03172281163724663224981097/38100922558256871086579832832000000, a[10,4 ] = -20462749524591049105403365239069/454251913499893469596231268750, \+ a[10,1] = -29055573360337415088538618442231036441314060511/22674759891 089577691327962602370597632000000000, a[11,1] = -234265984581408683695 1207140065609179073838476242943917/13584809613510567770222314001391587 60857532162795520000, a[17,16] = 272818447134154078329357400195312500/ 12220217271375375959649133447445146063-4837549950042758520976608750000 00000/12220217271375375959649133447445146063*7^(1/2), a[10,9] = -26984 04929400842518721166485087129798562269848229517793703413951226714583/4 6954567491393431507700044208087114188467603590271755032561672817587500 0000, a[15,12] = -2271644391263030621748072222279/74102764220560004300 800000000000, a[11,6] = 2098082234509676029222408679497810531264453392 5634933539/3775889992007550803878727839115494641972212962174156800, a[ 18,7] = -814586114736897681347580051914111367091406250/703910249065631 1413601778551137515262958026021*7^(1/2)+720476286898479329575534558408 9235457889776021875/16654516492892832804581808051991361112158689565686 , a[15,13] = -288610253631/128000000000000-4643982156663/4096000000000 000*7^(1/2), a[11,5] = -26053085959256534152588089363841/4377552804565 683061011299942400, a[11,4] = -996286030132538159613930889652/16353068 885996164905464325675, a[5,4] = 3982992/2907025, a[12,1] = -2866556991 825663971778295329101033887534912787724034363/868226711619262703011213 925016143612030669233795338240, a[11,10] = -31155237437111730665923206 875/392862141594230515010338956291, a[11,9] = 300760669768102517834232 497565452434946672266195876496371874262392684852243925359864884962513/ 4655443337501346455585065336604505603760824779615521285751892810315680 492364106674524398280000, a[11,8] = 1610214261431241783890751219292467 10833125/10997207722131034650667041364346422894371443, a[11,7] = 89072 2993756379186418929622095833835264322635782294899/13921242001395112657 501941955594013822830119803764736, a[12,6] = 2346305388553404258656258 473446184419154740172519949575/256726716407895402892744978301151486254 183185289662464, a[12,5] = -4583493974484572912949314673356033540575/4 51957703655250747157313034270335135744, a[6,1] = 5611/114400, a[20,2] \+ = 0, a[15,14] = -32507875096641/1024000000000000*7^(1/2), a[12,4] = -1 6957088714171468676387054358954754000/14369041511965468332636822810157 0221, a[12,9] = -32058909627170725427914343121527275340081027740232102 40571361570757249056167015230160352087048674542196011/9475695496839658 1478301512445127360498465774712725761537244920597319265730601723910349 1074738324033259120, a[12,8] = 345685379554677052215495825476969226377 187500/74771167436930077221667203179551347546362089, a[12,7] = 1657121 559319846802171283690913610698586256573484808662625/134314804112551464 77259155104956093505361644432088109056, a[20,9] = 14997958426599779095 4827772549859769701497240064737195980122858008025991905757512636147356 3579004627500002497186668723541/65652052283594593065389514665746547797 6134801711666046255394507802569129304793694528174865643432008312869020 528492900000*7^(1/2)-6755812650382155716186433992193716663959424666280 6827122510503022547617419025194068959010701808469924021246390403506719 787/131304104567189186130779029331493095595226960342333209251078901560 51382586095873890563497312868640166257380410569858000000, a[13,1] = 44 901867737754616851973/1014046409980231013380680, a[16,15] = -881802833 6501383042724502163418043935242930059774292246092303626208484148248065 3958/11852361034459769519597506546144613989755568051785365407543288652 815101388446044921875+198068155659884577440583331314306224717883619889 6959594619834515105995312818424208/94818888275678156156780052369156911 918044544414282923260346309222520811107568359375*7^(1/2), a[12,11] = - 6122933601070769591613093993993358877250/10505170015102355131982467213 02027675953, a[20,11] = 2845094817811868725022379378375854841597087/12 21918340972660579715133213675181325331535*7^(1/2)-71186143325418293848 61030894253928312032669/977534672778128463772106570940145060265228, a[ 12,10] = 40279545832706233433100438588458933210937500/8896460842799482 846916972126377338947215101, c[3] = 341/3200, a[13,6] = 79163867519161 5279648100000/2235604725089973126411512319, a[18,11] = -47288194420795 5026637075845907843049717091875/26393436165009468521846877415383916627 161156+21164021072452475373967469896889463184140625/219945301375078904 3487239784615326385596763*7^(1/2), a[13,8] = -137345124323977414765625 00000/875132892924995907746928783, a[13,7] = 3847749490868980348119500 000/15517045062138271618141237517, a[13,12] = -306814272936976936753/1 299331183183744997286, c[2] = 1/20, a[13,11] = 28203554318319084006875 0/12295407629873040425991, a[13,10] = -9798363684577739445312500000/30 8722986341456031822630699, a[13,9] = 122747654703131968784288120377406 35050319234276006986398294443554969616342274215316330684448207141/4893 4514749371551765038583414351093488882928068660965448289652679652335305 2166757299452852166040, a[20,3] = 0, a[20,4] = 0, a[20,5] = 0, a[20,17 ] = 0, a[20,18] = 0, a[20,19] = 0, a[17,4] = 0, a[17,5] = 0, a[8,1] = \+ -1221101821869329/690812928000000, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = -3899844/2907025, a[18,16] = -143628337584575192260742187500 00000000/109981955442378383636842201027006314567*7^(1/2)+2503126771431 5614819946289062500000000/109981955442378383636842201027006314567, a[4 ,3] = 3069/25600, a[5,1] = 4202367/11628100, a[8,4] = -125/2, a[7,6] = 5611/283500, a[14,6] = 3037913416047823635649583750/15649233075629811 884880586233-302465625814318865951896498250/53676869449410254765140410 77919*7^(1/2), a[16,8] = 586899259341979493965640551217963416854383856 7661624521691517151343133691358404104159327002855091833886481/25073018 1931602953351962624259572198106967555130814454605513204002197920632408 8409943760373235100000000000+59180686976880893900015471525059171817264 93055771480188498447662556733258145721786535132299414153253117/4643151 5172519065435548634122142999649438436135336010112132074815221837154149 785369328895800650000000000*7^(1/2), a[19,11] = -829733047290227712452 45622953720437681491775/3770490880715638360263839630769130946737308+64 1633041156809452351541809340157183397438/17455976299609436853073331623 9311617904505*7^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 1838743 28794901398385760606250/760335208044775309288920638333-114787229090554 407592495836250/37256425194193990155157111278317*7^(1/2), c[7] = 31/20 0, c[8] = 943/1000, c[9] = 7067558016280/7837150160667, a[16,10] = 366 6849023310249950220210533670427622576720819211578637372330790696658489 01352651139887527426189000273823/8845076120169924130459422155107198560 00077549488758852211462407717486163006277913807113787900300000000000-5 5414577616820064135992874904494513927438264537594825743111652486874357 8561608227565491722318912765023/54599235309690889694193963920414805925 93071293140486742046064245169667672878258727204406098150000000000*7^(1 /2), a[18,9] = 1803094024666158973583658298805771831136877916832146240 91405760862055998989936441237796383108094390439124899145193625/1909877 8846136608891749676993671722995669376049793921345611476590620192852503 089295365087000536203878192553324465248*7^(1/2)-6937015052858969611040 4350490394824354889640009122679205943733731521976904923295074242505442 070216860013335461409030103/420173134615005395618492893860777905904726 2730954662696034524849936442427550679644980319140117964853202361731382 354560, a[18,12] = 2566273931876403497786160206125629704906909/1115662 5521628357075657657403354534575801504-57538239511595085977501529345628 118125/500117694173765334214526510819191974888*7^(1/2), a[3,2] = 11628 1/1024000, a[7,4] = 8602624/76559175, a[4,1] = 1023/25600, a[2,1] = 1/ 20, a[3,1] = -7161/1024000, a[20,8] = -2793368497133571893133216050429 3597603876493750/17394151781603970808137088540209203535116461291*7^(1/ 2)+178148132680281061166207127585231501733817765625/347883035632079416 16274177080418407070232922582, a[20,10] = -603981888231213233806510514 0914067839480368750/2045393915334401112255143659131385548218882941*7^( 1/2)+29380145942317975737545613572395823054173078125/40907878306688022 24510287318262771096437765882, c[18] = 1/2, c[5] = 39/100, a[14,12] = \+ 11352128098668146659861/254668911904014019468056-521584263992860792480 1/127334455952007009734028*7^(1/2), c[11] = 47/50, a[6,5] = 923521/510 6400, a[7,1] = 21173/343200, a[6,4] = 31744/135025, a[18,8] = -1038962 029777403408608910724198710303293457031250/156547366034435737273233796 861882831816048151619*7^(1/2)+1165644281065690221311154705689337540375 9951171875/939284196206614423639402781171296990896288909714, a[20,6] = -2222162697028947028783013162206764717253288398/222174530441394409054 639744906646607539964167815*7^(1/2)+3397618205750050296208140024041778 8844155222553/88869812176557763621855897962658643015985667126\}: " } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "inter polation coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67899 "dd := \{d[5,8] = 0, d[8,1] = 0, d[4,2] = 0, d[4,8] = 0, d[4,5] = 0, d[16,8] = 195398509993934353753639090112 1314432517977633691173970644071010498687479218135509177272659891654094 88398437500000000000000000000000000/5220608815327855202818977789052154 6498609401203765965426640093021818622721672730278217121262940173467542 3069645782218707205760970841-13802411549352279737458613894097751985459 3575749589020903424845317948740855371739683606579784329294980468750000 000000000000000000/370687887753449956425748428065622185984160274729816 1291962200849771657521613676807671429989905980761095539480726706500604 64613*7^(1/2), d[15,4] = -10396957121472031619939811038933607422018457 5915498032176538811922969285115091229170642176190595575066153716787995 88819456000000000000/9900702794006894886175314765204702965450936777758 6589947018422496551200744779693856096223600398840148100763990729603924 64949348617*7^(1/2)-12964436448449927496558576626329464761432978622131 8916300811089508201759440217768731050109343726862778830568500494464205 209600000000000000/753740503707744907684526713075034036759779816890766 7092666512504662442912700078093264605502698363700474911162614244746783 5659391021221, d[19,7] = 251044937580126631977814391869579609064028672 6193644809376491171482120279406971516939799443734595709919241034670377 81613959750000/8571461668773467460895684372688561057291393739998751335 7684524153879167917193127384594627394222377959266228233019940104071397 -100315899978668348505071735336390981477458047030310233568521737103646 47549409026857654577454529570721372068389891274644437500000/8571461668 7734674608956843726885610572913937399987513357684524153879167917193127 384594627394222377959266228233019940104071397*7^(1/2), d[13,4] = -8292 1370723874621610645227273153492018716137776797773310317459250199092262 0613848096489949680394079015283756098965180953355466590649251/55501073 4774569281388652275478483716102728444201953442284846219619670527461147 8177372733263808776127038795448036449965180403636770302-77366430492096 6689505533743937862056552715463883347205286416663709038320330572020238 4676542496101885260527793588270202185564220513/46706673856934694509644 3019362683954340041946159568322787237307071228847723323277766600178728 15357589803797457156501907618541238*7^(1/2), d[9,3] = -269654356531086 2311564939116196605041407266964362138241618491887343083698468206300258 4402343940323794242978822242172052227106940509325049668582383136599524 6539968701584009374112846597108713915489411773326678361337020015/54771 3463535138643401171986418999347288456504333584487483417299555990513416 5911025080981711546089266959823603463009222657389547091787963620635086 799342037835867854162987278578515620772967204789205199659174368380116- 8782384771999897316056079141210076290936297477840113392323998410382164 1115583691500011849870786931920776782454424207398179467079796728671640 7005790142566868618319637661848839887491464534593842398909956877219157 019181693/386376128080766282138435455636881061293704642730974274322541 1167519976556601822016606040359558056450279266868095101071200701973981 0547699889045340573619375603024319041073302146267803061475172915154071 508740924768427*7^(1/2), d[20,2] = -2437141551304952649192026089700912 5106947024489849889322832049800283061956029496422934255313523940133798 5450825389141391065625/57095199218055355101902547769479732220751219298 6929812478920793484155328815823512951426298939311189590420204819864703 6676729-14255685113169849281386173633668670203573842919167746349749061 9484168060569866556480676214298851265779243484230275765005515625/22838 0796872221420407610191077918928883004877194771924991568317393662131526 32940518057051957572447583616808192794588146706916*7^(1/2), d[4,4] = 0 , d[11,1] = 0, d[6,1] = 0, d[18,5] = 509683320977153495000573988388300 6500347655013185254244325253818444946699080714613777117723819561154421 89098499179510427035427650/1784793533627722012420270209204847562145956 8783287443257595781869270943622584369814714292543991759220089634536832 2905584668147*7^(1/2)+210221531134936005976622357117343791600977923262 5089605157123390460488071455174909588416997373741320333136729945013636 08349455702/1784793533627722012420270209204847562145956878328744325759 57818692709436225843698147142925439917592200896345368322905584668147, \+ d[7,6] = 2151108717168087381768841370743436454189397967056557248989028 9537815656190624898695166209700388882623950070035147676605604597632332 2145857273066182000000000/91301280289827564029925424805955788333683823 7125750016806173347018965981574473290509116165584161930092993104142279 25015016650738820542603882059615975967*7^(1/2)+39674173609127147425139 8999621607739009527678979899145101428623942067574210220564194300163715 418447164902636685982760910401701671728387693823008570800000000/913012 8028982756402992542480595578833368382371257500168061733470189659815744 7329050911616558416193009299310414227925015016650738820542603882059615 975967, d[11,4] = 1258632247730767236788396287851163004654506840380503 6692370936090048135139110405215484003940900350765717147689941898522628 034743842956172432537119125000/723453759267506313768387418477880084695 3508775279030261134908470879115484663267347544671383036611279023646476 6513247237590257011405769724458745341+27383348963066076394429690337535 0782012016004564806341468892522832700795757838719442534439114936865432 630026761127196671509392723886382548649517875000/314545112725002745116 6901819469043846501525554469143591797786291686571949853594498932465818 711570121314628902891880314677837261365468248889510667*7^(1/2), d[15,2 ] = -95430249245870271570731557368989297685825376108074335319790134588 894998979620572519687281691486433379280042558557798400000000000000/243 3854834536939867882484785027072352222479953794977911029259099313004266 427743257213537893602752333131489929482464641022818282509-765312255141 8969262767526977508422657561360358942188828894733746456476768648997309 695962490058672865586009714983424000000000000/319697206690784167592602 7564727534943153132738467066742452724417854990498394513670318583861293 51416410283715944104116776936593*7^(1/2), d[8,8] = -486968665809770805 8574659224504690631223273456682215833115280693442078888759245765773849 551522974860699046710660090397390347887140689713093750000000000000/223 8793399461234110196227003541701413226720497576105950612544237988452811 0538778097106634752729817164688270111337724440613924541758105573003499 0201971*7^(1/2)-199077812151469265281948249769407489368375394538793444 9746938481915337935958690783391584532670393089563277680246642487569393 41298307871173002734375000000000/5149224818760838453451322108145913250 4214571444250436864088517473734414654239189623345259931278579478783021 25607676621341202644604364281790804774645333, d[16,6] = 13395570428355 3730161296484560259292245148558507231637208547566144503292217312266498 5993901926759972148363242187500000000000000000000000/52206088153278552 0281897778905215464986094012037659654266400930218186227216727302782171 212629401734675423069645782218707205760970841-946226130364215871469503 7386792292535610701144881628939162680771406179849332268170520078868551 83936310527343750000000000000000000/3706878877534499564257484280656221 8598416027472981612919622008497716575216136768076714299899059807610955 3948072670650060464613*7^(1/2), d[20,1] = 0, d[4,3] = 0, d[3,3] = 0, d [2,3] = 0, d[5,7] = 0, d[9,5] = -1730113207784114464204040659129627488 0528898262183650702397457263684416905243306269960656955007636865429733 3430506135036332580170365810930609014412836123080967356296966837493587 6459294048689064577348441477257628446984568608/44433254729288122445920 0773982413220487760339140620415470922284264797304009209531909694641349 1764917821156898309366231880807270078212985487240214165966228194347796 689723429746820797352069644885242718223505206348369105-790335040321686 4662563316771035114134505837900455617836367307403237887424748381649041 1021339546874890653888560482019573540179952253418072708504351364109159 34363083651245392760317300579788728713068809901451603734258873900/3863 7612808076628213843545563688106129370464273097427432254111675199765566 0182201660604035955805645027926686809510107120070197398105476998890453 4057361937560302431904107330214626780306147517291515407150874092476842 7*7^(1/2), d[11,5] = -588365078178231646007854060948090885162693751699 3815699165594145109751953197427677226648812495404740363751722260622443 71663110241072491075264725000000/3145451127250027451166901819469043846 5015255544691435917977862916865719498535944989324658187115701213146289 02891880314677837261365468248889510667*7^(1/2)-25759662442418468490091 3056734201723274658543249307791037853096354403458772729102416681481282 72002737470315453312749062733246568740635068153622590400000/7234537592 6750631376838741847788008469535087752790302611349084708791154846632673 475446713830366112790236464766513247237590257011405769724458745341, d[ 17,4] = -9211470773400427862668590377375063625382975296268805132574456 3976527174090102375131761060349950180085979662839059095393082015625000 0000/65488128014915989192389920409370961263948791333952313890482723680 41760358255874176057166095974250510702775147164098867220057094791837*7 ^(1/2)-118965031906802054699885506874167396691856137438594967592822068 5642082886089881184186886242587657522112265414370354456678779362500000 0000/65488128014915989192389920409370961263948791333952313890482723680 41760358255874176057166095974250510702775147164098867220057094791837, \+ d[16,7] = -80479633425145694979535497072990347527394902872888625090381 0357350318045105404629507631861823856776630017187500000000000000000000 000000/522060881532785520281897778905215464986094012037659654266400930 218186227216727302782171212629401734675423069645782218707205760970841+ 5684859223897618546525624281784722268021553621501920992895615566215911 80398449424996333110914849056414843750000000000000000000000/3706878877 5344995642574842806562218598416027472981612919622008497716575216136768 0767142998990598076109553948072670650060464613*7^(1/2), d[16,3] = 9024 2573098694489902118330581195780256541215932972642868960778553350848914 210883132331230082422146548033007812500000000000000000/370687887753449 9564257484280656221859841602747298161291962200849771657521613676807671 42998990598076109553948072670650060464613*7^(1/2)-12775495252009857263 8045465482694120832088354060111253472605825889684271490394681993803573 581180972581177706640625000000000000000000000/522060881532785520281897 7789052154649860940120376596542664009302181862272167273027821712126294 01734675423069645782218707205760970841, d[10,5] = 47013700420612497405 1103158903998482181422171497184844177550470067529511603029052907544286 267915626682907561440628289229353576346909204303127281250000000000/181 6505900124627262640201696649812174169629188359339232374432176372532726 1133327662711539505263930738515410294330163515644365157739894353484984 10427849*7^(1/2)+68840898058460162718873118871189732943062401046853431 5986751052316230282776579725984682930576876991949726118943041279172185 96370050182306877634000000000000/1397312230865097894338616689730624749 3612532218148763325957170587481020970102559740547338080972254414242623 3033308950120341270444153033488346031571373, d[16,1] = 0, d[19,6] = 16 6972517897707163697547218010724005084059591136967141360302878080955415 10066136160183907920238703782067359620085428246873437500/8571461668773 4674608956843726885610572913937399987513357684524153879167917193127384 594627394222377959266228233019940104071397*7^(1/2)-4387528225629889178 9118612946294739832067939381712320693575518740978851571917550273760613 9952843120589990930475622519083225800000/85714616687734674608956843726 8856105729139373999875133576845241538791679171931273845946273942223779 59266228233019940104071397, d[7,4] = 859245651934450811084899343286497 8572914211436940293475190227730050731704992713782501329367518959847997 3114368746000085394361194627085479383144130610000000/91301280289827564 0299254248059557883336838237125750016806173347018965981574473290509116 16558416193009299310414227925015016650738820542603882059615975967*7^(1 /2)+171712456371207265467167296597685364852350680823702449070473874112 8893862853580658389126779723973576089959195536801230776807590702383056 19442133329970000000/9130128028982756402992542480595578833368382371257 5001680617334701896598157447329050911616558416193009299310414227925015 016650738820542603882059615975967, d[11,7] = -119687217542847681847754 1995728047627805488585352545398550253429984677800410242965033977884811 10122420363517558871177390848167316265090008864939525000000/5064176314 8725441963787119293451605928674561426953211827944359296153808392642871 4328126996812562789531655253365592730663131799079840388071211217387-41 1866392211820148693368327135585756474056232997955589408745656658792156 4747212272942634837558927353689900244499226698033188635729760457867370 00000000/3145451127250027451166901819469043846501525554469143591797786 2916865719498535944989324658187115701213146289028918803146778372613654 68248889510667*7^(1/2), d[3,2] = 0, d[5,5] = 0, d[5,4] = 0, d[2,4] = 0 , d[3,4] = 0, d[3,5] = 0, d[3,7] = 0, d[10,1] = 0, d[2,7] = 0, d[15,1] = 0, d[14,2] = 214382158733035474438957175414276691559614007389155767 4829883431834929066858157350198700881817074241476862904182773793087019 667/319697206690784167592602756472753494315313273846706674245272441785 499049839451367031858386129351416410283715944104116776936593*7^(1/2)-1 0844643232734393713292817403956749717060683816218118985209091913994321 99508427813244465808250363966178459545992615406748444000/3196972066907 8416759260275647275349431531327384670667424527244178549904983945136703 1858386129351416410283715944104116776936593, d[1,3] = 4213874270803753 2121246979803882440252551164988949607528569720573249805893290031999397 5740176455578712815549474220415029872005415543040719763484960111027/10 2752344188182942069322599560446133917511688778235816746261671588403589 3488182222230994465268083502804794731556625550929145374243885779990060 4189917364+81150418524196608412620003413618046134786447563429998688294 5701133737127907347999014386498467656739192804734051570418037753008238 2850787836831585606/16797227906501217962323735645042951548651735134812 2920237103332180225087598784069178503234688450849132437711716412981150 2387732336312632840150193411*7^(1/2), d[9,2] = 14726305284415201050094 3442431412250374747127089993578556840439460283525127524638988697181412 6737077695449770745137409375833975843507419189582352150574984397586932 401859688284240929974840607290798552784868588628002433/269486401451275 5237234074668783826059589919042587440448631498634713148426574941249594 4483763264561117902471617053887157459124491585386364351557343033038797 2819698825046718759520612401474979410044666584193485089928+13291818432 0148658365460982766992724804133422719375241575532733096573383966122423 2864389720330268967522980972391958143707112571818972611271402388007803 5868617026913265498360754013571907254497706173480056258753618529/61246 9094207444372098653333814505922634072509678963738325340598798442824221 5775567260109946196491163159652640239519808513437384451224173716263032 507508817564084064205607244534559372760794986591924240549851933862*7^( 1/2), d[10,7] = 415812444291202033577507639917262766158892950733998117 6845635068830689842971132282796661424601112119781849514762631806204372 862855277778102052468750000000000/127155413008723908384814118765486852 1918740431851537462662102523460772908279332936389807765368475151696078 7206031114460951055610417926047439488872994943+32910456270994581289583 0867483061940758517487445179585688895793458371217795950072336162373450 451428768765340095616264626457107935303851348136250000000000000/181650 5900124627262640201696649812174169629188359339232374432176372532726113 3327662711539505263930738515410294330163515644365157739894353484984104 27849*7^(1/2), d[1,8] = -315558082540469430895909422645449273814073607 7128360066170349405940374172024304797821426047585565847556641377835276 745462174813233606611295033155888146250/151559707677569839552250834351 6580475283297409478978297007359655928952942895068777790716836270423166 63707222904602268762048942700973152548533911801281119-1241178565075009 1073512780439774882169457948245327405587678471888500808055781896287373 864728823978609222029670117813108179951651930346571735684200000/167972 2790650121796232373564504295154865173513481229202371033321802250875987 8406917850323468845084913243771171641298115023877323363126328401501934 11*7^(1/2), d[16,4] = 535076890995029437268096097204550734635696854122 1509393404281888103482680864429323637776808226044594308252791015625000 00000000000000000/5220608815327855202818977789052154649860940120376596 5426640093021818622721672730278217121262940173467542306964578221870720 5760970841-37796355050458493631102809827579259334097684638765166756254 6923659436185831073022771514115561178040804338769531250000000000000000 /370687887753449956425748428065622185984160274729816129196220084977165 752161367680767142998990598076109553948072670650060464613*7^(1/2), d[1 9,1] = 0, d[2,1] = 0, d[7,1] = 0, d[13,1] = 0, d[16,5] = 8121013767932 2212652755250448027013526215919284718529896464888870924333520694191457 9567490864133389370449218750000000000000000000/37068788775344995642574 8428065622185984160274729816129196220084977165752161367680767142998990 598076109553948072670650060464613*7^(1/2)-1149678796506143627298355266 1200167737954297103479097471391813390195887384596133335486438941955921 36218076835937500000000000000000000000/5220608815327855202818977789052 1546498609401203765965426640093021818622721672730278217121262940173467 5423069645782218707205760970841, d[14,3] = -69537292130138629369023922 5272526307045271178664190744876827898620675962288203956612776865588439 092002702739230177018876641637163246/990070279400689488617531476520470 2965450936777758658994701842249655120074477969385609622360039884014810 076399072960392464949348617*7^(1/2)+3517583407959113267515523305597344 0579072320835698664652192058865185604043791920256044511025727863449276 7409983773502387900819672000/99007027940068948861753147652047029654509 3677775865899470184224965512007447796938560962236003988401481007639907 2960392464949348617, d[6,6] = 8162592265890581274904267140876836238713 7494266861392419450427131265108865578644029154881712938474354859943950 545138008226945582205110546522070202640000000/131541522761148302511191 2602577221958464883069039860624042748458072382214029609403382137397916 4204758075652253810560465309795576182203094819198747826469+44257061409 3563190694841211655127009120980446804634250385047831348465218998718873 8440494408272363186036804086774576263009557809175194439641764503560000 0000/13154152276114830251119126025772219584648830690398606240427484580 7238221402960940338213739791642047580756522538105604653097955761822030 94819198747826469*7^(1/2), d[9,7] = -160772631502208585650983829936857 4351397894637214925990923378503481718582681762547589463919745422824546 6464649552120059858620164446919952403977750765962269947068297157127663 66974024465553037284694221254682820428633803893100/6220655662100337142 4288108357537850868286447479686858165929119797071622561289334467357249 7888847088494961965763311272463313017810949817968213629983235271947208 6915365612801645549116292897502839339805512907288887716747-55324908593 1082256946776962556284850523363391454381561072169790859324601579202052 6506656641213841224185072335245003346515932899463224200185115505809412 305356424725993493254199982749602115856219837329783416738781282748000/ 3863761280807662821384354556368810612937046427309742743225411167519976 5566018220166060403595580564502792668680951010712007019739810547699889 0453405736193756030243190410733021462678030614751729151540715087409247 68427*7^(1/2), d[14,7] = 221591271125389424581121911567394697854994772 5855186972281633357636637201842311667852354187102328116657845599068755 034236015392000000/990070279400689488617531476520470296545093677775865 8994701842249655120074477969385609622360039884014810076399072960392464 949348617-438052354888582719562698261701047240821083499565976777048096 8047158187008612922660400815609630794712428619505097414253365445796456 000/990070279400689488617531476520470296545093677775865899470184224965 5120074477969385609622360039884014810076399072960392464949348617*7^(1/ 2), d[6,5] = -37983753935780826845491560567435699912337813884541364724 8758684546754409945250829352818740717874747201967186260699114771684352 05916546212854066374800000000/1315415227611483025111912602577221958464 8830690398606240427484580723822140296094033821373979164204758075652253 810560465309795576182203094819198747826469*7^(1/2)-7230416712426172469 1930066912817750380192743571742784794551465291383971580846818515186344 973622679103810295735104343879946877647708372919532508368614400000/131 5415227611483025111912602577221958464883069039860624042748458072382214 0296094033821373979164204758075652253810560465309795576182203094819198 747826469, d[2,8] = 0, d[5,2] = 0, d[5,3] = 0, d[14,4] = -147327172576 7622218282223438041412461091536412132115133311588519623701982172428530 038148835093719226576389233401247041408970391836000/990070279400689488 6175314765204702965450936777758658994701842249655120074477969385609622 360039884014810076399072960392464949348617+291243488782590462114448020 5285619960202415191887672718786681701459505685629467794725596578010663 098124551310380813803704380834095123/990070279400689488617531476520470 2965450936777758658994701842249655120074477969385609622360039884014810 076399072960392464949348617*7^(1/2), d[1,2] = -34526622656681398212098 5282899981269245058458792437036355181222634328552609814391281742925584 254890968273060306956735227745352027544581331833732294791/391513339604 3006607956364993423308405911194832197302585184822644396533159985969912 5983191869816220520832373768504573931879994239652078542680464154808-35 7407440204845554472881112140114408021040394498956705404353902548452587 9937195265997709738666590698990636816002511628009933721760047442122841 /774840642785698969122289831077296261637293290286244402176846580129553 9207354085383932468629387491137793909656957094474670005177233974217720 717*7^(1/2), d[1,5] = 730280225263209507765580583530815330217522135086 4237210485591115091790362450072914059927648370501747154506660339648300 0835191028731264643821938053800/16797227906501217962323735645042951548 6517351348122920237103332180225087598784069178503234688450849132437711 7164129811502387732336312632840150193411*7^(1/2)+138676489300620688342 1309272867642023659816748829651588360531520324793611684581929110473352 80418389043453765374932033685619411161630006418467158243649434172/7577 9853838784919776125417175829023764164870473948914850367982796447647144 7534388895358418135211583318536114523011343810244713504865762742669559 006405595, d[2,2] = 0, d[18,1] = 0, d[15,3] = 309538870169024301470563 3856567395447874515922155612719497055529881588346474260993859858599921 4945992746627898897849140019200000000000000/75374050370774490768452671 3075034036759779816890766709266651250466244291270007809326460550269836 37004749111626142447467835659391021221+2482377366245665316964026392734 4971146715600084186708084366057450366893551535014530056424942108070556 07760606178384294912000000000000/9900702794006894886175314765204702965 4509367777586589947018422496551200744779693856096223600398840148100763 99072960392464949348617*7^(1/2), d[12,1] = 0, d[14,1] = 0, d[2,6] = 0, d[5,6] = 0, d[5,1] = 0, d[19,3] = -1592434320753434931592281346468546 6907447851127071014387214356429033522763090853817696362854518576180383 59061918598632466078125/8571461668773467460895684372688561057291393739 9987513357684524153879167917193127384594627394222377959266228233019940 104071397*7^(1/2)+5091796679412412114278737399280824710898381458635953 1478620137903211951855418414175730080645271866282268124971252604810903 662500/857146166877346746089568437268856105729139373999875133576845241 53879167917193127384594627394222377959266228233019940104071397, d[20,6 ] = -68011840009188320025394891017762120454841491283830969423875193642 4232260024515482561518943089541903554601499199787917725791816587500/17 6818122458395629215082000187301782714444451046122293626596980534108063 780972383725927210518515282304257233230663900078841620401-121210628923 6359750558110384054977777309753288304350946337692919639143578015357004 25627842228284898678661529858166089302283880468750/1768181224583956292 1508200018730178271444445104612229362659698053410806378097238372592721 0518515282304257233230663900078841620401*7^(1/2), d[15,6] = -324562739 3591955096247548400353940087213319609143918412746471729898776061010704 32516872949892668598249686819652735732203520000000000000000/7537405037 0774490768452671307503403675977981689076670926665125046624429127000780 932646055026983637004749111626142447467835659391021221-260286276057029 9379110542370631288611189303360540254254317208065405545467610208577742 2915252909860847316669815138806067200000000000000/99007027940068948861 7531476520470296545093677775865899470184224965512007447796938560962236 0039884014810076399072960392464949348617*7^(1/2), d[14,5] = -625772611 9533156704909756747360746277870199010127944476405579664448471299756182 784405064551389990170123265077552459755577661337025800/990070279400689 4886175314765204702965450936777758658994701842249655120074477969385609 622360039884014810076399072960392464949348617*7^(1/2)+3165506290987968 5508064298456491406169566440031122352980412966971051701464295368002501 75145054417596954251063186896627242843165600000/9900702794006894886175 3147652047029654509367777586589947018422496551200744779693856096223600 39884014810076399072960392464949348617, d[12,3] = 81793288064001405035 0518020773261181499853542347766137189202482139155110545615505001277478 763564907623796961751224239072887644092271365112874456940/382258993481 4074517009340409360962220010170665236967566985909215747458647765894333 746038996390701657590889787189289978886713973532215119457016193*7^(1/2 )+16105597157527356608797264349063792957397470513682855346349967627106 2133884829652530090255062307578402569190331617510431175371811190260279 157345525/347508175892188592455394582669178383637288242294269778816900 8377952235134332631212496399087627910597809899806535718162624285430483 83192677910563, d[20,7] = 41005093718975289778836814119900709888082369 3889281728633739794265900726272290516959224636707468292399586615990636 550375134586625000/176818122458395629215082000187301782714444451046122 2936265969805341080637809723837259272105185152823042572332306639000788 41620401+7282248288849647423818750048139265893342513472500573180792592 2425847616811105599124817119652350853092996924788489181048401968750000 /176818122458395629215082000187301782714444451046122293626596980534108 063780972383725927210518515282304257233230663900078841620401*7^(1/2), \+ d[1,1] = 1, d[3,6] = 0, d[4,7] = 0, d[4,6] = 0, d[10,6] = -10103100080 4449468739520738617393533920252360035964385638772689683529625552202236 3340391186817130641794372274265723171823266527662376078799434731250000 000000/181650590012462726264020169664981217416962918835933923237443217 6372532726113332766271153950526393073851541029433016351564436515773989 435348498410427849-547783726198824397777037201117257022442448102930264 5103978710767297651493563716028972238016514230083186399077353183298808 62379241636585227543156250000000000/1816505900124627262640201696649812 1741696291883593392323744321763725327261133327662711539505263930738515 41029433016351564436515773989435348498410427849*7^(1/2), d[8,7] = 5828 5050131912126406709353348490172763725290684361195686227019945490541408 1455277770442517203693434228794599954407666266920827889552060277962934 3750000000000/36044573731325869174159254757021392752950200010975305804 8619622316140902579674327363416819518950056351481148792537363493884185 12230549972535633422517331+2005699005336285327497127182582905772654829 7187972594510969261863439867121513735387533884634513580436726314483466 386983345753276732134992128750000000000000/223879339946123411019622700 3541701413226720497576105950612544237988452811053877809710663475272981 71646882701113377244406139245417581055730034990201971*7^(1/2), d[12,4] = -684605801141385513385347864687884460954942747756571403232221753013 2466124428377211969348888663442360106913008869448126554216730415321581 869463218190/382258993481407451700934040936096222001017066523696756698 5909215747458647765894333746038996390701657590889787189289978886713973 532215119457016193-342575355539828849211109505827556367940910699371237 6686730135531344332033895307841084703163416505227062430432094368610994 856466429304421286052243470/382258993481407451700934040936096222001017 0665236967566985909215747458647765894333746038996390701657590889787189 289978886713973532215119457016193*7^(1/2), d[17,2] = -1749083901770424 9100159525714812151887627475867159173341855360067337533408349166015757 81774702831361806842464954124362375000000000/3020906990627308838441663 8024831726317999470130938456378259699183246658447645222070259965476878 955041229133037019041471056573139-880583752871865309316034990835128208 1895854029779995546328939123123089412319527210225730649032978512128091 309187620781250000000/274627908238846258040151254771197511981813364826 7132398023609016658787131604111097296360497898086821929921185183549224 641506649*7^(1/2), d[12,7] = 65101274498595179624055006508397379942285 6644362995051736727005635479601247294413932254794020420091445812054382 26470451847478382395009443841361326000/2675812954369852161906538286552 6735540071194656658772968901364510232210534361260336222272974734911603 136228510325029852206997814725505836199113351+515259386049443372267043 7538937924265346057301121017297317828917043468691398032685480792242816 884146530237629541886497718708096817362817205693840000/382258993481407 4517009340409360962220010170665236967566985909215747458647765894333746 038996390701657590889787189289978886713973532215119457016193*7^(1/2), \+ d[17,8] = -28504487591710923397038399865217159843956508784729598662931 7653633129626615600872741889573537127451580160105262561269495426250000 0000000/65488128014915989192389920409370961263948791333952313890482723 6804176035825587417605716609597425051070277514716409886722005709479183 7-33638299359705254815872536649901897552842162393759582986976547450330 2015550605939430622910793059779163293088010967113843750000000000000/65 4881280149159891923899204093709612639487913339523138904827236804176035 8255874176057166095974250510702775147164098867220057094791837*7^(1/2), d[12,8] = -2223592373365157587973339004172678190485217792037090356741 6415589231365933667448165989441705124500081466937606377029491283817957 30341131912874475000/3822589934814074517009340409360962220010170665236 9675669859092157474586477658943337460389963907016575908897871892899788 86713973532215119457016193-1251011119329888698111117061495326606347049 5557458106266686649249444532974742261693786944098973214928398782061861 24372126243512599934278342258000000/3822589934814074517009340409360962 2200101706652369675669859092157474586477658943337460389963907016575908 89787189289978886713973532215119457016193*7^(1/2), d[17,6] = -23060780 1552335810322868833020897876237613539083120310166995037492694852915188 5177530240477807145832543211828904839642766218750000000000/65488128014 9159891923899204093709612639487913339523138904827236804176035825587417 6057166095974250510702775147164098867220057094791837*7^(1/2)-232638016 7061552197717657181403505134473073459426969897643089432387120344376201 0616997769059520870257224345221992259076887455500000000000/65488128014 9159891923899204093709612639487913339523138904827236804176035825587417 6057166095974250510702775147164098867220057094791837, d[20,8] = -10058 8052013148262644607260455844694689171805732730313407816433798589755428 967916485597741040090477881490177601592195261933660937500/176818122458 3956292150820001873017827144444510461222936265969805341080637809723837 25927210518515282304257233230663900078841620401-1768075231568468319964 1293272926727330406519464682075018195265143815649070678254732343608396 805579067425652914274461763671093750000/176818122458395629215082000187 3017827144444510461222936265969805341080637809723837259272105185152823 04257233230663900078841620401*7^(1/2), d[12,5] = 736065468529743240339 5681898384222748095329548154901761463973685799223409676638193306845021 718324588125457006459478446445551482807819591972519362000/382258993481 4074517009340409360962220010170665236967566985909215747458647765894333 746038996390701657590889787189289978886713973532215119457016193*7^(1/2 )+10778008842118839789806251301506325555086046569852970034319930283981 3280962948719684185424140681812532835983374782072997318094656143153588 3818323072/29404537960108265515456464687392017077001312809515135130660 8401211342972905068794903541461260823204430068445168406921452824151810 170393804385861, d[8,2] = -2471639358420968741360873596016663862254554 4790491991034086084400006220361764728156448171638703388027372722233003 504210331991014276048306679687500000/722914333514557819172794408454164 2975965386346269191612943731596074954990648318672577943993260943900251 306180805878278476516691452131348446349259-981586054119171519660337068 8137727188591415082841115684551743806152881529697956124727393075059399 588320067976984998968561696002875212248906250000000/722914333514557819 1727944084541642975965386346269191612943731596074954990648318672577943 993260943900251306180805878278476516691452131348446349259*7^(1/2), d[8 ,3] = 1586124234059706409096223420780209907415744625136804971910262208 5182264931272400092946616400914240077881162345729492871888143053658400 5100819782421875000000/51492248187608384534513221081459132504214571444 2504368640885174737344146542391896233452599312785794787830212560767662 1341202644604364281790804774645333+31838860378840274206743061989665991 9867179698964695183095110702985781946083571408736137229709465589327732 0131186355049986706358866322088853911562500000000/22387933994612341101 9622700354170141322672049757610595061254423798845281105387780971066347 527298171646882701113377244406139245417581055730034990201971*7^(1/2), \+ d[10,4] = -43726887644062781540811485738789865262417943341027981084779 7975851755447481935186225863616156585927386131448133129778114765656047 710054763967097968750000000/181650590012462726264020169664981217416962 9188359339232374432176372532726113332766271153950526393073851541029433 016351564436515773989435348498410427849-218808459647003806575748777788 1822842805652460937914437874454853591617806228460301150180255772242665 01153891456039904834446426497851553075035257968750000000/1816505900124 6272626402016966498121741696291883593392323744321763725327261133327662 71153950526393073851541029433016351564436515773989435348498410427849*7 ^(1/2), d[17,5] = 1979198281787298725488101613331052490470495772562545 8429937830915253209516950544718648931755305224437902073770347478092695 93750000000000/6548812801491598919238992040937096126394879133395231389 0482723680417603582558741760571660959742505107027751471640988672200570 94791837*7^(1/2)+22359181545887551831462319487707094827367763626020747 4524844109581989396546655893511694249202876085011278230549677582946387 06500000000000/6548812801491598919238992040937096126394879133395231389 0482723680417603582558741760571660959742505107027751471640988672200570 94791837, d[18,7] = 20297584237676749329406254044151416959910735998312 0293055649727308681729566027986900258842686774658206717662387827601542 077891000/178479353362772201242027020920484756214595687832874432575957 818692709436225843698147142925439917592200896345368322905584668147+356 7877128795337168364196312555077512698884093839357223957122825251019683 44233850747881335628023549492445997911845923748588698000/1784793533627 7220124202702092048475621459568783287443257595781869270943622584369814 7142925439917592200896345368322905584668147*7^(1/2), d[13,7] = 4376089 9151494195174066878704552080964209969973173515341184163005591874780786 9029088973518182295147401302720403960493031415536805158000/27750536738 7284640694326137739241858051364222100976721142423109809835263730573908 8686366631904388063519397724018224982590201818385151+58182497414878959 2183237643069777493134087487664596222656748837192827045735383734916135 3369846295590975624032400403981594646268000/23353336928467347254822150 9681341977170020973079784161393618653535614423861661638883300089364076 78794901898728578250953809270619*7^(1/2), d[17,7] = 127106543127824078 5346120971010321792458874748922521535190380188528631949078687322549459 8935063734419610369058198179685323830000000000000/65488128014915989192 3899204093709612639487913339523138904827236804176035825587417605716609 5974250510702775147164098867220057094791837+13854752534184780030654768 1318035731763716608962946537927720996587569439577670513314807555739625 2707184185374222343503238750000000000000/65488128014915989192389920409 3709612639487913339523138904827236804176035825587417605716609597425051 0702775147164098867220057094791837*7^(1/2), d[3,1] = 0, d[9,1] = 0, d[ 18,2] = -5912034700498324467872734005922020507441411750635564776981464 2905697528786124985128848509627815563673602984974002137944229/57631616 5723052734160053669542073545205191281064530441977325127361908476947410 9533635019711321566476182516237796600005963-69844546427325205156872709 6320590248837682603410867204567800014095511556478009012751992241911776 409999449378126103018473619/230526466289221093664021467816829418082076 5124258121767909300509447633907789643813454007884528626590473006495118 6400023852*7^(1/2), d[7,3] = -4443556874769042450831206410459729378618 5912960795864005682142311283817881786724825154276381470942933737584010 302517957481751494388002036040172988825000000/913012802898275640299254 2480595578833368382371257500168061733470189659815744732905091161655841 6193009299310414227925015016650738820542603882059615975967-20515348226 2807413759759807703547187236822333040165273830091511393399988045265781 8403113077320064429171921791402884578719657184476976781356798837122000 0000/91301280289827564029925424805955788333683823712575001680617334701 8965981574473290509116165584161930092993104142279250150166507388205426 03882059615975967*7^(1/2), d[8,4] = -612926302095430866779818876513633 7780742753517391949508477793069152586718714921094775099309267735864733 18078548852057191067389178215649262438344843750000000/5149224818760838 4534513221081459132504214571444250436864088517473734414654239189623345 25993127857947878302125607676621341202644604364281790804774645333-1333 5090411977974961484005083057523261962629278053337151675291330605458005 6578722550534180834156305567251819089424324274018024423014758437889089 06250000000/2238793399461234110196227003541701413226720497576105950612 5442379884528110538778097106634752729817164688270111337724440613924541 7581055730034990201971*7^(1/2), d[16,2] = 3938661065322760533777541868 7523877304681748113670260664815043928900244999946710525301045762213969 9754101562500000000000000000000/16857531128960751728563976198947833801 0944496766979771470309319066868877657246699209587397923536999798321892 74622435942626683489-2782161490314464763833280811113944771938142757256 1137721370976674036788078177026061878813202600407626953125000000000000 0000/11969643441940326017170345444335373631184741991340247641067521875 978099136600073646780425554283253450532918340039092320077*7^(1/2), d[1 7,3] = 332008941727435812521342153646660651862104281144722950507480792 5285832675850584335260906065882780964516981035235628300315714750000000 000/654881280149159891923899204093709612639487913339523138904827236804 1760358255874176057166095974250510702775147164098867220057094791837+41 4968048870843552092706961151127487083676604470158466124314193025877090 0840278641601193762511598860012266004724192792197812500000000/12356250 5688520734325264000772398040120658096856513799793363629585693591665205 173133154077282533028503825946172907525796604850845129*7^(1/2), d[6,8] = 1147457518039621551255824818977190128144087495374027109532121924022 1078639525850680273720863149232774083290557059491215099556682274497692 598064415000000000/131541522761148302511191260257722195846488306903986 0624042748458072382214029609403382137397916420475807565225381056046530 9795576182203094819198747826469+64556891416829325595492504161155319441 0790759908791577715524829159218795349906063329726930501932077850004605 5904439398984312251861818996684573200000000000/13154152276114830251119 1260257722195846488306903986062404274845807238221402960940338213739791 64204758075652253810560465309795576182203094819198747826469*7^(1/2), d [9,6] = 39063332725217265717071490143611295954614729115106886733209602 4585059584864020056016326828734054000201163377320437416095541808172011 3187584016321081644077505836717591541152616139475501436583990476139792 19224584416958221460/8886650945857624489184015479648264409755206782812 4083094184456852959460801841906381938928269835298356423137966187324637 6161454015642597097448042833193245638869559337944685949364159470413928 977048543644701041269673821+920864916948971501109575170465865475326490 5614233761875382396154224998872242994678229261867564101464981448963983 8474563202345125792497043206777834104869843781697837622134127335384302 09563365640188716034976391938816258300/3863761280807662821384354556368 8106129370464273097427432254111675199765566018220166060403595580564502 7926686809510107120070197398105476998890453405736193756030243190410733 02146267803061475172915154071508740924768427*7^(1/2), d[13,5] = 831156 6636032892598082197081966910014320872331189118452552901897702519972166 300222644759947729568743126576641058043415508315809900/233533369284673 4725482215096813419771700209730797841613936186535356144238616616388833 0008936407678794901898728578250953809270619*7^(1/2)+792933684811743045 8815830336139833751577210010662997799890138299729019110773540943233164 66587444276204111497462281081616679858486737604/2775053673872846406943 2613773924185805136422210097672114242310980983526373057390886863666319 04388063519397724018224982590201818385151, d[15,5] = 27855693161625913 0703679359583669620129955511285396960929986580417832755735840570849485 837604183798299224759863025182556160000000000000000/753740503707744907 6845267130750340367597798168907667092666512504662442912700078093264605 5026983637004749111626142447467835659391021221+22339146675745678541633 4650644077607120644135325165528927872023104780315224181156562052284425 07653266682317225165342617600000000000000/9900702794006894886175314765 2047029654509367777586589947018422496551200744779693856096223600398840 14810076399072960392464949348617*7^(1/2), d[4,1] = 0, d[3,8] = 0, d[19 ,8] = -582986430347992097911710859056207410310312161050982104244163220 72742177632237400380577658856590526947451093362815756807028125000/8571 4616687734674608956843726885610572913937399987513357684524153879167917 193127384594627394222377959266228233019940104071397+243559476482504379 9825685342102338620810810873899076449329797603446951342270204059861728 140813479442444278566214005004687500000/857146166877346746089568437268 8561057291393739998751335768452415387916791719312738459462739422237795 9266228233019940104071397*7^(1/2), d[18,3] = 1648600132090977282526501 9864052422706442598727037013033733537159019665916899003914719364978689 253867887112758933844599050717104/178479353362772201242027020920484756 2145956878328744325759578186927094362258436981471429254399175922008963 45368322905584668147+1132743661440127252865165909038201446251082009481 6120656098557373277301308820731122678921680089892835747342703821918861 9947361911/35695870672554440248405404184096951242919137566574886515191 5637385418872451687396294285850879835184401792690736645811169336294*7^ (1/2), d[8,6] = -14161665980926452803750393133517619102224065541199277 5125459861525068870789025594846342064581049267165852603512722425468205 7967445763638566385856250000000000/51492248187608384534513221081459132 5042145714442504368640885174737344146542391896233452599312785794787830 2125607676621341202644604364281790804774645333-33384200623943001178124 7880237911802439562719057594945472704077792415624250158255677569420121 09098699078083037059970401121014107150159855466468750000000000/2238793 3994612341101962270035417014132267204975761059506125442379884528110538 7780971066347527298171646882701113377244406139245417581055730034990201 971*7^(1/2), d[17,1] = 0, d[2,5] = 0, d[7,5] = -1846195422833748992131 8663265626420160802986139333665646532211620339724753470146722383476511 8816177325945218969476347783698687223240747265935403006000000000/91301 2802898275640299254248059557883336838237125750016806173347018965981574 4732905091161655841619300929931041422792501501665073882054260388205961 5975967*7^(1/2)-270333434216489366239362995789451067435259467150343402 6005916783494109613653516913231466470912308995921359054252418624940312 6820163997530206000292736000000/70231754069098126176865711389196760256 6798643942884628312441036168435370441902531160858588910893792379225464 7248301924232050056832349431067850739690459, d[12,6] = -15817821247389 3002664759839735869048307364155339506409214276794582464561922773640658 11448951198787801960879566494334415203137390380235046630632171600/3822 5899348140745170093404093609622200101706652369675669859092157474586477 6589433374603899639070165759088978718928997888671397353221511945701619 3-85763231030571426752502147312731159389039043426294851936745018780996 9623598243020347105024294650499837180650688340215306397714203608242860 8312314000/38225899348140745170093404093609622200101706652369675669859 0921574745864776589433374603899639070165759088978718928997888671397353 2215119457016193*7^(1/2), d[1,6] = -2696059269776327206084263156125380 1620798758267391811759935890785632971468151803651018578207043144014525 781306920514921820002789333771676890104946885356240/151559707677569839 5522508343516580475283297409478978297007359655928952942895068777790716 83627042316663707222904602268762048942700973152548533911801281119-8508 9158986644615193355263069018586582904428305112255804832220556118448013 9527871433308931807136338369170330688024860591107499104322072883268067 98600/1679722790650121796232373564504295154865173513481229202371033321 8022508759878406917850323468845084913243771171641298115023877323363126 32840150193411*7^(1/2), d[9,8] = 5491333311176146435762383876397485099 7328362760041893298923587697605989569412874021195949651548316788146956 2574172850696928656695333537754072313391477759945159947269551224168530 97197045843387736173492959471872595661778750/8886650945857624489184015 4796482644097552067828124083094184456852959460801841906381938928269835 2983564231379661873246376161454015642597097448042833193245638869559337 944685949364159470413928977048543644701041269673821+134324725953166104 4857304111560614308867168451820141330207764657219339488758168404640732 4686935522370205786546394920923177698477112368717081368048121189687332 55100758576818467733646684807200561840812005685408434465100000/3863761 2808076628213843545563688106129370464273097427432254111675199765566018 2201660604035955805645027926686809510107120070197398105476998890453405 73619375603024319041073302146267803061475172915154071508740924768427*7 ^(1/2), d[11,3] = -325707528469545807806170992670687460523990132573907 1792690826354163800163591657325343015058638708151782063177712234125474 120434700299898965310311562500/723453759267506313768387418477880084695 3508775279030261134908470879115484663267347544671383036611279023646476 6513247237590257011405769724458745341-65380480926998113445490508799317 8183109676233638353406518551024493169666845312628477142902351718833149 43437270535759867183264854301545874673160750000/3145451127250027451166 9018194690438465015255544691435917977862916865719498535944989324658187 11570121314628902891880314677837261365468248889510667*7^(1/2), d[6,2] \+ = 32766246554655149990288207623619736902717015147048170067486812143127 8410739370058142491907001854659429840142498164244380695225345307244344 90928500000/4247522450229206707068076471882275690092941551357359372413 5375958939010430740721475738234941923228900111893357262296055118975673 0349804476063119501+13012776542135754112180426231437067007756777097637 9454146846627867172374978907766692117732064525575901702499032628752283 19163439616755569233302000000/4247522450229206707068076471882275690092 9415513573593724135375958939010430740721475738234941923228900111893357 2622960551189756730349804476063119501*7^(1/2), d[10,8] = -142024466801 5206351384502571342190610698550324609220580352800877902795431425211987 0690238906121544298667798603036423187084246574732066936707304687500000 0000/18165059001246272626402016966498121741696291883593392323744321763 7253272611333276627115395052639307385154102943301635156443651577398943 5348498410427849-79904117910204467207891004842203730468394823946241547 8550097835848742850956252400165340587393047019052653937538296467987660 89916666288208031250000000000000/1816505900124627262640201696649812174 1696291883593392323744321763725327261133327662711539505263930738515410 29433016351564436515773989435348498410427849*7^(1/2), d[19,4] = -19682 6548991650437242542262461615294664842541145911546791968576707071459227 370613802318648911504395238812735248641634201808175000/857146166877346 7460895684372688561057291393739998751335768452415387916791719312738459 4627394222377959266228233019940104071397+13339205857064209657992406748 9007079599155014209412414333594964038001055340534724335168930257412034 82897169276671730570911390625/1714292333754693492179136874537712211458 2787479997502671536904830775833583438625476918925478844475591853245646 6039880208142794*7^(1/2), d[13,3] = 1141292339734172012166041107011526 7504930966378426873005008617355445586670369074294845276232112092790772 9132485770322236685645140655590/27750536738728464069432613773924185805 1364222100976721142423109809835263730573908868636663190438806351939772 4018224982590201818385151+92360040402671143063122892980378734219969770 1777603448500801894253723034930402845279880050908849376015497763224251 615485426652013/233533369284673472548221509681341977170020973079784161 3936186535356144238616616388833000893640767879490189872857825095380927 0619*7^(1/2), d[7,7] = -1292370802283018277405743021465848627032102105 0998450871387107692290929845159049815460445597804439261716248622654732 1428903285431628815347910135920000000000/91301280289827564029925424805 9557883336838237125750016806173347018965981574473290509116165584161930 09299310414227925015016650738820542603882059615975967*7^(1/2)-16328666 4214832968427608815220808436506686649492041163460709373907162938380456 7277021164937398034410400515952686591879890431037132214876681277077138 000000000/639108962028792948209477973641690518335786765988025011764321 3429132761871021313033563813159089133510650951728995954751051165551717 43798227174417311831769, d[20,5] = 58785477955290480228019526274117504 7283054750246131963742599896431075402868587223539394837561497330299593 054067533345877754525106250/176818122458395629215082000187301782714444 4510461222936265969805341080637809723837259272105185152823042572332306 63900078841620401+1040293810032153470946266269048260902993901370046610 0116949120823746011837685419890027756425508750235808314826915563275158 0164843750/17681812245839562921508200018730178271444445104612229362659 6980534108063780972383725927210518515282304257233230663900078841620401 *7^(1/2), d[7,2] = 122507627289048641947100243988443731883035716908511 2229107098869298704502345467558092421182268692989283529716893887294609 9442523769754760499227500000/22678082621039790169803904352480467647221 3711757849665249908307071082492312280839278264906490649937802068347290 784394853952341271347135234197124211+632484312400165298216823947811501 1458373917100516605971276804369670015750224022701673198193871344276911 5214823334416471917126266305351488464690000000/29481507407351727220745 0756582246079413877825285204564824880799192407240005965091061744378437 8449191426888514780197133101380436527512758044562614743*7^(1/2), d[13, 8] = -9141006833869224588315095647107193862305946586668790904224523017 5009278363443137132577478243566789260376443130423677085792324595475000 /277505367387284640694326137739241858051364222100976721142423109809835 2637305739088686366631904388063519397724018224982590201818385151-14126 2737151597615430668619802623050272805600935498396460363842863464135017 3619430632300927493571029994844396944726140850889100000/23353336928467 3472548221509681341977170020973079784161393618653535614423861661638883 30008936407678794901898728578250953809270619*7^(1/2), d[6,7] = -265893 2740303209102589635768245951136016130235628524389304968966507718544114 4717376975370768966488324766577127009970609387323362799244860271309136 000000000/131541522761148302511191260257722195846488306903986062404274 8458072382214029609403382137397916420475807565225381056046530979557618 2203094819198747826469*7^(1/2)-335947126605415679480598702942334468326 8140167654040959848386091331707957182441379742566325045498351382522220 08159633118252068332576383788851733020400000000/9207906593280381175783 3882180405537092541814832790243682992392065066754982072658236749617854 149433306529565776673923257168569033275421663734391234785283, d[19,5] \+ = -1433046576500324291727908566371343476297679816375344615015904302071 2463201528340587793456664492394680311192714307500577292187500/85714616 6877346746089568437268856105729139373999875133576845241538791679171931 27384594627394222377959266228233019940104071397*7^(1/2)+39743786391643 1507921665408289346112123154258835907749235530037281825985188654489397 632669420451571093469036756742124232661962500/857146166877346746089568 4372688561057291393739998751335768452415387916791719312738459462739422 2377959266228233019940104071397, d[1,7] = 2037065375027368807025271943 2011929526687727691365952933666962265575023386848673317598678819769110 36412936346321230035983714578513768849836145638690060100/2165138681108 1405650321547764522578218332820135413975671533709370413613469929553968 438811946720330951958174700657466966006991814424736078361987400183017+ 5112096092006325468958656605425185500036219228080713836499947782851212 3985981686761001423585713337656035530371142833591324630186076256759439 937416000/167972279065012179623237356450429515486517351348122920237103 3321802250875987840691785032346884508491324377117164129811502387732336 312632840150193411*7^(1/2), d[14,6] = -3688313865681068397757598697972 2545324073793690928451048285040466129629099556724777811513317414320218 33697311432126211539306023200000/9900702794006894886175314765204702965 4509367777586589947018422496551200744779693856096223600398840148100763 99072960392464949348617+7291237449130203878877762139194289422291772990 0618890755721398838829912335877189839538429370231752286039706943523027 35037721592282600/9900702794006894886175314765204702965450936777758658 9947018422496551200744779693856096223600398840148100763990729603924649 49348617*7^(1/2), d[13,6] = -82513151875964588993809896845883696955673 4837613182220627066402275652598384129940345657141805307572721378853571 722640948275822161193500/277505367387284640694326137739241858051364222 1009767211424231098098352637305739088686366631904388063519397724018224 982590201818385151-968428543531483625328606192541434711499279959590284 8524627738059526100983462763368210611158967876373760938418285943680673 176650300/233533369284673472548221509681341977170020973079784161393618 65353561442386166163888330008936407678794901898728578250953809270619*7 ^(1/2), d[15,7] = 1563781415259811620684144966926318425000096355109776 9111467296477045145829508553775562648757608718882855070448267948032000 000000000000/990070279400689488617531476520470296545093677775865899470 1842249655120074477969385609622360039884014810076399072960392464949348 617*7^(1/2)+1949949830564707149071768871455493524765846931862335535308 3767033348550077217639477866906691184038288522603605186038661120000000 0000000000/75374050370774490768452671307503403675977981689076670926665 1250466244291270007809326460550269836370047491116261424474678356593910 21221, d[10,2] = -3119678890638239310012160520858034913209835904825256 2458722473581631680672936448446057852147071959414840371338255688715643 136806608601639882812500000/451197078002227354560565949733806306100052 7049032504545176521872673002347541916025879859886006088157267791437673 446030557695451714715796065292117-161063274321616334424282915519625320 5776544880591779891318260035611340301665744366825424744221586923221318 14634682795903271961350348277277968750000000/5865562014028955609287357 3465394819793006851637422559087294784344749030518044908336438178518079 146044481288689754798397250040872291305348848797521*7^(1/2), d[13,2] = -12476503768645101289658202040732429734578239270094672764970228183592 9824119362356220396914903741991057842871446404823016889335239/25602133 6901218860052980296184887060379609307096014651649274260260108277614549 02724718881387418645036920770761713095423443736994-8135550307160072092 8657320305175578691144586926203264977172893795712329067590647837462357 0807815776436821380439371075882711/21545358195492586830906621799803672 5361325355844124457539215393767605784458801325642047659977098562109592 530120703661761386*7^(1/2), d[9,4] = 338137392973937475185729389431818 6540714660120187334941000847271459375486493183596150892865907430694671 3533479753374209171175888188130458283600379234256093257777443219406092 6017404485049563918113164951290655851522853174503/17773301891715248978 3680309592965288195104135656248166188368913705918921603683812763877856 5396705967128462759323746492752322908031285194194896085666386491277739 118675889371898728318940827857954097087289402082539347642+735666374825 5870253107574867042287692516679854014047419166639033502076342840095353 8952658242234186252800282851415222994359560172056936619230038777317750 91541621460958549736380197540267383395148992252404297705904153910193/7 7275225616153256427687091127376212258740928546194854864508223350399531 1320364403321208071911611290055853373619020214240140394796210953997780 9068114723875120604863808214660429253560612295034583030814301748184953 6854*7^(1/2), d[12,2] = -976859036304749167634648894456031167117954082 1112133206647277177374614578196954917730159057202141616626512200358501 27721472213831479625871785/1898965931099374569114693059988505736510789 0348099799884181497704888305912691323252513252688871013234529267665627 364232156295121584190242138-229242875269874892324026686661447283885584 9971261956645727920019531720702177901357500225358450589123005012419712 79335206379701261768061476330/1122116232013266790840500444538662480665 4662478422609022470885007433998948408509194666922043423780547676385438 779806137183265299117930597627*7^(1/2), d[6,3] = -91422050365506498669 2185420173847439212009889923028518119879144639881739178180652685314693 8153228423910546087772207539535091311174943330104078559435000000/13154 1522761148302511191260257722195846488306903986062404274845807238221402 9609403382137397916420475807565225381056046530979557618220309481919874 7826469-42208421128995358842156208592494534485204404559920822814891166 5752889396036357668954665980202365137178014898649883641509363244848609 8620541583170876000000/13154152276114830251119126025772219584648830690 3986062404274845807238221402960940338213739791642047580756522538105604 65309795576182203094819198747826469*7^(1/2), d[11,2] = 507546338056293 6673306904935589653169772101650739549311989045477829289708150119540454 20445330484300575640078033630308292882163258363565838406250/1015677331 2829046631040401109073731300660420273399669320280881822747172817506521 0337190927014484488401776902802540615282309317748893030091043+20156678 8276746271987271579228149331675537257467093229677082267012806961207565 557162113491185766389277801850070194015195858242837743145405875000/101 5677331282904663104040110907373130066042027339966932028088182274717281 75065210337190927014484488401776902802540615282309317748893030091043*7 ^(1/2), d[18,6] = -593861419646616270181977900067188203635589684385714 1235472019752316267003811522307301927906077959523263045545886062796742 93597050/1784793533627722012420270209204847562145956878328744325759578 18692709436225843698147142925439917592200896345368322905584668147*7^(1 /2)-293169309397321889731944270635417780999693557702329121031086847433 460867749942907967397050459644916254775493491143831425988008700/178479 3533627722012420270209204847562145956878328744325759578186927094362258 43698147142925439917592200896345368322905584668147, d[20,4] = -2801654 3119366658958826284306348615895267093288507180448920884916275781506511 3259550361197011834959146313257916899640088743553603125/17681812245839 5629215082000187301782714444451046122293626596980534108063780972383725 927210518515282304257233230663900078841620401-193667023967038570529407 1807289942364254336997921541416806614052647627416283178748325382154834 04427710081957648717062052700191515625/7072724898335825168603280007492 0713085777780418448917450638792213643225512388953490370884207406112921 7028932922655600315366481604*7^(1/2), d[19,2] = 4718167869032807360918 8668930778287418848168761395721793299744150543036473297146709363320005 28729809970591986906711265625/2659920764875628003815632320963416362485 4982202978328091261159724395775859111274770623167898407229890073151866 75353848906*7^(1/2)-65982720444966500804870923793913193304821923219818 3691715556289563657396175213661510184793099756940430884839659808908657 5000/10240694944771167814690184435709152995569168148146656315135546493 8923737057578407866899196408867835076781634687001123182881, d[8,5] = 2 8652088987871969730461484093398674485225567996576007701775493863924161 3251767101169156353515809823323493612271889996256380919645069107795028 43750000000000/2238793399461234110196227003541701413226720497576105950 6125442379884528110538778097106634752729817164688270111337724440613924 5417581055730034990201971*7^(1/2)+125443906848281132789256985493180093 3806073125786780629631174239438193931241474520151950311661207508633275 253528493006418641801407161213398088322000000000000/514922481876083845 3451322108145913250421457144425043686408851747373441465423918962334525 993127857947878302125607676621341202644604364281790804774645333, d[10, 3] = 11315597965879973974644533121777215900726160873880109633888855374 9312442222969467857998443623249332012729005478902442853841149710994275 042928758984375000000/181650590012462726264020169664981217416962918835 9339232374432176372532726113332766271153950526393073851541029433016351 564436515773989435348498410427849+522427053824280208745285749713098908 0443789586863775531930360165218524545609239151469088413996475381591039 1576364819164592640745000306658367547187500000000/18165059001246272626 4020169664981217416962918835933923237443217637253272611333276627115395 0526393073851541029433016351564436515773989435348498410427849*7^(1/2), d[20,3] = 43622581585067442305577813217385009636834707044899007307457 4509785443670123799063695239094666358884339884792179503974368371390625 /667238197956209921566347170518119934771488494513669032553196152958898 3538904618253808573981830765369971971065308071701088363034*7^(1/2)+705 1375022627666027753839535396466798074963133495973198382538619125336283 4881692846076916400546946903261336086721333143319830737500/17681812245 8395629215082000187301782714444451046122293626596980534108063780972383 725927210518515282304257233230663900078841620401, d[18,8] = -549965585 5061989644464185782389657418859233767257081017458026707461142199119291 0284753845222261665164711045525020601300886512500/17847935336277220124 2027020920484756214595687832874432575957818692709436225843698147142925 439917592200896345368322905584668147-866253790105137284737836853228784 0099220608912433482861847389785210565408266215677638345337996707422720 4435858497192550948850000/17847935336277220124202702092048475621459568 7832874432575957818692709436225843698147142925439917592200896345368322 905584668147*7^(1/2), d[11,8] = 40880241772120074452319139480276259105 7673892560237852548270444566162545051583806471623274014663760902270923 1347986993404683889077141021865585312500000/72345375926750631376838741 8477880084695350877527903026113490847087911548466326734754467138303661 12790236464766513247237590257011405769724458745341+9999806898924295571 3165900344354603506565236821337160696942111686873063871805250436269290 432419169745610787956901446499763439953269976032525000000000/314545112 7250027451166901819469043846501525554469143591797786291686571949853594 498932465818711570121314628902891880314677837261365468248889510667*7^( 1/2), d[1,4] = -339882820903098897021856353785710196660904219969461756 3149743302558653084599078745969102563427995054187029435525124912699177 2272823509966390500216303/16797227906501217962323735645042951548651735 1348122920237103332180225087598784069178503234688450849132437711716412 9811502387732336312632840150193411*7^(1/2)-169023264501229588249421376 0387529082357569218631323265960376566769786443613461061130235197379353 5801296627666611080451180220999384173194579769504617980019/15155970767 7569839552250834351658047528329740947897829700735965592895294289506877 7790716836270423166637072229046022687620489427009731525485339118012811 19, d[14,8] = 10635582420262279381257355975097882620230201405539935477 7274736488434143316859234776091394396639267564176834986729816747953443 2200000/99007027940068948861753147652047029654509367777586589947018422 49655120074477969385609622360039884014810076399072960392464949348617*7 ^(1/2)-538006975961628363438682629650841349599633924389002786019464040 408549480795090161895306413963167578026196813726206604386890400000000/ 9900702794006894886175314765204702965450936777758658994701842249655120 074477969385609622360039884014810076399072960392464949348617, d[18,4] \+ = -7968660885827415981490828196930203818729245579271200734364509932953 1366118331158857213702226267642497024098283438504812608714705/17847935 3362772201242027020920484756214595687832874432575957818692709436225843 698147142925439917592200896345368322905584668147-948855515503175960512 9294093766551165413947581209400453139581139724792639378108598137294419 50898606598178330140938940662311931411/7139174134510888049681080836819 3902485838275133149773030383127477083774490337479258857170175967036880 3585381473291622338672588*7^(1/2), d[6,4] = 35328241040167355316548528 7842653748913763949016351206660699987623516928169268675618764436576775 30054408201728161521917397113366618167386721518719126000000/1315415227 6114830251119126025772219584648830690398606240427484580723822140296094 0338213739791642047580756522538105604653097955761822030948191987478264 69+1767818022393979995747005190755505733665038478733060055220727402900 0964414590249521432871850604032521865375346674368489623684501026634997 341154316438000000/131541522761148302511191260257722195846488306903986 0624042748458072382214029609403382137397916420475807565225381056046530 9795576182203094819198747826469*7^(1/2), d[15,8] = -473433184570421347 2729799339604404118959129048478492998981796287397128520807150480846823 5852140022793357112022557245440000000000000000000/75374050370774490768 4526713075034036759779816890766709266651250466244291270007809326460550 26983637004749111626142447467835659391021221-3796743915119540595295058 8381924902015434281261245664060231016768134728904206713926024125600031 33811238773650809958400000000000000000/9900702794006894886175314765204 7029654509367777586589947018422496551200744779693856096223600398840148 10076399072960392464949348617*7^(1/2), d[11,6] = 290807058275182405006 9345725709790855088935948723106858442020701090367458365427824477989930 4687587881000462976582575696730187843539765360572245115000000/72345375 9267506313768387418477880084695350877527903026113490847087911548466326 73475446713830366112790236464766513247237590257011405769724458745341+6 8553807083100363319207627962617251777647665334853058758145597837239084 0067714190246085432984717058232523573733668192414291872068751656685425 825000000/314545112725002745116690181946904384650152555446914359179778 6291686571949853594498932465818711570121314628902891880314677837261365 468248889510667*7^(1/2), d[7,8] = 557720235151732634699281612058896493 8288238337522293725943527768787851064891777906574032357987392416213512 0922788489417248086761403863712181951925000000000/91301280289827564029 9254248059557883336838237125750016806173347018965981574473290509116165 58416193009299310414227925015016650738820542603882059615975967+3137779 3160631577132315162086232913988296575745420045208152458041739558656955 5411953137884423227729062363793019659053146394054983488886605110540000 00000000/9130128028982756402992542480595578833368382371257500168061733 4701896598157447329050911616558416193009299310414227925015016650738820 542603882059615975967*7^(1/2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Suppose that we are given the i nitial value problem: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=f(x,y)" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$%\"xG%\"yG" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y(x[k])=y[k]" "6#/-%\"yG6#&%\"xG6#%\" kG&F%6#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "When a Rung e-Kutta step of width " }{TEXT 271 1 "h" }{TEXT -1 69 " has been made \+ using the basic scheme, we wish to obtain the result " }{XPPEDIT 18 0 "y[k](u)" "6#-&%\"yG6#%\"kG6#%\"uG" }{TEXT -1 21 " of a step of widt h " }{XPPEDIT 18 0 "h*u" "6#*&%\"hG\"\"\"%\"uGF%" }{TEXT -1 7 " for \+ " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 107 "ee2 \+ := map(_U->lhs(_U)=rhs(_U)/u,ee):\nsubs(ee2,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(8-i)],i=2..8)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7)7*,$*&\"\"\"F**&\"#?F*%\"uGF*!\"\"F*F(%!GF/F/F/F/F/7*,$* (\"$T$F*\"%+KF.F-F.F*,$*(\"%hrF*\"(+S-\"F.F-F.F.,$*(\"'\"G;\"F*F8F.F-F .F*F/F/F/F/F/7*,$*(\"%B5F*\"%+kF.F-F.F*,$*(F?F*\"&+c#F.F-F.F*\"\"!,$*( \"%pIF*FCF.F-F.F*F/F/F/F/7*,$*(\"#RF*\"$+\"F.F-F.F*,$*(\"(nB?%F*\")+\" G;\"F.F-F.F*FD,$*(\"(W)**QF*\"(Dq!HF.F-F.F.,$*(\"(#*H)RF*FTF.F-F.F*F/F /F/7*,$*(\"#$*F*\"$+#F.F-F.F*,$*(\"%6cF*\"'+W6F.F-F.F*FDFD,$*(\"&W<$F* \"'D]8F.F-F.F*,$*(\"'@N#*F*\"(+k5&F.F-F.F*F/F/7*,$*(\"#JF*FfnF.F-F.F*, $*(\"&t6#F*\"'+KMF.F-F.F*FDFD,$*(\"(CEg)F*\")v\"fl(F.F-F.F*,$*(\")4@yE F*\"*+SO*oF.F-F.F.,$*(FinF*\"'+NGF.F-F.F*F/7*,$*(\"$V*F*\"%+5F.F-F.F*, $*(\"1H$p=#=5@7F*\"0+++GH\"3pF.F-F.F.FDFD,$*(\"$D\"F*\"\"#F.F-F.F.,$*( \"1*))fzgIS-\"F*\"0+++!GH*o\"F.F-F.F.,$*(\"1*oGNN39]\"F*\"0+++!G(pl#F. F-F.F*,$*(\"1$GK@@R,2'F*\"/+++;?]#*F.F-F.F*Q(pprint26\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The new polynomials (of degree " } {XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 55 " ) are obtained a s follows (re-using the weight symbol " }{XPPEDIT 18 0 "b[j]" "6#&%\"b G6#%\"jG" }{TEXT -1 3 "). " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "[seq(b[j]=add(d[j,i]*u^(i-1),i=1..8),j=1..20)]:\npols := eval(subs (dd,%)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The first few non-zero polynomials with rough approximations for t he coefficients are . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct to 12 do\n if rhs(pols[ct])<>0 then print(evalf[6](pols[c t])) end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" \",2$F'\"\"!F'*&$\"'#R+\"!\"%F'%\"uGF'!\"\"*&$\"'@z`F.F')F/\"\"#F'F'*& $\"'e];!\"$F')F/\"\"$F'F0*&$\"'E!)HF9F')F/\"\"%F'F'*&$\"'8>JF9F')F/\" \"&F'F0*&$\"'1Ye\"!\"$\"\"\"%\"uGF-F-* &$\"''Ra\"!\"#F-)F.\"\"#F-!\"\"*&$\"'RTiF2F-)F.\"\"$F-F-*&$\"'l88F5F-) F.\"\"%F-F5*&$\"'p5:F5F-)F.\"\"&F-F-*&$\"'Z'**)F2F-)F.F'F-F5*&$\"'yq@F 2F-)F.\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(,0*&$ \"'\"y5\"!\"$\"\"\"%\"uGF-F-*&$\"'>\"3\"!\"#F-)F.\"\"#F-!\"\"*&$\"'mqV F2F-)F.\"\"$F-F-*&$\"'5*>*F2F-)F.\"\"%F-F5*&$\"'!z0\"F5F-)F.\"\"&F-F-* &$\"'(**H'F2F-)F.\"\"'F-F5*&$\"'8?:F2F-)F.F'F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"),0*&$\"'W6q!\"#\"\"\"%\"uGF-!\"\"*&$\"'&H %oF/F-)F.\"\"#F-F-*&$\"'CmF\"\"!F-)F.\"\"$F-F/*&$\"'?AeF8F-)F.\"\"%F-F -*&$\"'^&p'F8F-)F.\"\"&F-F/*&$\"'K()RF8F-)F.\"\"'F-F-*&$\"'0@'*F/F-)F. \"\"(F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*,0*&$\"'k?6 !\"\"\"\"\"%\"uGF-F-*&$\"'r$4\"\"\"!F-)F.\"\"#F-F,*&$\"'G@WF2F-)F.\"\" $F-F-*&$\"'l0$*F2F-)F.\"\"%F-F,*&$\"'9q5F-F-)F.\"\"&F-F-*&$\"'#HP'F2F- )F.\"\"'F-F,*&$\"'tP:F2F-)F.\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5,0*&$\"'#zT\"!\"\"\"\"\"%\"uGF-F,*&$\"'&QQ\"\"\"!F-)F. \"\"#F-F-*&$\"';%f&F2F-)F.\"\"$F-F,*&$\"'Ux6F-F-)F.\"\"%F-F-*&$\"'.a8F -F-)F.\"\"&F-F,*&$\"'bj!)F2F-)F.\"\"'F-F-*&$\"'mX>F2F-)F.\"\"(F-F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6,0*&$\"'yC5!\"\"\"\"\"%\" uGF-F-*&$\"':+5\"\"!F-)F.\"\"#F-F,*&$\"'1VSF2F-)F.\"\"$F-F-*&$\"'g4&)F 2F-)F.\"\"%F-F,*&$\"','y*F2F-)F.\"\"&F-F-*&$\"'wFeF2F-)F.\"\"'F-F,*&$ \"'>19F2F-)F.\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"# 7,0*&$\"'$\\0\"!\"$\"\"\"%\"uGF-!\"\"*&$\"'eH5!\"#F-)F.\"\"#F-F-*&$\"' /iTF3F-)F.\"\"$F-F/*&$\"'+g()F3F-)F.\"\"%F-F-*&$\"'S25F/F-)F.\"\"&F-F/ *&$\"'C**fF3F-)F.\"\"'F-F-*&$\"'dZ9F3F-)F.\"\"(F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 74 "The whole scheme, including the weights, \+ is given by the set of equations:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ee3 := `union`(ee2,\{op(pols)\}):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "We can now check that \+ this scheme satisfies the order conditions (and row sum conditions) fo r a 20 stage, order 8 Runge-Kutta scheme. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 88 "RK8_20eqs := [op(RowSumConditions(20,'expanded')),o p(OrderConditions(8,20,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "simplify(subs(ee3,RK8_20 eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7gx\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$>#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 43 "#-------------- ----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 97 "S tage by stage construction of the interpolation scheme B .. [7 stage s cheme] .. (shorter method)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Start with linking coefficients using the weight s of the 12 stage scheme as the linking coefficients for the first new stage." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5614 "e1 := \{c[2] = 1/20,\nc[3] = 341/3200,\nc[4] = 1023/6400,\nc[5] = 39/100,\nc[6] = 93/200,\nc[7] \+ = 31/200,\nc[8] = 943/1000,\nc[9] = 7067558016280/7837150160667,\nc[10 ] = 909/1000,\nc[11] = 47/50,\nc[12] = 1,\nc[13] = 1,\na[2,1] = 1/20, \na[3,1] = -7161/1024000,\na[3,2] = 116281/1024000,\na[4,1] = 1023/256 00,\na[4,2] = 0,\na[4,3] = 3069/25600,\na[5,1] = 4202367/11628100,\na[ 5,2] = 0,\na[5,3] = -3899844/2907025,\na[5,4] = 3982992/2907025,\na[6, 1] = 5611/114400,\na[6,2] = 0,\na[6,3] = 0,\na[6,4] = 31744/135025,\na [6,5] = 923521/5106400,\na[7,1] = 21173/343200,\na[7,2] = 0,\na[7,3] = 0,\na[7,4] = 8602624/76559175,\na[7,5] = -26782109/689364000,\na[7,6] = 5611/283500,\na[8,1] = -1221101821869329/690812928000000,\na[8,2] = 0,\na[8,3] = 0,\na[8,4] = -125/2,\na[8,5] = -1024030607959889/1689292 80000000,\na[8,6] = 1501408353528689/265697280000000,\na[8,7] = 607013 9212132283/92502016000000,\na[9,1] = -14725142644862158038813847088772 64246346044433307094207829051978044531801133057155/\n 1246894 8016200320011570596216439860248033015583934879004404536361680460696864 36608,\na[9,2] = 0,\na[9,3] = 0,\na[9,4] = -51722943110856684583751756 55246981230039025336933699114138315270772319372469280000/\n 1 2461938100480914589727863057121529836525707941023625292185093674907648 7132995191,\na[9,5] = -12070679258469254807978936441733187949484571516 120469966534514296406891652614970375/\n 272203115476165722171 0478184531100699497284085048389015085076961673446140398628096,\na[9,6] = 7801251558438936413230905525304310365677955925684971827014606748031 26770111481625/\n 1831104254127319721978898745071587868592261 02980861859505241443073629143100805376,\na[9,7] = 66411312295991164213 4782135839106469928140328160577035357155340392950009492511875/\n \+ 151784655985862481363330231072953491752797651500890783011399432530 16877823170816,\na[9,8] = 10332848184452015604056836767286656859124007 796970668046446015775000000/\n 131270355003603364807383424874 0727914537972028638950165249582733679393783,\na[10,1] = -2905557336033 7415088538618442231036441314060511/\n 22674759891089577691327 962602370597632000000000,\na[10,2] = 0,\na[10,3] = 0,\na[10,4] = -2046 2749524591049105403365239069/454251913499893469596231268750,\na[10,5] \+ = -180269259803172281163724663224981097/381009225582568710865798328320 00000,\na[10,6] = 21127670214172802870128286992003940810655221489/\n \+ 4679473877997892906145822697976708633673728000,\na[10,7] = 31 8607235173649312405151265849660869927653414425413/\n 6714716 715558965303132938072935465423910912000000,\na[10,8] = 212083202434519 082281842245535894/20022426044775672563822865371173879,\na[10,9] = -26 9840492940084251872116648508712979856226984822951779370341395122671458 3/\n 4695456749139343150770004420808711418846760359027175503 25616728175875000000,\na[11,1] = -234265984581408683695120714006560917 9073838476242943917/\n 1358480961351056777022231400139158760 857532162795520000,\na[11,2] = 0,\na[11,3] = 0,\na[11,4] = -9962860301 32538159613930889652/16353068885996164905464325675,\na[11,5] = -260530 85959256534152588089363841/4377552804565683061011299942400,\na[11,6] = 20980822345096760292224086794978105312644533925634933539/\n \+ 3775889992007550803878727839115494641972212962174156800,\na[11,7] = 8 90722993756379186418929622095833835264322635782294899/\n 139 21242001395112657501941955594013822830119803764736,\na[11,8] = 1610214 26143124178389075121929246710833125/1099720772213103465066704136434642 2894371443,\na[11,9] = 30076066976810251783423249756545243494667226619 5876496371874262392684852243925359864884962513/\n 4655443337 5013464555850653366045056037608247796155212857518928103156804923641066 74524398280000,\na[11,10] = -31155237437111730665923206875/39286214159 4230515010338956291,\na[12,1] = -2866556991825663971778295329101033887 534912787724034363/\n 86822671161926270301121392501614361203 0669233795338240,\na[12,2] = 0,\na[12,3] = 0,\na[12,4] = -169570887141 71468676387054358954754000/143690415119654683326368228101570221,\na[12 ,5] = -4583493974484572912949314673356033540575/4519577036552507471573 13034270335135744,\na[12,6] = 2346305388553404258656258473446184419154 740172519949575/\n 25672671640789540289274497830115148625418 3185289662464,\na[12,7] = 16571215593198468021712836909136106985862565 73484808662625/\n 134314804112551464772591551049560935053616 44432088109056,\na[12,8] = 3456853795546770522154958254769692263771875 00/74771167436930077221667203179551347546362089,\na[12,9] = \n -32058 9096271707254279143431215272753400810277402321024057136157075724905616 7015230160352087048674542196011/\n 9475695496839658147830151244512736 0498465774712725761537244920597319265730601723910349107473832403325912 0,\na[12,10] = 40279545832706233433100438588458933210937500/8896460842 799482846916972126377338947215101,\na[12,11] = -6122933601070769591613 093993993358877250/1050517001510235513198246721302027675953,\na[13,1] \+ = 44901867737754616851973/1014046409980231013380680,\na[13,2] = 0,\na[ 13,3] = 0,\na[13,4] = 0,\na[13,5] = 0,\na[13,6] = 79163867519161527964 8100000/2235604725089973126411512319,\na[13,7] = 384774949086898034811 9500000/15517045062138271618141237517,\na[13,8] = -1373451243239774147 6562500000/875132892924995907746928783,\na[13,9] = 1227476547031319687 8428812037740635050319234276006986398294443554969616342274215316330684 448207141/\n 489345147493715517650385834143510934888829280686609 654482896526796523353052166757299452852166040,\na[13,10] = -9798363684 577739445312500000/308722986341456031822630699,\na[13,11] = 2820355431 83190840068750/12295407629873040425991,\na[13,12] = -30681427293697693 6753/1299331183183744997286\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "subs(e1,matrix([se q([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..13)])):\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7/$\"+++++]!#6F(%!GF+F+F +F+F+F+F+F+F+F+7/$\"++]il5!#5$!+iS;$*p!#7$\"+TmbN6F/F+F+F+F+F+F+F+F+F+ F+7/$\"++vV)f\"F/$\"+]P4'*RF*$\"\"!F;$\"+D\"G))>\"F/F+F+F+F+F+F+F+F+F+ 7/$\"+++++RF/$\"+Gc(Rh$F/F:$!+nS_T8!\"*$\"+/l7q8FEF+F+F+F+F+F+F+F+7/$ \"++++]YF/$\"+!G?Z!\\F*F:F:$\"+U?(4N#F/$\"+Ifb3=F/F+F+F+F+F+F+F+7/$\"+ +++]:F/$\"+W!*GphF*F:F:$\"+JolB6F/$!+rg/&)QF*$\"+8()=z>F*F+F+F+F+F+F+7 /$\"++++I%*F/$!+S-jnB3l&FE$\"+U'p @c'F]oF+F+F+F+F+7/$\"+$QWV%FE$ \"+*=3/E%FE$\"+ASOvVF]o$\"+!\\D9(yF2F+F+F+F+7/$\"++++!4*F/$!+**fS\"G\" FEF:F:$!+'*Rr/XF]o$!+p?OJZFE$\"+,$)R'F]o$\"+DG?k9F*$\"+s(3/Y'F*$!+pJKIzF*F+F+7/$\"\"\"F;$ !+oEi,LFEF:F:$!+CF6!=\"!\"($!+RA995F]o$\"+K8JR\"*FE$\"+G%fPB\"F^s$\"+z VCBYFE$!+QxF$Q$FE$\"++@fFXFE$!+'[&\\GeFEF+7/Fhr$\"+>%*)zU%F*F:F:F:F:$ \"+#R\\5a$F/$\"+b@pzCF/$!+/-Up:F]o$\"+(\\1%3DF]o$!+zn$Q<$F]o$\"+F$GQH# F]o$!+LYKhBF/Q)pprint146\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "convert(ListTools[Enumerate](Simpl eOrderConditions(7)),matrix):\nlinalg[augment](linalg[delcols](%,2..2) ,matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7\\o7%\"\"\"%#~~G/*&%\"bGF (%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"$F)/*&F,F(-%!G6#*&%\"aGF(F 2F(F(#F(\"\"'7%\"\"%F)/*&F,F()F2F/F(#F(F57%\"\"&F)/*&F,F(-F96#*&FF)/*(F,F(F2F(F8F(#F(\"\")7%\"\"(F)/*&F,F(-F96#*&FCF(FF)/*&F,F(-F96#*(F< F(F2F(FIF(F(#F(\"$!=7%F]rF)/*(F,F(F2F(F_oF(#F(\"$W\"7%\"#@F)/*&F,F(-F9 6#*&FF(#F(FTQ)pprin t236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "seq(c[i]=subs(e23,c[i]),i=2..20);" }}{PARA 12 "" 1 " " {XPPMATH 20 "65/&%\"cG6#\"\"##\"\"\"\"#?/&F%6#\"\"$#\"$T$\"%+K/&F%6# \"\"%#\"%B5\"%+k/&F%6#\"\"&#\"#R\"$+\"/&F%6#\"\"'#\"#$*\"$+#/&F%6#\"\" (#\"#JFF/&F%6#\"\")#\"$V*\"%+5/&F%6#\"\"*#\".!G;!ev1(\".n1;]r$y/&F%6# \"#5#\"$4*FS/&F%6#\"#6#\"#Z\"#]/&F%6#\"#7F)/&F%6#\"#8F)/&F%6#\"#9,&#F) F'F)*&F]p!\"\"FJF_pFap/&F%6#\"#:#FWF*/&F%6#\"#;#\"$r$\"$+&/&F%6#\"#<#F C\"#D/&F%6#\"#=#F.F5/&F%6#\"#>#FJFhn/&F%6#F*#FWFhn" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 14" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The stan dard (simple) order conditions can be adapted to give a method of stag e by stage construction for an interpolation scheme that avoids dealin g with the weight polynomials for a given stage (corresponding to an \+ \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_13 := SimpleOrderCon ditions(7,13,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "whch := [1,2,4,8,16,17,25,32,64]: \ninterp_order_eqns14 := []:\nfor ct in whch do\n temp_eqn := conver t(SO7_13[ct],'interpolation_order_condition'):\n interp_order_eqns14 := [op(interp_order_eqns14),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order \+ conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 347 "interp_order_eqns14 := [add(a[14,i],i=1..13 )=c[14],seq(op(StageOrderConditions(i,14..14,'expanded')),i=2..7),\n \+ add(a[14,i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*add(a[l,m]*c[m],\n \+ m=2..l-1),l=2..k-1),k=2..j-1),j=2..i-1),i=2..13)=c[14]^6/720, ##17\n add(a[14,i]*add(a[i,j]*c[j]^2*add(a[j,k]*c[k],k=2..j-1),j=2..i-1), i=2..13)=c[14]^6/60]: ##25" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "e2 := `union`(e1,\{seq(a[14 ,i]=0,i=2..5)\}):\neqs_14 := expand(subs(e2,interp_order_eqns14)):\nno ps(eqs_14);\nindets(eqs_14);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<,&%\"aG6$\"#9\"\")&%\" cG6#F'&F%6$F'\"\"*&F%6$F'\"\"(&F%6$F'\"#8&F%6$F'\"#7&F%6$F'\"#5&F%6$F' \"\"\"&F%6$F'\"#6&F%6$F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We s olve for the linking coefficients in terms of " }{XPPEDIT 18 0 "c[14] ;" "6#&%\"cG6#\"#9" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "sol_14 := solve(\{op(eqs_14)\},indets(eqs_14) minus \+ \{c[14]\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 20 "We choose the node " }{XPPEDIT 18 0 "c[1 4];" "6#&%\"cG6#\"#9" }{TEXT -1 63 " so that an additional (adapted) \+ order condition is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 6 "EITHER" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "extra_eqn := add(a[14,i]*ad d(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..13)=c[14]^6/120:" }}}{PARA 0 "" 0 "" {TEXT -1 6 "OR ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "extra_eqn := convert(SO7_13[27],'interpolation_order_ condition'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "expand(subs(e2,extra_eqn)):\neq_14 := subs(sol_1 4,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eq_14G/,,*&#\"APBpMi4C#)37 g9'[,.$\"C])RR#HzYro67#[D&HUB\"\"\"*$)&%\"cG6#\"#9\"\"&F+F+!\"\"*&#\"B LL;JU16;_ph1qaVb$\"D+#ys3::wX-aay0Vv5GF+*$)F.\"\"'F+F+F+*&#\"@\">8yu8? .%eW^t$yGV\"C+(zy%ee$HuLUU'40f%o%F+*$)F.\"\"#F+F+F+*&#\"@(R/Ee/SMh[r6z #HW\"\"B&)RR#HzYro67#[D&HUBF+*$)F.\"\"$F+F+F3*&#\"At&RMCTg4_PV07N')H\" \"BSfdprre[n%[G>5=p$*F+*$)F.\"\"%F+F+F+,$*&#F+\"$?\"F+F8F+F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lhs(eq_14)-rhs(eq_14);\nfactor(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*&#\"APBpMi4C#)37g9'[,.$\"C])RR#HzYro67#[D&HUB\"\"\"*$)&%\"cG6 #\"#9\"\"&F(F(!\"\"*&#\"Az2By?.3%HS+#Q&\\+,\"F'F(*$)F+\"\"'F(F(F(*&#\" @\">8yu8?.%eW^t$yGV\"C+(zy%ee$HuLUU'40f%o%F(*$)F+\"\"#F(F(F(*&#\"@(R/E e/SMh[r6z#HW\"\"B&)RR#HzYro67#[D&HUBF(*$)F+\"\"$F(F(F0*&#\"At&RMCTg4_P V07N')H\"\"BSfdprre[n%[G>5=p$*F(*$)F+\"\"%F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"@(R/Ee/SMh[r6z#HW\"\"C+(zy%ee$HuLUU'40f%o%\"\"\" *()&%\"cG6#\"#9\"\"#F(,(*&F.F(F*F(F(*&F.F(F+F(!\"\"\"\"$F(F(),&F+F(F(F 3F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "14*c[14]^2-14*c[14]+3;\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"#9\"\"\")&%\"cG6#F%\"\"#F&F&*&F%F&F(F&!\"\"\" \"$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"\"\"\"\"#F%*&\"#9!\"\"\" \"(F$F%,&F$F%*&F(F)F*F$F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We can now obtain values for the linking coeffici ents for this stage." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "e3 : = \{c[14]=1/2-1/14*7^(1/2)\}:\ne4 := solve(\{op(subs(e3,eqs_14))\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "e5 := `union`(e2,e3,e4):\n[c[14]=subs(e5,c[14]),seq(a[14,i]=s ubs(e5,a[14,i]),i=1..13)]:\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&%\"cG6#\"#9$\"I@fp*4H)=tTFR'Q'Q&\\jx,6$!#S/&%\"aG6$ F(\"\"\"$\"IiGV;\"Q-:8/t,J'\\vY1q?Y!#T/&F.6$F(\"\"#$\"\"!F9/&F.6$F(\" \"$F8/&F.6$F(\"\"%F8/&F.6$F(\"\"&F8/&F.6$F(\"\"'$\"H:(pz1S%Q?&Go'3[U3; /R]%F+/&F.6$F(\"\"($\"Ikl7@-Ml5q)y5WU8xp;oL#F+/&F.6$F(\"\")$\"I]TDb='3 AQ.y5u1@%o8!Ry$!#Q/&F.6$F(\"\"*$!IN3P(R2.\\RhE5YUX*G8\"\\f\"FX/&F.6$F( \"#5$\"IS>4!fHjf5D9&G5;=No$GI#FX/&F.6$F(\"#6$!IZF+[p,)*48;[_7%px]yb[%F X/&F.6$F(\"#7$!IB8yS,L-u14&4SWZ'o(e)zjF3/&F.6$F(\"#8$!ILD;U)>XYK5Coi;' QaN]f7F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "These linking coefficients essentially agree to digits with those \+ of the published scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 70 "#------------------------------------------------- --------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check whic h of the (adapted) simple order conditions are satisfied at this stage ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "recd := []:\nfor ct to nops(SO7_13) do\n tt := con vert(SO7_13[ct],'interpolation_order_condition'):\n if expand(subs(e 5,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop(r ecd);\nnops(recd);\nop(\{seq(i,i=1..nops(SO7_13))\} minus \{op(recd)\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6J\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"' \"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"# B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#Z\"#^\"#`\"#d\"#f\"#g\"#i\"#k " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6:\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#[\"# \\\"#]\"#_\"#a\"#b\"#c\"#e\"#h\"#j" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 70 "#--------------------------------------- ------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6888 "e5 := \{a[13,5] = 0, a[14,10] = 30448415149 825325326308593750/15127426330731345559308904251+588780394238322448263 2816406250/741243890205835932406136308299*7^(1/2), a[14,3] = 0, c[14] \+ = 1/2-1/14*7^(1/2), a[12,3] = 0, a[4,2] = 0, c[13] = 1, a[11,3] = 0, a [12,2] = 0, a[11,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/392-3/39 2*7^(1/2), a[10,2] = 0, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[14,2] = \+ 0, a[6,3] = 0, a[9,3] = 0, a[14,5] = 0, a[6,2] = 0, a[7,2] = 0, a[7,3] = 0, a[5,2] = 0, a[13,2] = 0, a[14,4] = 0, a[13,4] = 0, a[13,3] = 0, \+ a[14,11] = -1466584824552592368357500/602474973863778980873559-9466779 79546641857718938375/59042547438650340125608782*7^(1/2), a[14,8] = 742 00703416028798327128906250/42881511753324799479599510367+2867664719921 7261041085964843750/2101194075912915174500376007983*7^(1/2), a[14,1] = 8941065567926479206438689/198753096356125278622613280+152838094177334 666489948287/331122658529304714185273724480*7^(1/2), a[14,9] = -128719 9574154792351913181734948009779654190448189357505217021276723163171317 10725250664367373074947/9591164890876824145947562349212814323821053901 4575492278647719252118577198224684430692759024543840-88218508206817762 2906787605005472555636444759904636504531314529608120034066591927204665 390933748491981/159788807082007890271486388737885486634858757998282770 136227100274029549612242324261534136534890037440*7^(1/2), a[9,1] = -14 7251426448621580388138470887726424634604443330709420782905197804453180 1133057155/12468948016200320011570596216439860248033015583934879004404 53636168046069686436608, a[8,7] = 6070139212132283/92502016000000, a[8 ,6] = 1501408353528689/265697280000000, a[8,5] = -1024030607959889/168 929280000000, c[6] = 93/200, a[9,5] = -1207067925846925480797893644173 3187949484571516120469966534514296406891652614970375/27220311547616572 21710478184531100699497284085048389015085076961673446140398628096, a[9 ,4] = -517229431108566845837517565524698123003902533693369911413831527 0772319372469280000/12461938100480914589727863057121529836525707941023 6252921850936749076487132995191, a[9,8] = 1033284818445201560405683676 7286656859124007796970668046446015775000000/13127035500360336480738342 48740727914537972028638950165249582733679393783, a[9,7] = 664113122959 911642134782135839106469928140328160577035357155340392950009492511875/ 1517846559858624813633302310729534917527976515008907830113994325301687 7823170816, a[9,6] = 7801251558438936413230905525304310365677955925684 97182701460674803126770111481625/1831104254127319721978898745071587868 59226102980861859505241443073629143100805376, a[10,7] = 31860723517364 9312405151265849660869927653414425413/67147167155589653031329380729354 65423910912000000, a[10,8] = 212083202434519082281842245535894/2002242 6044775672563822865371173879, a[10,6] = 211276702141728028701282869920 03940810655221489/4679473877997892906145822697976708633673728000, a[10 ,5] = -180269259803172281163724663224981097/38100922558256871086579832 832000000, a[10,4] = -20462749524591049105403365239069/454251913499893 469596231268750, a[10,1] = -290555733603374150885386184422310364413140 60511/22674759891089577691327962602370597632000000000, a[11,1] = -2342 659845814086836951207140065609179073838476242943917/135848096135105677 7022231400139158760857532162795520000, a[10,9] = -26984049294008425187 21166485087129798562269848229517793703413951226714583/4695456749139343 15077000442080871141884676035902717550325616728175875000000, a[11,6] = 20980822345096760292224086794978105312644533925634933539/377588999200 7550803878727839115494641972212962174156800, a[11,5] = -26053085959256 534152588089363841/4377552804565683061011299942400, a[11,4] = -9962860 30132538159613930889652/16353068885996164905464325675, a[5,4] = 398299 2/2907025, a[12,1] = -286655699182566397177829532910103388753491278772 4034363/868226711619262703011213925016143612030669233795338240, a[11,1 0] = -31155237437111730665923206875/392862141594230515010338956291, a[ 11,9] = 30076066976810251783423249756545243494667226619587649637187426 2392684852243925359864884962513/46554433375013464555850653366045056037 60824779615521285751892810315680492364106674524398280000, a[11,8] = 16 1021426143124178389075121929246710833125/10997207722131034650667041364 346422894371443, a[11,7] = 8907229937563791864189296220958338352643226 35782294899/13921242001395112657501941955594013822830119803764736, a[1 2,6] = 2346305388553404258656258473446184419154740172519949575/2567267 16407895402892744978301151486254183185289662464, a[12,5] = -4583493974 484572912949314673356033540575/451957703655250747157313034270335135744 , a[6,1] = 5611/114400, a[12,4] = -16957088714171468676387054358954754 000/143690415119654683326368228101570221, a[12,9] = -32058909627170725 4279143431215272753400810277402321024057136157075724905616701523016035 2087048674542196011/94756954968396581478301512445127360498465774712725 7615372449205973192657306017239103491074738324033259120, a[12,8] = 345 685379554677052215495825476969226377187500/747711674369300772216672031 79551347546362089, a[12,7] = 16571215593198468021712836909136106985862 56573484808662625/1343148041125514647725915510495609350536164443208810 9056, a[13,1] = 44901867737754616851973/1014046409980231013380680, a[1 2,11] = -6122933601070769591613093993993358877250/10505170015102355131 98246721302027675953, a[12,10] = 4027954583270623343310043858845893321 0937500/8896460842799482846916972126377338947215101, c[3] = 341/3200, \+ a[13,6] = 791638675191615279648100000/2235604725089973126411512319, a[ 13,8] = -13734512432397741476562500000/875132892924995907746928783, a[ 13,7] = 3847749490868980348119500000/15517045062138271618141237517, a[ 13,12] = -306814272936976936753/1299331183183744997286, c[2] = 1/20, a [13,11] = 282035543183190840068750/12295407629873040425991, a[13,10] = -9798363684577739445312500000/308722986341456031822630699, a[13,9] = \+ 1227476547031319687842881203774063505031923427600698639829444355496961 6342274215316330684448207141/48934514749371551765038583414351093488882 9280686609654482896526796523353052166757299452852166040, a[8,1] = -122 1101821869329/690812928000000, c[10] = 909/1000, c[4] = 1023/6400, a[5 ,3] = -3899844/2907025, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100 , a[8,4] = -125/2, a[7,6] = 5611/283500, a[14,6] = 3037913416047823635 649583750/15649233075629811884880586233-302465625814318865951896498250 /5367686944941025476514041077919*7^(1/2), a[7,5] = -26782109/689364000 , a[14,7] = 183874328794901398385760606250/760335208044775309288920638 333-114787229090554407592495836250/37256425194193990155157111278317*7^ (1/2), c[7] = 31/200, c[8] = 943/1000, c[9] = 7067558016280/7837150160 667, a[3,2] = 116281/1024000, a[7,4] = 8602624/76559175, a[4,1] = 1023 /25600, a[2,1] = 1/20, a[3,1] = -7161/1024000, c[5] = 39/100, a[14,12] = 11352128098668146659861/254668911904014019468056-521584263992860792 4801/127334455952007009734028*7^(1/2), c[11] = 47/50, a[6,5] = 923521/ 5106400, a[7,1] = 21173/343200, a[6,4] = 31744/135025\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 " calculation for stage 15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 253 "The standard (simple) order conditions can be ada pted to give a method of stage by stage construction for an interpolat ion scheme that avoids dealing with the weight polynomials for a given stage (corresponding to an \"approximate\" interpolation scheme)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_14 := SimpleOrderConditions(7,14,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "whch : = [1,2,4,8,16,17,25,27,32,64]:\ninterp_order_eqns15 := []:\nfor ct in \+ whch do\n temp_eqn := convert(SO7_14[ct],'interpolation_order_condit ion'):\n interp_order_eqns15 := [op(interp_order_eqns15),temp_eqn]; \nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 439 "interp_orde r_eqns15 := [add(a[15,i],i=1..14)=c[15],seq(op(StageOrderConditions(i, 15..15,'expanded')),i=2..7),\n add(a[15,i]*add(a[i,j]*add(a[j,k]*ad d(a[k,l]*add(a[l,m]*c[m],\n m=2..l-1),l=2..k-1),k=2..j-1),j=2..i -1),i=2..14)=c[15]^6/720, ##17\n add(a[15,i]*add(a[i,j]*c[j]^2*add( a[j,k]*c[k],k=2..j-1),j=2..i-1),i=2..14)=c[15]^6/60, ##25\n add(a[1 5,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..14)=c[15]^6 /120]: ##27" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We specify the node for this stage immediately, namely \+ " }{XPPEDIT 18 0 "c[15] = 9/20;" "6#/&%\"cG6#\"#:*&\"\"*\"\"\"\"#?!\" \"" }{TEXT -1 80 ", and have enough equations to determine the corresp onding linking coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "e6 := `union`(e5,\{c[15]=9/20,seq(a[15,i]=0,i=2..5)\}):\neqs_15 : = expand(subs(e6,interp_order_eqns15)):\nnops(eqs_15);\nindets(eqs_15) ;\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<,&%\"aG6$\"#:\"\"\"&F%6$F'\"\"'&F%6$F'\"\"(&F%6$F'\" \")&F%6$F'\"\"*&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\"#7&F%6$F'\"#8&F%6$F'\"#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "e7 := solve(\{op(eqs_1 5)\}):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "e8 := `union`(e6,e7):\n[seq(a[15,i]=subs(e8,a[15,i]),i=1..14)]:\ne valf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&%\"aG6$\"#:\"\"\"$ \"IvcmmC-+u\\yhbMV4#H*p@\\!#T/&F&6$F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1/&F&6 $F(\"\"%F1/&F&6$F(\"\"&F1/&F&6$F(\"\"'$\"I&Rjl&>G'=(>8ojkOBEpUiwF,/&F& 6$F(\"\"($\"Ih4%y**=;Q1.5Yfrj)G\"H(=@!#S/&F&6$F(\"\")$!H/G3t\"[Z.h:+5 \\?Y2<^ " 0 "" {MPLTEXT 1 0 253 "rec d := []:\nfor ct to nops(SO7_14) do\n tt := convert(SO7_14[ct],'inte rpolation_order_condition'):\n if expand(subs(e8,lhs(tt)=rhs(tt))) t hen recd := [op(recd),ct] end if: \nend do:\nop(recd);\nnops(recd);\no p(\{seq(i,i=1..nops(SO7_14))\} minus \{op(recd)\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6J\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"# 6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G \"#H\"#I\"#J\"#K\"#Z\"#^\"#`\"#d\"#f\"#g\"#i\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6:\"#L\"#M\"#N\"# O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#[\"#\\\"#]\"#_\"#a\"#b\"#c \"#e\"#h\"#j" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#-------------------------------------------------------- -------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e8" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8335 "e8 := \{a[9,3] = 0, a[13,7] = 3847749490868980348119500000/15517 045062138271618141237517, a[13,8] = -13734512432397741476562500000/875 132892924995907746928783, a[13,11] = 282035543183190840068750/12295407 629873040425991, a[13,12] = -306814272936976936753/1299331183183744997 286, c[3] = 341/3200, a[13,6] = 791638675191615279648100000/2235604725 089973126411512319, a[14,2] = 0, c[2] = 1/20, a[5,3] = -3899844/290702 5, c[4] = 1023/6400, a[8,1] = -1221101821869329/690812928000000, c[10] = 909/1000, a[13,10] = -9798363684577739445312500000/3087229863414560 31822630699, a[13,9] = 12274765470313196878428812037740635050319234276 006986398294443554969616342274215316330684448207141/489345147493715517 6503858341435109348888292806866096544828965267965233530521667572994528 52166040, a[9,2] = 0, a[8,2] = 0, a[11,2] = 0, a[11,3] = 0, a[14,4] = \+ 0, a[14,13] = 3/392-3/392*7^(1/2), a[12,2] = 0, a[7,2] = 0, a[4,2] = 0 , c[14] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[10,3] = 0, a[14,3] = 0, a[1 4,5] = 0, a[13,5] = 0, a[8,3] = 0, c[12] = 1, c[13] = 1, a[6,3] = 0, a [5,2] = 0, a[13,4] = 0, a[7,3] = 0, a[13,2] = 0, a[4,3] = 3069/25600, \+ a[5,1] = 4202367/11628100, a[8,4] = -125/2, a[6,2] = 0, a[13,3] = 0, a [10,2] = 0, a[7,6] = 5611/283500, a[14,6] = 30379134160478236356495837 50/15649233075629811884880586233-302465625814318865951896498250/536768 6944941025476514041077919*7^(1/2), a[12,10] = 402795458327062334331004 38588458933210937500/8896460842799482846916972126377338947215101, a[15 ,1] = 339349033530268807690405611/7402593223688213516896000000+7846990 17603056191374015321/615131617555510903855616000000*7^(1/2), a[7,5] = \+ -26782109/689364000, a[14,7] = 183874328794901398385760606250/76033520 8044775309288920638333-114787229090554407592495836250/3725642519419399 0155157111278317*7^(1/2), c[7] = 31/200, c[8] = 943/1000, c[9] = 70675 58016280/7837150160667, a[3,2] = 116281/1024000, a[7,4] = 8602624/7655 9175, a[15,11] = -5009581586726585160535323/182154187109230228533200+1 64995836726103803189698643/13162109004021797158528000*7^(1/2), a[15,8] = 4171780920238230167282154375/180381659089210106588738944-3217313543 3579259286326757125/3056994432985560753767049472*7^(1/2), a[15,6] = 39 667151342526762946267729161/264960560010663481648771830400-15445972765 4214653639901999/5591570455455665877423584000*7^(1/2), a[15,7] = 33170 4901880412220279128473073/1418701262824070547944341715840-110013424964 1546169459352427/132690135289901616433045894400*7^(1/2), a[15,14] = 70 8939/16000000*7^(1/2), a[15,13] = -79893/16000000+101277/64000000*7^(1 /2), a[15,10] = 6659828881684523938253413125/4878585463173626181888485 12-105274812306435785357342625/16947939574300438348992256*7^(1/2), a[1 5,9] = -15372775602531728215983123367589049442461995162196337284574757 119670408788921748236494825887253876104043/166377350147863276001131183 6087937178622019554334472825241848191108179400377366974818139697364536 000000+405732438148607561541740662873496460776199401666888744396064357 8167392109927444312471311939993/96852668533426148027403526835032792014 9725248811661666881873633747980768250399197689344000000*7^(1/2), a[3,1 ] = -7161/1024000, c[5] = 39/100, a[14,12] = 11352128098668146659861/2 54668911904014019468056-5215842639928607924801/12733445595200700973402 8*7^(1/2), a[2,1] = 1/20, a[4,1] = 1023/25600, a[6,5] = 923521/5106400 , a[7,1] = 21173/343200, c[11] = 47/50, a[6,4] = 31744/135025, a[15,2] = 0, a[15,4] = 0, a[15,5] = 0, a[15,3] = 0, a[15,12] = -1764488883660 554363266503/153994806895851258937600000, a[14,10] = 30448415149825325 326308593750/15127426330731345559308904251+588780394238322448263281640 6250/741243890205835932406136308299*7^(1/2), c[15] = 9/20, a[10,5] = - 180269259803172281163724663224981097/381009225582568710865798328320000 00, a[10,7] = 318607235173649312405151265849660869927653414425413/6714 716715558965303132938072935465423910912000000, a[10,8] = 2120832024345 19082281842245535894/20022426044775672563822865371173879, a[9,6] = 780 1251558438936413230905525304310365677955925684971827014606748031267701 11481625/1831104254127319721978898745071587868592261029808618595052414 43073629143100805376, a[9,8] = 103328481844520156040568367672866568591 24007796970668046446015775000000/1312703550036033648073834248740727914 537972028638950165249582733679393783, a[9,7] = 66411312295991164213478 2135839106469928140328160577035357155340392950009492511875/15178465598 586248136333023107295349175279765150089078301139943253016877823170816, a[9,4] = -51722943110856684583751756552469812300390253369336991141383 15270772319372469280000/1246193810048091458972786305712152983652570794 10236252921850936749076487132995191, c[6] = 93/200, a[9,5] = -12070679 2584692548079789364417331879494845715161204699665345142964068916526149 70375/2722031154761657221710478184531100699497284085048389015085076961 673446140398628096, a[8,5] = -1024030607959889/168929280000000, a[8,7] = 6070139212132283/92502016000000, a[9,1] = -147251426448621580388138 4708877264246346044433307094207829051978044531801133057155/12468948016 2003200115705962164398602480330155839348790044045363616804606968643660 8, a[8,6] = 1501408353528689/265697280000000, a[14,9] = -1287199574154 7923519131817349480097796541904481893575052170212767231631713171072525 0664367373074947/95911648908768241459475623492128143238210539014575492 278647719252118577198224684430692759024543840-882185082068177622906787 6050054725556364447599046365045313145296081200340665919272046653909337 48491981/1597888070820078902714863887378854866348587579982827701362271 00274029549612242324261534136534890037440*7^(1/2), a[14,8] = 742007034 16028798327128906250/42881511753324799479599510367+2867664719921726104 1085964843750/2101194075912915174500376007983*7^(1/2), a[14,1] = 89410 65567926479206438689/198753096356125278622613280+152838094177334666489 948287/331122658529304714185273724480*7^(1/2), a[14,11] = -14665848245 52592368357500/602474973863778980873559-946677979546641857718938375/59 042547438650340125608782*7^(1/2), a[10,6] = 21127670214172802870128286 992003940810655221489/4679473877997892906145822697976708633673728000, \+ a[5,4] = 3982992/2907025, a[12,1] = -286655699182566397177829532910103 3887534912787724034363/86822671161926270301121392501614361203066923379 5338240, a[11,5] = -26053085959256534152588089363841/43775528045656830 61011299942400, a[11,4] = -996286030132538159613930889652/163530688859 96164905464325675, a[11,6] = 20980822345096760292224086794978105312644 533925634933539/377588999200755080387872783911549464197221296217415680 0, a[10,9] = -26984049294008425187211664850871297985622698482295177937 03413951226714583/4695456749139343150770004420808711418846760359027175 50325616728175875000000, a[10,1] = -2905557336033741508853861844223103 6441314060511/22674759891089577691327962602370597632000000000, a[11,1] = -2342659845814086836951207140065609179073838476242943917/1358480961 351056777022231400139158760857532162795520000, a[10,4] = -204627495245 91049105403365239069/454251913499893469596231268750, a[11,10] = -31155 237437111730665923206875/392862141594230515010338956291, a[13,1] = 449 01867737754616851973/1014046409980231013380680, a[12,7] = 165712155931 9846802171283690913610698586256573484808662625/13431480411255146477259 155104956093505361644432088109056, a[12,8] = 3456853795546770522154958 25476969226377187500/74771167436930077221667203179551347546362089, a[1 2,9] = -32058909627170725427914343121527275340081027740232102405713615 70757249056167015230160352087048674542196011/9475695496839658147830151 2445127360498465774712725761537244920597319265730601723910349107473832 4033259120, a[6,1] = 5611/114400, a[12,4] = -1695708871417146867638705 4358954754000/143690415119654683326368228101570221, a[12,5] = -4583493 974484572912949314673356033540575/451957703655250747157313034270335135 744, a[11,7] = 890722993756379186418929622095833835264322635782294899/ 13921242001395112657501941955594013822830119803764736, a[12,6] = 23463 05388553404258656258473446184419154740172519949575/2567267164078954028 92744978301151486254183185289662464, a[11,9] = 30076066976810251783423 2497565452434946672266195876496371874262392684852243925359864884962513 /465544333750134645558506533660450560376082477961552128575189281031568 0492364106674524398280000, a[11,8] = 161021426143124178389075121929246 710833125/10997207722131034650667041364346422894371443, a[12,11] = -61 22933601070769591613093993993358877250/1050517001510235513198246721302 027675953\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order conditions can be adapted to give a method of stage by stage co nstruction for an interpolation scheme that avoids dealing with the we ight polynomials for a given stage (corresponding to an \"approximate \" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_15 := SimpleOrderConditions(7,1 5,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 222 "whch := [1,2,4,8,16,17,25,27,32,63,64]:\ninte rp_order_eqns16 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_ 15[ct],'interpolation_order_condition'):\n interp_order_eqns16 := [o p(interp_order_eqns16),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order condit ions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 509 "interp_order_eqns16 := [add(a[16,i],i=1..15)=c[16 ],seq(op(StageOrderConditions(i,16..16,'expanded')),i=2..7),\n add( a[16,i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*add(a[l,m]*c[m],\n m=2.. l-1),l=2..k-1),k=2..j-1),j=2..i-1),i=2..15)=c[16]^6/720, ##17\n add (a[16,i]*add(a[i,j]*c[j]^2*add(a[j,k]*c[k],k=2..j-1),j=2..i-1),i=2..15 )=c[16]^6/60, ##25\n add(a[16,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2.. j-1),j=2..i-1),i=2..15)=c[16]^6/120, ##27\n add(a[16,i]*add(a[i,j] *c[j]^5,j=2..i-1),i=2..15)=c[16]^7/42]: ##63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the node " } {XPPEDIT 18 0 "c[16] = 371/500;" "6#/&%\"cG6#\"#;*&\"$r$\"\"\"\"$+&!\" \"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "e9 := `union`(e8,\{c[16]=371/500,seq(a[16,i]=0,i=2..5)\}):\neqs_16 := expan d(subs(e9,interp_order_eqns16)):\nnops(eqs_16);\nindets(eqs_16);\nnops (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<-&%\"aG6$\"#;\"\"*&F%6$F'\"\"(&F%6$F'\"\"'&F%6$F'\"#7& F%6$F'\"#6&F%6$F'\"#5&F%6$F'\"#9&F%6$F'\"#8&F%6$F'\"\")&F%6$F'\"\"\"&F %6$F'\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[so lve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e10 := solve(\{ op(eqs_16)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "e11 := `union`(e9,e10): \nseq(a[16,i]=subs(e11,a[16,i]),i=1..15):\nevalf[40](%);" }}{PARA 12 " " 1 "" {XPPMATH 20 "61/&%\"aG6$\"#;\"\"\"$\"Iu]'\\3qY#\\)o$R2Sq30UlZS! #T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/& F%6$F'\"\"'$\"I%Q'=()QG&RPf&>M)*>(R-Q)4R!#S/&F%6$F'\"\"($\"IG\"QxC)z34 ![$4^m__=(yAr#FD/&F%6$F'\"\")$\"IJVQK6\\dN)>c3(o**4M==SQ!#R/&F%6$F'\" \"*$!IN.VZE2v&)H&p\"Rl*\\sG:Fj\"FQ/&F%6$F'\"#5$\"I.H(G;A*)3N\"[*)=n')) )3WFUIFQ/&F%6$F'\"#6$!Ic)=YR.?c$)HS=FubGQ+DJ&FQ/&F%6$F'\"#7$\"IijI\\1D %)\\WJ4)R.v.KtvA\"FD/&F%6$F'\"#8$\"I+5#)zqHYEGzIj@UrU>(ft\"F+/&F%6$F' \"#9$!Il&F+/&F%6$F'\"#:$\"IRrPH<`u))=MGc\"3Y1Cwt%= F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#- --------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check which of the (adapted) sim ple order conditions are satisfied at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "recd := []: \nfor ct to nops(SO7_15) do\n tt := convert(SO7_15[ct],'interpolatio n_order_condition'):\n if expand(subs(e11,lhs(tt)=rhs(tt))) then rec d := [op(recd),ct] end if: \nend do:\nop(recd);\nnops(recd);\nop(\{seq (i,i=1..nops(SO7_15))\} minus \{op(recd)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6R\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6 \"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\" #H\"#I\"#J\"#K\"#O\"#R\"#V\"#Y\"#Z\"#\\\"#^\"#`\"#a\"#c\"#d\"#f\"#g\"# i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"#L\"#M\"#N\"#P\"#Q\"#S\"#T\"#U\"#W\"#X\"#[\"#]\"#_\"# b\"#e\"#h" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#-------------------------------------------------------------- -------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e11" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12938 "e11 := \{a[9,3] = 0, a[13,7] = 3847749490868980348119500000/155 17045062138271618141237517, a[13,8] = -13734512432397741476562500000/8 75132892924995907746928783, a[13,11] = 282035543183190840068750/122954 07629873040425991, a[13,12] = -306814272936976936753/12993311831837449 97286, c[3] = 341/3200, a[13,6] = 791638675191615279648100000/22356047 25089973126411512319, a[14,2] = 0, c[2] = 1/20, a[5,3] = -3899844/2907 025, c[4] = 1023/6400, a[8,1] = -1221101821869329/690812928000000, c[1 0] = 909/1000, a[13,10] = -9798363684577739445312500000/30872298634145 6031822630699, a[13,9] = 122747654703131968784288120377406350503192342 76006986398294443554969616342274215316330684448207141/4893451474937155 1765038583414351093488882928068660965448289652679652335305216675729945 2852166040, a[9,2] = 0, a[8,2] = 0, a[11,2] = 0, a[11,3] = 0, a[14,4] \+ = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12,2] = 0, a[7,2] = 0, a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[10,3] = 0, a[14,3] = 0, a [14,5] = 0, a[13,5] = 0, a[8,3] = 0, c[12] = 1, c[13] = 1, a[6,3] = 0, a[5,2] = 0, a[13,4] = 0, a[7,3] = 0, a[13,2] = 0, a[4,3] = 3069/25600 , a[5,1] = 4202367/11628100, a[8,4] = -125/2, a[6,2] = 0, a[13,3] = 0, a[10,2] = 0, a[7,6] = 5611/283500, a[14,6] = 303791341604782363564958 3750/15649233075629811884880586233-302465625814318865951896498250/5367 686944941025476514041077919*7^(1/2), a[12,10] = 4027954583270623343310 0438588458933210937500/8896460842799482846916972126377338947215101, a[ 15,1] = 339349033530268807690405611/7402593223688213516896000000+78469 9017603056191374015321/615131617555510903855616000000*7^(1/2), a[7,5] \+ = -26782109/689364000, a[14,7] = 183874328794901398385760606250/760335 208044775309288920638333-114787229090554407592495836250/37256425194193 990155157111278317*7^(1/2), c[7] = 31/200, c[8] = 943/1000, c[9] = 706 7558016280/7837150160667, a[3,2] = 116281/1024000, a[7,4] = 8602624/76 559175, a[15,11] = -5009581586726585160535323/182154187109230228533200 +164995836726103803189698643/13162109004021797158528000*7^(1/2), a[15, 8] = 4171780920238230167282154375/180381659089210106588738944-32173135 433579259286326757125/3056994432985560753767049472*7^(1/2), a[15,6] = \+ 39667151342526762946267729161/264960560010663481648771830400-154459727 654214653639901999/5591570455455665877423584000*7^(1/2), a[15,7] = 331 704901880412220279128473073/1418701262824070547944341715840-1100134249 641546169459352427/132690135289901616433045894400*7^(1/2), a[15,14] = \+ 708939/16000000*7^(1/2), a[15,13] = -79893/16000000+101277/64000000*7^ (1/2), a[15,10] = 6659828881684523938253413125/48785854631736261818884 8512-105274812306435785357342625/16947939574300438348992256*7^(1/2), a [15,9] = -153727756025317282159831233675890494424619951621963372845747 57119670408788921748236494825887253876104043/1663773501478632760011311 8360879371786220195543344728252418481911081794003773669748181396973645 36000000+4057324381486075615417406628734964607761994016668887443960643 578167392109927444312471311939993/968526685334261480274035268350327920 149725248811661666881873633747980768250399197689344000000*7^(1/2), a[1 6,3] = 0, a[16,4] = 0, a[3,1] = -7161/1024000, c[5] = 39/100, a[14,12] = 11352128098668146659861/254668911904014019468056-521584263992860792 4801/127334455952007009734028*7^(1/2), a[2,1] = 1/20, a[4,1] = 1023/25 600, a[6,5] = 923521/5106400, a[7,1] = 21173/343200, c[11] = 47/50, a[ 6,4] = 31744/135025, a[16,8] = 983793822755252686947250523701385862698 8567133654547552254520645532228606707500790611669969401/32259731011851 8639188599326558455791230650166200511678790534528086950881088374551145 0895875000+28561628149763756200561443807577266534751285592123262315461 1070914192208615136563873961039661/95584388183264781981807207869172086 2905630122075590159379361564702076684706294966355821000000*7^(1/2), a[ 16,7] = 46524774257782474024979595329864014557423886513194065784729256 922453947551327371902009709751399763/178749887535033474724915051815187 443064002962286337680548268494761712979027730922844930286328125000+591 7947991674583424374460853714087059815712755844405621829850895301134910 279168916803272958541371/142999910028026779779932041452149954451202369 8290701444386147958093703832221847382759442290625000000*7^(1/2), a[16, 13] = 7657600557578519236927356742918368477397412237942153506954797647 3129717157/55437992885728406227840443875374037377596717643306127691712 96320800781250000+1189094049890461233623674097341247034240151862866286 37854661183118141741311/8870078861716544996454471020059845980415474822 9289804306740741132812500000000*7^(1/2), a[16,11] = -96158633263000021 7361760815186779719927548025597247352446620287932654896979515634838072 19489943/2266207487042601159749295745587544379598026513632294759378289 2225347467590606715447437500000000-84809796822306017166887302540539985 441826761552922862300464496874472373645260113147330694619/209834026578 0186259027125690358837388516691216326198851276193724569209962093214393 28125000000*7^(1/2), a[16,15] = 12635888347363109520131358222163810669 5002907774309390964517191171456273477/20622933353490967116756645121639 14190446597896330987950131722231337890625000-6671871692162193835322902 29658720215318230956208501949338288072118597379/4124586670698193423351 3290243278283808931957926619759002634444626757812500*7^(1/2), a[16,6] \+ = 49533733144813936314720097158218450549258878950176451985891897198080 23283132738773826477363486812/1609577127855048565864775134835217928822 9798843001789591740513389498327596076419765602058349609375+12003592096 3661763421724642843654652677566912614024166810909682222079969055372098 2691932935469701/38152939326934484524202077270168128683211375035263501 254496031738070109857366328333278953125000000*7^(1/2), a[16,10] = 1593 8572853277337601245712630986530453929893615425271539167654240226372075 35538065172875193509/5690176073295282674253116222513454642673066630220 26387368840720992228188711915769497165687500+3074120446596300747566954 0689503689743154973507615199658230759885748954946271746253192825123/33 7195619158238973289073553926723238084329874383423044366720427254653741 458913048590913000000*7^(1/2), a[16,9] = -3184052033230814076928413758 2614147196058544487888858929436357101564990679775786021969042971007961 2049302255526981662229357155593151511164687774987839435430833882792338 90381/1916597683915550813698243244628054389580334115392664767756692974 2465917121958272305909225307798213596172216491160467719850036366151584 437340953031830090690571875000000000000+414201402564698416863698436554 0791069521463268326411298504009416767913623345226168698251274678255070 2093947564073313337424266169065614041755234408498232180327610377394167 /383319536783110162739648648925610877916066823078532953551338594849318 3424391654461181845061559642719234443298232093543970007273230316887468 190606366018138114375000000000000*7^(1/2), a[16,14] = 9578806188437261 689373090827210246131323841838285462486649801851406702570303/499949899 478568899800161093857918591623417671837815260637993268203125000000000- 2860115831441802109832054006645866354942015998662300480233819046110729 90151/9998997989571377996003221877158371832468353436756305212759865364 062500000000*7^(1/2), a[16,12] = 2734753906564893127311360222281230937 36719752249770182523001316211267752774954161575132912540789/2993550665 5395543992957483547554285885024664375096226229774236480115214112504353 60468750000000000+2631889880176553367478291455572062331520747622973074 72568607403164903392978255666269313019069/2217444937436706962441295077 5966137692610862500071278688721656651937195638892113781250000000000*7^ (1/2), a[16,1] = 16746171708057136796041288508983162021370539915059038 36096367839971885317604473217871843621390982227/3971673185491125692358 0528752957310738091829268688077817226330743141912382660576772065625000 000000000-625472443421101973099323918176333336959929577122180411397436 07093656896553066548038906825979321/9806600458002779487303834259989459 4415041553749847105721546495662078796006569325363125000000000000*7^(1/ 2), a[16,2] = 0, c[16] = 371/500, a[16,5] = 0, a[15,2] = 0, a[15,4] = \+ 0, a[15,5] = 0, a[15,3] = 0, a[15,12] = -1764488883660554363266503/153 994806895851258937600000, a[14,10] = 30448415149825325326308593750/151 27426330731345559308904251+5887803942383224482632816406250/74124389020 5835932406136308299*7^(1/2), c[15] = 9/20, a[10,5] = -1802692598031722 81163724663224981097/38100922558256871086579832832000000, a[10,7] = 31 8607235173649312405151265849660869927653414425413/67147167155589653031 32938072935465423910912000000, a[10,8] = 21208320243451908228184224553 5894/20022426044775672563822865371173879, a[9,6] = 7801251558438936413 23090552530431036567795592568497182701460674803126770111481625/1831104 2541273197219788987450715878685922610298086185950524144307362914310080 5376, a[9,8] = 1033284818445201560405683676728665685912400779697066804 6446015775000000/13127035500360336480738342487407279145379720286389501 65249582733679393783, a[9,7] = 664113122959911642134782135839106469928 140328160577035357155340392950009492511875/151784655985862481363330231 07295349175279765150089078301139943253016877823170816, a[9,4] = -51722 9431108566845837517565524698123003902533693369911413831527077231937246 9280000/12461938100480914589727863057121529836525707941023625292185093 6749076487132995191, c[6] = 93/200, a[9,5] = -120706792584692548079789 36441733187949484571516120469966534514296406891652614970375/2722031154 7616572217104781845311006994972840850483890150850769616734461403986280 96, a[8,5] = -1024030607959889/168929280000000, a[8,7] = 6070139212132 283/92502016000000, a[9,1] = -1472514264486215803881384708877264246346 044433307094207829051978044531801133057155/124689480162003200115705962 1643986024803301558393487900440453636168046069686436608, a[8,6] = 1501 408353528689/265697280000000, a[14,9] = -12871995741547923519131817349 4800977965419044818935750521702127672316317131710725250664367373074947 /959116489087682414594756234921281432382105390145754922786477192521185 77198224684430692759024543840-8821850820681776229067876050054725556364 44759904636504531314529608120034066591927204665390933748491981/1597888 0708200789027148638873788548663485875799828277013622710027402954961224 2324261534136534890037440*7^(1/2), a[14,8] = 7420070341602879832712890 6250/42881511753324799479599510367+28676647199217261041085964843750/21 01194075912915174500376007983*7^(1/2), a[14,1] = 894106556792647920643 8689/198753096356125278622613280+152838094177334666489948287/331122658 529304714185273724480*7^(1/2), a[14,11] = -1466584824552592368357500/6 02474973863778980873559-946677979546641857718938375/590425474386503401 25608782*7^(1/2), a[10,6] = 211276702141728028701282869920039408106552 21489/4679473877997892906145822697976708633673728000, a[5,4] = 3982992 /2907025, a[12,1] = -2866556991825663971778295329101033887534912787724 034363/868226711619262703011213925016143612030669233795338240, a[11,5] = -26053085959256534152588089363841/4377552804565683061011299942400, \+ a[11,4] = -996286030132538159613930889652/1635306888599616490546432567 5, a[11,6] = 20980822345096760292224086794978105312644533925634933539/ 3775889992007550803878727839115494641972212962174156800, a[10,9] = -26 9840492940084251872116648508712979856226984822951779370341395122671458 3/46954567491393431507700044208087114188467603590271755032561672817587 5000000, a[10,1] = -29055573360337415088538618442231036441314060511/22 674759891089577691327962602370597632000000000, a[11,1] = -234265984581 4086836951207140065609179073838476242943917/13584809613510567770222314 00139158760857532162795520000, a[10,4] = -2046274952459104910540336523 9069/454251913499893469596231268750, a[11,10] = -311552374371117306659 23206875/392862141594230515010338956291, a[13,1] = 4490186773775461685 1973/1014046409980231013380680, a[12,7] = 1657121559319846802171283690 913610698586256573484808662625/134314804112551464772591551049560935053 61644432088109056, a[12,8] = 34568537955467705221549582547696922637718 7500/74771167436930077221667203179551347546362089, a[12,9] = -32058909 6271707254279143431215272753400810277402321024057136157075724905616701 5230160352087048674542196011/94756954968396581478301512445127360498465 7747127257615372449205973192657306017239103491074738324033259120, a[6, 1] = 5611/114400, a[12,4] = -16957088714171468676387054358954754000/14 3690415119654683326368228101570221, a[12,5] = -45834939744845729129493 14673356033540575/451957703655250747157313034270335135744, a[11,7] = 8 90722993756379186418929622095833835264322635782294899/1392124200139511 2657501941955594013822830119803764736, a[12,6] = 234630538855340425865 6258473446184419154740172519949575/25672671640789540289274497830115148 6254183185289662464, a[11,9] = 300760669768102517834232497565452434946 672266195876496371874262392684852243925359864884962513/465544333750134 6455585065336604505603760824779615521285751892810315680492364106674524 398280000, a[11,8] = 161021426143124178389075121929246710833125/109972 07722131034650667041364346422894371443, a[12,11] = -612293360107076959 1613093993993358877250/1050517001510235513198246721302027675953\}: " } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 17" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order conditio ns can be adapted to give a method of stage by stage construction for \+ an interpolation scheme that avoids dealing with the weight polynomial s for a given stage (corresponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_16 := SimpleOrderConditions(7,16,'expanded'):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,17,25,27,32,61,63,64]:\ninterp_order_eqns17 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_16[ct],'interpo lation_order_condition'):\n interp_order_eqns17 := [op(interp_order_ eqns17),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be spec ified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 585 "interp_order_eqns17 := [add(a[17,i],i=1..16)=c[17],seq(op(StageOr derConditions(i,17..17,'expanded')),i=2..7),\n add(a[17,i]*add(a[i, j]*add(a[j,k]*add(a[k,l]*add(a[l,m]*c[m],\n m=2..l-1),l=2..k-1),k =2..j-1),j=2..i-1),i=2..16)=c[17]^6/720, ##17\n add(a[17,i]*add(a[i ,j]*c[j]^2*add(a[j,k]*c[k],k=2..j-1),j=2..i-1),i=2..16)=c[17]^6/60, ## 25\n add(a[17,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i =2..16)=c[17]^6/120, ##27\n add(a[17,i]*c[i]*add(a[i,j]*c[j]^4,j=2 ..i-1),i=2..16)=c[17]^7/35, ##61\n add(a[17,i]*add(a[i,j]*c[j]^5,j =2..i-1),i=2..16)=c[17]^7/42]: ##63" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the node " }{XPPEDIT 18 0 "c[17] = 6/25;" "6#/&%\"cG6#\"#<*&\"\"'\"\"\"\"#D!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "e12 := `union`(e11, \{c[17]=6/25,seq(a[17,i]=0,i=2..5)\}):\neqs_17 := expand(subs(e12,inte rp_order_eqns17)):\nnops(eqs_17);\nindets(eqs_17);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&% \"aG6$\"#<\"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$F'\"#5&F%6$F' \"#6&F%6$F'\"#7&F%6$F'\"#8&F%6$F'\"#9&F%6$F'\"#:&F%6$F'\"#;&F%6$F'\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:= 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e13 := solve(\{op(eqs _17)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e14 := `union`(e12,e13):\nse q(a[17,i]=subs(e14,a[17,i]),i=1..16):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "62/&%\"aG6$\"#<\"\"\"$\"I)*o^RQc')z/6,V`?un)*QWZ!#T/&F %6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F '\"\"'$\"Im\"=&\\FYf@VS>Eds*H7<)y>!#S/&F%6$F'\"\"($\"I$3nk:$yPF&Q8o$4D !G+v%)>#FD/&F%6$F'\"\")$!H#zVtm)[bu;dec3dA[SUQ)!#R/&F%6$F'\"\"*$\"Ib?w 9n\"G%>-I&*['y%RpLqKCFQ/&F%6$F'\"#5$!I(e!z@u))zsnhHyA\"QP[/l$GFQ/&F%6$ F'\"#6$\"ItG`$\\Zg@#eHVz*R0.l!=)H\"FQ/&F%6$F'\"#7$\"H\"RMLorL$>xTUQ!R* 4&zF;;F+/&F%6$F'\"#8$!H;B.s%H)>0$e8\\vo#og\\`g$F+/&F%6$F'\"#9$!I&4ypDV .N#=Vl3ZZU!HIkz&F+/&F%6$F'\"#:$!I]$[2ooTE\\sD(4\\#*RMmqH9FD/&F%6$F'\"# ;$!H&[EipMHd5?4,yYP*4c/#yFD" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#---------------------------------------------- -----------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check w hich of the (adapted) simple order conditions are satisfied at this st age." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_16) do\n tt := con vert(SO7_16[ct],'interpolation_order_condition'):\n if expand(subs(e 14,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop( recd);\nnops(recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\" \"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#= \"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\" #O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"# `\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 70 "#------------------------------------------------------ ---------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e14" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15213 "e14 := \{a[9,3] = 0, a[17,4] = 0, a[17,3] = 0, a[13,7] = 3847 749490868980348119500000/15517045062138271618141237517, a[13,8] = -137 34512432397741476562500000/875132892924995907746928783, a[13,11] = 282 035543183190840068750/12295407629873040425991, a[13,12] = -30681427293 6976936753/1299331183183744997286, c[3] = 341/3200, a[13,6] = 79163867 5191615279648100000/2235604725089973126411512319, a[14,2] = 0, c[2] = \+ 1/20, a[5,3] = -3899844/2907025, c[4] = 1023/6400, a[8,1] = -122110182 1869329/690812928000000, c[10] = 909/1000, a[13,10] = -979836368457773 9445312500000/308722986341456031822630699, a[13,9] = 12274765470313196 8784288120377406350503192342760069863982944435549696163422742153163306 84448207141/4893451474937155176503858341435109348888292806866096544828 96526796523353052166757299452852166040, a[9,2] = 0, a[8,2] = 0, a[11,2 ] = 0, a[11,3] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12, 2] = 0, a[7,2] = 0, a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[10,3] = 0, a[14,3] = 0, a[14,5] = 0, a[13,5] = 0, a[8,3] = 0, c[12] = 1, c[13] = 1, a[6,3] = 0, a[5,2] = 0, a[13,4] = 0, a[7,3] = 0, a[13 ,2] = 0, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100, a[8,4] = -125 /2, a[6,2] = 0, a[13,3] = 0, a[10,2] = 0, a[7,6] = 5611/283500, a[14,6 ] = 3037913416047823635649583750/15649233075629811884880586233-3024656 25814318865951896498250/5367686944941025476514041077919*7^(1/2), a[12, 10] = 40279545832706233433100438588458933210937500/8896460842799482846 916972126377338947215101, a[15,1] = 339349033530268807690405611/740259 3223688213516896000000+784699017603056191374015321/6151316175555109038 55616000000*7^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 183874328 794901398385760606250/760335208044775309288920638333-11478722909055440 7592495836250/37256425194193990155157111278317*7^(1/2), c[7] = 31/200, c[8] = 943/1000, a[17,5] = 0, c[9] = 7067558016280/7837150160667, a[3 ,2] = 116281/1024000, a[7,4] = 8602624/76559175, a[15,11] = -500958158 6726585160535323/182154187109230228533200+164995836726103803189698643/ 13162109004021797158528000*7^(1/2), a[15,8] = 417178092023823016728215 4375/180381659089210106588738944-32173135433579259286326757125/3056994 432985560753767049472*7^(1/2), a[15,6] = 39667151342526762946267729161 /264960560010663481648771830400-154459727654214653639901999/5591570455 455665877423584000*7^(1/2), a[15,7] = 331704901880412220279128473073/1 418701262824070547944341715840-1100134249641546169459352427/1326901352 89901616433045894400*7^(1/2), a[15,14] = 708939/16000000*7^(1/2), a[15 ,13] = -79893/16000000+101277/64000000*7^(1/2), a[15,10] = 66598288816 84523938253413125/487858546317362618188848512-105274812306435785357342 625/16947939574300438348992256*7^(1/2), a[15,9] = -1537277560253172821 5983123367589049442461995162196337284574757119670408788921748236494825 887253876104043/166377350147863276001131183608793717862201955433447282 5241848191108179400377366974818139697364536000000+40573243814860756154 1740662873496460776199401666888744396064357816739210992744431247131193 9993/96852668533426148027403526835032792014972524881166166688187363374 7980768250399197689344000000*7^(1/2), a[16,3] = 0, a[16,4] = 0, a[3,1] = -7161/1024000, c[5] = 39/100, a[14,12] = 11352128098668146659861/25 4668911904014019468056-5215842639928607924801/127334455952007009734028 *7^(1/2), a[2,1] = 1/20, a[4,1] = 1023/25600, a[6,5] = 923521/5106400, a[7,1] = 21173/343200, c[11] = 47/50, a[6,4] = 31744/135025, a[16,8] \+ = 98379382275525268694725052370138586269885671336545475522545206455322 28606707500790611669969401/3225973101185186391885993265584557912306501 662005116787905345280869508810883745511450895875000+285616281497637562 0056144380757726653475128559212326231546110709141922086151365638739610 39661/9558438818326478198180720786917208629056301220755901593793615647 02076684706294966355821000000*7^(1/2), a[16,7] = 465247742577824740249 7959532986401455742388651319406578472925692245394755132737190200970975 1399763/17874988753503347472491505181518744306400296228633768054826849 4761712979027730922844930286328125000+59179479916745834243744608537140 87059815712755844405621829850895301134910279168916803272958541371/1429 9991002802677977993204145214995445120236982907014443861479580937038322 21847382759442290625000000*7^(1/2), a[16,13] = 76576005575785192369273 567429183684773974122379421535069547976473129717157/554379928857284062 2784044387537403737759671764330612769171296320800781250000+11890940498 9046123362367409734124703424015186286628637854661183118141741311/88700 7886171654499645447102005984598041547482292898043067407411328125000000 00*7^(1/2), a[16,11] = -9615863326300002173617608151867797199275480255 9724735244662028793265489697951563483807219489943/22662074870426011597 4929574558754437959802651363229475937828922253474675906067154474375000 00000-8480979682230601716688730254053998544182676155292286230046449687 4472373645260113147330694619/20983402657801862590271256903588373885166 9121632619885127619372456920996209321439328125000000*7^(1/2), a[16,15] = 1263588834736310952013135822216381066950029077743093909645171911714 56273477/2062293335349096711675664512163914190446597896330987950131722 231337890625000-667187169216219383532290229658720215318230956208501949 338288072118597379/412458667069819342335132902432782838089319579266197 59002634444626757812500*7^(1/2), a[16,6] = 495337331448139363147200971 5821845054925887895017645198589189719808023283132738773826477363486812 /160957712785504856586477513483521792882297988430017895917405133894983 27596076419765602058349609375+1200359209636617634217246428436546526775 669126140241668109096822220799690553720982691932935469701/381529393269 3448452420207727016812868321137503526350125449603173807010985736632833 3278953125000000*7^(1/2), a[16,10] = 159385728532773376012457126309865 3045392989361542527153916765424022637207535538065172875193509/56901760 7329528267425311622251345464267306663022026387368840720992228188711915 769497165687500+307412044659630074756695406895036897431549735076151996 58230759885748954946271746253192825123/3371956191582389732890735539267 23238084329874383423044366720427254653741458913048590913000000*7^(1/2) , a[16,9] = -318405203323081407692841375826141471960585444878888589294 3635710156499067977578602196904297100796120493022555269816622293571555 9315151116468777498783943543083388279233890381/19165976839155508136982 4324462805438958033411539266476775669297424659171219582723059092253077 9821359617221649116046771985003636615158443734095303183009069057187500 0000000000+41420140256469841686369843655407910695214632683264112985040 0941676791362334522616869825127467825507020939475640733133374242661690 65614041755234408498232180327610377394167/3833195367831101627396486489 2561087791606682307853295355133859484931834243916544611818450615596427 1923444329823209354397000727323031688746819060636601813811437500000000 0000*7^(1/2), a[16,14] = 957880618843726168937309082721024613132384183 8285462486649801851406702570303/49994989947856889980016109385791859162 3417671837815260637993268203125000000000-28601158314418021098320540066 4586635494201599866230048023381904611072990151/99989979895713779960032 21877158371832468353436756305212759865364062500000000*7^(1/2), a[16,12 ] = 273475390656489312731136022228123093736719752249770182523001316211 267752774954161575132912540789/299355066553955439929574835475542858850 2466437509622622977423648011521411250435360468750000000000+26318898801 7655336747829145557206233152074762297307472568607403164903392978255666 269313019069/221744493743670696244129507759661376926108625000712786887 21656651937195638892113781250000000000*7^(1/2), a[16,1] = 167461717080 5713679604128850898316202137053991505903836096367839971885317604473217 871843621390982227/397167318549112569235805287529573107380918292686880 77817226330743141912382660576772065625000000000000-6254724434211019730 9932391817633333695992957712218041139743607093656896553066548038906825 979321/980660045800277948730383425998945944150415537498471057215464956 62078796006569325363125000000000000*7^(1/2), a[16,2] = 0, c[16] = 371/ 500, a[16,5] = 0, a[15,2] = 0, a[15,4] = 0, a[15,5] = 0, a[15,3] = 0, \+ a[17,2] = 0, a[15,12] = -1764488883660554363266503/1539948068958512589 37600000, a[17,11] = 118364493841744414303786169254455656328/227702025 01285002774702584389480484375-353315848337304788532673730285272096/239 686342118789502891606151468215625*7^(1/2), a[17,8] = -3650321340106846 54668509433415077086432/103723686305239653444288515533579762663+553156 3793954378477130959983247997120/5459141384486297549699395554398934877* 7^(1/2), a[17,16] = -10265122569721941440000/49674155541527163507147+2 411555846605400000000/49674155541527163507147*7^(1/2), a[17,1] = 75904 016597212274352620201077585695621748/155516935187999420106352599763572 0322265625-160311321137339876236889599190517742013/3110338703759988402 12705199527144064453125*7^(1/2), a[17,15] = -10969058169805055883776/5 6011535488023799921875+8953379007044874752/448092283904190399375*7^(1/ 2), a[17,13] = -43884412335325588358/2956370631395177734375+1255812007 0786700736/2956370631395177734375*7^(1/2), a[17,9] = 24635436365841922 2299343662240541497501411933822141129747478653250964620305936849758369 4524067255325664382625575544/38231649647625848841557333871463334154707 0640922873677355229421588715318582472671110079488595970841204373017578 125-115920268168677580544994926840509851653431625841698719263191663083 353612400657431413037172143283745436569876116611/764632992952516976831 1466774292666830941412818457473547104588431774306371649453422201589771 9194168240874603515625*7^(1/2), a[17,6] = 2652112948108236779675852103 101319384090464/12420533056451560429826348580335898898078125-773102852 863002395047055595586658963648/130742453225805899261329985056167356821 875*7^(1/2), a[17,12] = -1271868983299399472812234477872349348/3007833 8034769176569907176844579296875+3752601621753783651501005551584508/226 152917554655462931632908605859375*7^(1/2), a[17,10] = -875118359158215 911384486198234977080928/109772652071629208992639244024476495017+30439 516745616652500404896664815444160/156818074388041727132341777177823564 31*7^(1/2), a[17,7] = 32655056064704801750397863652882675227617248/155 176782436333689589257284317258630419371875+110378602051450377273915565 181129491329216/31035356487266737917851456863451726083874375*7^(1/2), \+ a[17,14] = 196969534835590563883592/2582599130067348931640625-52411289 24571054652928/103303965202693957265625*7^(1/2), a[14,10] = 3044841514 9825325326308593750/15127426330731345559308904251+58878039423832244826 32816406250/741243890205835932406136308299*7^(1/2), c[15] = 9/20, a[10 ,5] = -180269259803172281163724663224981097/38100922558256871086579832 832000000, a[10,7] = 3186072351736493124051512658496608699276534144254 13/6714716715558965303132938072935465423910912000000, a[10,8] = 212083 202434519082281842245535894/20022426044775672563822865371173879, a[9,6 ] = 780125155843893641323090552530431036567795592568497182701460674803 126770111481625/183110425412731972197889874507158786859226102980861859 505241443073629143100805376, a[9,8] = 10332848184452015604056836767286 656859124007796970668046446015775000000/131270355003603364807383424874 0727914537972028638950165249582733679393783, a[9,7] = 6641131229599116 42134782135839106469928140328160577035357155340392950009492511875/1517 8465598586248136333023107295349175279765150089078301139943253016877823 170816, a[9,4] = -5172294311085668458375175655246981230039025336933699 114138315270772319372469280000/124619381004809145897278630571215298365 257079410236252921850936749076487132995191, c[6] = 93/200, a[9,5] = -1 2070679258469254807978936441733187949484571516120469966534514296406891 652614970375/272203115476165722171047818453110069949728408504838901508 5076961673446140398628096, a[8,5] = -1024030607959889/168929280000000, a[8,7] = 6070139212132283/92502016000000, a[9,1] = -14725142644862158 03881384708877264246346044433307094207829051978044531801133057155/1246 8948016200320011570596216439860248033015583934879004404536361680460696 86436608, a[8,6] = 1501408353528689/265697280000000, a[14,9] = -128719 9574154792351913181734948009779654190448189357505217021276723163171317 10725250664367373074947/9591164890876824145947562349212814323821053901 4575492278647719252118577198224684430692759024543840-88218508206817762 2906787605005472555636444759904636504531314529608120034066591927204665 390933748491981/159788807082007890271486388737885486634858757998282770 136227100274029549612242324261534136534890037440*7^(1/2), a[14,8] = 74 200703416028798327128906250/42881511753324799479599510367+286766471992 17261041085964843750/2101194075912915174500376007983*7^(1/2), a[14,1] \+ = 8941065567926479206438689/198753096356125278622613280+15283809417733 4666489948287/331122658529304714185273724480*7^(1/2), a[14,11] = -1466 584824552592368357500/602474973863778980873559-94667797954664185771893 8375/59042547438650340125608782*7^(1/2), a[10,6] = 2112767021417280287 0128286992003940810655221489/46794738779978929061458226979767086336737 28000, a[5,4] = 3982992/2907025, a[12,1] = -28665569918256639717782953 29101033887534912787724034363/8682267116192627030112139250161436120306 69233795338240, a[11,5] = -26053085959256534152588089363841/4377552804 565683061011299942400, a[11,4] = -996286030132538159613930889652/16353 068885996164905464325675, a[11,6] = 2098082234509676029222408679497810 5312644533925634933539/37758899920075508038787278391154946419722129621 74156800, a[10,9] = -2698404929400842518721166485087129798562269848229 517793703413951226714583/469545674913934315077000442080871141884676035 902717550325616728175875000000, a[10,1] = -290555733603374150885386184 42231036441314060511/22674759891089577691327962602370597632000000000, \+ a[11,1] = -2342659845814086836951207140065609179073838476242943917/135 8480961351056777022231400139158760857532162795520000, a[10,4] = -20462 749524591049105403365239069/454251913499893469596231268750, a[11,10] = -31155237437111730665923206875/392862141594230515010338956291, c[17] \+ = 6/25, a[13,1] = 44901867737754616851973/1014046409980231013380680, a [12,7] = 1657121559319846802171283690913610698586256573484808662625/13 431480411255146477259155104956093505361644432088109056, a[12,8] = 3456 85379554677052215495825476969226377187500/7477116743693007722166720317 9551347546362089, a[12,9] = -32058909627170725427914343121527275340081 02774023210240571361570757249056167015230160352087048674542196011/9475 6954968396581478301512445127360498465774712725761537244920597319265730 6017239103491074738324033259120, a[6,1] = 5611/114400, a[12,4] = -1695 7088714171468676387054358954754000/14369041511965468332636822810157022 1, a[12,5] = -4583493974484572912949314673356033540575/451957703655250 747157313034270335135744, a[11,7] = 8907229937563791864189296220958338 35264322635782294899/1392124200139511265750194195559401382283011980376 4736, a[12,6] = 234630538855340425865625847344618441915474017251994957 5/256726716407895402892744978301151486254183185289662464, a[11,9] = 30 0760669768102517834232497565452434946672266195876496371874262392684852 243925359864884962513/465544333750134645558506533660450560376082477961 5521285751892810315680492364106674524398280000, a[11,8] = 161021426143 124178389075121929246710833125/109972077221310346506670413643464228943 71443, a[12,11] = -6122933601070769591613093993993358877250/1050517001 510235513198246721302027675953\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 18" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The stan dard (simple) order conditions can be adapted to give a method of stag e by stage construction for an interpolation scheme that avoids dealin g with the weight polynomials for a given stage (corresponding to an \+ \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_17 := SimpleOrderCon ditions(7,17,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,17,25,27,32,61 ,63,64]:\ninterp_order_eqns18 := []:\nfor ct in whch do\n temp_eqn : = convert(SO7_17[ct],'interpolation_order_condition'):\n interp_orde r_eqns18 := [op(interp_order_eqns18),temp_eqn];\nend do:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 590 "interp_order_eqns18 := [add(a[18,i ],i=1..17)=c[18],seq(op(StageOrderConditions(i,18..18,'expanded')),i=2 ..7),\n add(a[18,i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*add(a[l,m]*c[m ],\n m=2..l-1),l=2..k-1),k=2..j-1),j=2..i-1),i=2..17)=c[18]^ 6/720, ##17\n add(a[18,i]*add(a[i,j]*c[j]^2*add(a[j,k]*c[k],k=2..j- 1),j=2..i-1),i=2..17)=c[18]^6/60, ##25\n add(a[18,i]*add(a[i,j]*ad d(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..17)=c[18]^6/120, ##27\n ad d(a[18,i]*c[i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..17)=c[18]^7/35, ##61 \n add(a[18,i]*add(a[i,j]*c[j]^5,j=2..i-1),i=2..17)=c[18]^7/42]: # #63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "W e specify " }{XPPEDIT 18 0 "c[18] = 3/4;" "6#/&%\"cG6#\"#=*&\"\"$\"\" \"\"\"%!\"\"" }{TEXT -1 10 " and also " }{XPPEDIT 18 0 "a[18,17]=0" "6 #/&%\"aG6$\"#=\"#<\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "e15 := `union`(e14,\{c[18]=3/4,seq(a[18,i]=0,i=2..5) ,a[18,17]=0\}):\neqs_18 := expand(subs(e15,interp_order_eqns18)):\nnop s(eqs_18);\nindets(eqs_18);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#=\"\"(&F%6$F' \"#8&F%6$F'\"#9&F%6$F'\"\")&F%6$F'\"\"*&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\" #7&F%6$F'\"#:&F%6$F'\"#;&F%6$F'\"\"'&F%6$F'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "e16 := solve(\{op(eqs_18)\}):\ninfolevel[sol ve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e17 := `union`(e15,e16):\nseq(a[18,i]=subs(e17,a[18,i ]),i=1..17):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "63/&%\"aG 6$\"#=\"\"\"$\"In1sAI[i9cOx)R&RW**H+,V!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F' \"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"IiH'zc&==NFNHAhP; ]G4'y\"!#S/&F%6$F'\"\"($\"I1JR,yZ1_\\qp'pCm'=B$Qb#FD/&F%6$F'\"\")$\"Ig :zfLS$e%=_l;TJyB'o2![!#R/&F%6$F'\"\"*$!I)QNend56m$G;o#RUm^m7#fFQ/&F%6$ F'\"#5$\"I%GR-Yb%z*z+-$[R?yYWKzyFQ/&F%6$F'\"#6$!I`S2LmX*ybGH^znG6uR+!p FQ/&F%6$F'\"#7$\"IJRlg#4!QC)QNUcB`Lt#oe&e([![q4h&\\ " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_17) do\n tt := convert(SO7_17[c t],'interpolation_order_condition'):\n if expand(subs(e17,lhs(tt)=rh s(tt))) then recd := [op(recd),ct] end if: \nend do:\nop(recd);\nnops( recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"& \"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\" #A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R \"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c \"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#- --------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e17" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17381 "e17 := \{ a[9,3] = 0, a[17,4] = 0, a[17,3] = 0, a[13,7] = 3847749490868980348119 500000/15517045062138271618141237517, a[13,8] = -137345124323977414765 62500000/875132892924995907746928783, a[13,11] = 282035543183190840068 750/12295407629873040425991, a[13,12] = -306814272936976936753/1299331 183183744997286, c[3] = 341/3200, a[13,6] = 79163867519161527964810000 0/2235604725089973126411512319, a[14,2] = 0, c[2] = 1/20, a[5,3] = -38 99844/2907025, c[4] = 1023/6400, a[8,1] = -1221101821869329/6908129280 00000, c[10] = 909/1000, a[13,10] = -9798363684577739445312500000/3087 22986341456031822630699, a[13,9] = 12274765470313196878428812037740635 050319234276006986398294443554969616342274215316330684448207141/489345 1474937155176503858341435109348888292806866096544828965267965233530521 66757299452852166040, a[9,2] = 0, a[8,2] = 0, a[11,2] = 0, a[11,3] = 0 , a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12,2] = 0, a[7,2] = 0 , a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[10,3] = 0, a[14 ,3] = 0, a[14,5] = 0, a[13,5] = 0, a[8,3] = 0, c[12] = 1, c[13] = 1, a [6,3] = 0, a[5,2] = 0, a[13,4] = 0, a[7,3] = 0, a[13,2] = 0, a[4,3] = \+ 3069/25600, a[5,1] = 4202367/11628100, a[8,4] = -125/2, a[6,2] = 0, a[ 13,3] = 0, a[10,2] = 0, a[7,6] = 5611/283500, a[14,6] = 30379134160478 23635649583750/15649233075629811884880586233-3024656258143188659518964 98250/5367686944941025476514041077919*7^(1/2), a[12,10] = 402795458327 06233433100438588458933210937500/8896460842799482846916972126377338947 215101, a[15,1] = 339349033530268807690405611/740259322368821351689600 0000+784699017603056191374015321/615131617555510903855616000000*7^(1/2 ), a[7,5] = -26782109/689364000, a[14,7] = 183874328794901398385760606 250/760335208044775309288920638333-114787229090554407592495836250/3725 6425194193990155157111278317*7^(1/2), c[7] = 31/200, c[8] = 943/1000, \+ a[17,5] = 0, c[9] = 7067558016280/7837150160667, a[3,2] = 116281/10240 00, a[7,4] = 8602624/76559175, a[15,11] = -5009581586726585160535323/1 82154187109230228533200+164995836726103803189698643/131621090040217971 58528000*7^(1/2), a[15,8] = 4171780920238230167282154375/1803816590892 10106588738944-32173135433579259286326757125/3056994432985560753767049 472*7^(1/2), a[15,6] = 39667151342526762946267729161/26496056001066348 1648771830400-154459727654214653639901999/5591570455455665877423584000 *7^(1/2), a[15,7] = 331704901880412220279128473073/1418701262824070547 944341715840-1100134249641546169459352427/1326901352899016164330458944 00*7^(1/2), a[15,14] = 708939/16000000*7^(1/2), a[15,13] = -79893/1600 0000+101277/64000000*7^(1/2), a[15,10] = 6659828881684523938253413125/ 487858546317362618188848512-105274812306435785357342625/16947939574300 438348992256*7^(1/2), a[15,9] = -1537277560253172821598312336758904944 2461995162196337284574757119670408788921748236494825887253876104043/16 6377350147863276001131183608793717862201955433447282524184819110817940 0377366974818139697364536000000+40573243814860756154174066287349646077 61994016668887443960643578167392109927444312471311939993/9685266853342 6148027403526835032792014972524881166166688187363374798076825039919768 9344000000*7^(1/2), a[16,3] = 0, a[16,4] = 0, a[3,1] = -7161/1024000, \+ c[5] = 39/100, a[14,12] = 11352128098668146659861/25466891190401401946 8056-5215842639928607924801/127334455952007009734028*7^(1/2), a[2,1] = 1/20, a[4,1] = 1023/25600, a[6,5] = 923521/5106400, a[7,1] = 21173/34 3200, c[11] = 47/50, a[6,4] = 31744/135025, a[18,2] = 0, a[16,8] = 983 7938227552526869472505237013858626988567133654547552254520645532228606 707500790611669969401/322597310118518639188599326558455791230650166200 5116787905345280869508810883745511450895875000+28561628149763756200561 4438075772665347512855921232623154611070914192208615136563873961039661 /955843881832647819818072078691720862905630122075590159379361564702076 684706294966355821000000*7^(1/2), a[16,7] = 46524774257782474024979595 3298640145574238865131940657847292569224539475513273719020097097513997 63/1787498875350334747249150518151874430640029622863376805482684947617 12979027730922844930286328125000+5917947991674583424374460853714087059 815712755844405621829850895301134910279168916803272958541371/142999910 0280267797799320414521499544512023698290701444386147958093703832221847 382759442290625000000*7^(1/2), a[16,13] = 7657600557578519236927356742 9183684773974122379421535069547976473129717157/55437992885728406227840 44387537403737759671764330612769171296320800781250000+1189094049890461 23362367409734124703424015186286628637854661183118141741311/8870078861 7165449964544710200598459804154748229289804306740741132812500000000*7^ (1/2), a[16,11] = -961586332630000217361760815186779719927548025597247 35244662028793265489697951563483807219489943/2266207487042601159749295 7455875443795980265136322947593782892225347467590606715447437500000000 -848097968223060171668873025405399854418267615529228623004644968744723 73645260113147330694619/2098340265780186259027125690358837388516691216 32619885127619372456920996209321439328125000000*7^(1/2), a[16,15] = 12 6358883473631095201313582221638106695002907774309390964517191171456273 477/206229333534909671167566451216391419044659789633098795013172223133 7890625000-66718716921621938353229022965872021531823095620850194933828 8072118597379/41245866706981934233513290243278283808931957926619759002 634444626757812500*7^(1/2), a[16,6] = 49533733144813936314720097158218 45054925887895017645198589189719808023283132738773826477363486812/1609 5771278550485658647751348352179288229798843001789591740513389498327596 076419765602058349609375+120035920963661763421724642843654652677566912 6140241668109096822220799690553720982691932935469701/38152939326934484 5242020772701681286832113750352635012544960317380701098573663283332789 53125000000*7^(1/2), a[16,10] = 15938572853277337601245712630986530453 92989361542527153916765424022637207535538065172875193509/5690176073295 2826742531162225134546426730666302202638736884072099222818871191576949 7165687500+30741204465963007475669540689503689743154973507615199658230 759885748954946271746253192825123/337195619158238973289073553926723238 084329874383423044366720427254653741458913048590913000000*7^(1/2), a[1 6,9] = -31840520332308140769284137582614147196058544487888858929436357 1015649906797757860219690429710079612049302255526981662229357155593151 51116468777498783943543083388279233890381/1916597683915550813698243244 6280543895803341153926647677566929742465917121958272305909225307798213 5961722164911604677198500363661515844373409530318300906905718750000000 00000+4142014025646984168636984365540791069521463268326411298504009416 7679136233452261686982512746782550702093947564073313337424266169065614 041755234408498232180327610377394167/383319536783110162739648648925610 8779160668230785329535513385948493183424391654461181845061559642719234 443298232093543970007273230316887468190606366018138114375000000000000* 7^(1/2), a[16,14] = 95788061884372616893730908272102461313238418382854 62486649801851406702570303/4999498994785688998001610938579185916234176 71837815260637993268203125000000000-2860115831441802109832054006645866 35494201599866230048023381904611072990151/9998997989571377996003221877 158371832468353436756305212759865364062500000000*7^(1/2), a[16,12] = 2 7347539065648931273113602222812309373671975224977018252300131621126775 2774954161575132912540789/29935506655395543992957483547554285885024664 37509622622977423648011521411250435360468750000000000+2631889880176553 3674782914555720623315207476229730747256860740316490339297825566626931 3019069/22174449374367069624412950775966137692610862500071278688721656 651937195638892113781250000000000*7^(1/2), a[16,1] = 16746171708057136 7960412885089831620213705399150590383609636783997188531760447321787184 3621390982227/39716731854911256923580528752957310738091829268688077817 226330743141912382660576772065625000000000000-625472443421101973099323 9181763333369599295771221804113974360709365689655306654803890682597932 1/98066004580027794873038342599894594415041553749847105721546495662078 796006569325363125000000000000*7^(1/2), a[16,2] = 0, a[18,17] = 0, c[1 6] = 371/500, a[18,5] = 0, a[18,3] = 0, a[18,4] = 0, a[16,5] = 0, a[15 ,2] = 0, a[15,4] = 0, a[15,5] = 0, a[15,3] = 0, a[17,2] = 0, a[15,12] \+ = -1764488883660554363266503/153994806895851258937600000, a[18,15] = 1 791582608825903125/11471162467947274224+925013716834375/13035411895394 6298*7^(1/2), a[18,7] = 400496411534857632473620487549793366403125/158 9010252148056981393994591408728375494368+10035263151351682177508738931 80856009375/794505126074028490696997295704364187747184*7^(1/2), a[18,9 ] = -18027034777101113229452644098683106369586923840053703055239154013 70316026047873432829768143660422541770189474551/4008879026090892207488 4822921603537074606114437634318910643704597180875389793486358192270983 4008720786716641280-11549073395893898707178189409937016283047719903485 54758621285722737496241965692115980914214471862767372483/2145047368019 0979760760245557067546190061594755007929215390713573321673385303379719 72404675659525500490752*7^(1/2), a[18,8] = 127743831371014360950335687 24910537109375/3319157961767668910217232497074552405216+59720772884380 2066516925307526298828125/1659578980883834455108616248537276202608*7^( 1/2), a[18,6] = 4684662399542643647977751441516649198125/2543725169961 2795760284361892527920943264-26709437901843964919449276414071475625/12 718625849806397880142180946263960471632*7^(1/2), a[18,10] = 2127113527 2151061077797757004198544921875/35127248662921346877644558087832478405 44+172966360250610619535692995992880859375/250908919020866763411746843 484517702896*7^(1/2), a[18,13] = 13060395422189465/774994822796457472+ 3321435324453/2201689837489936*7^(1/2), a[18,16] = 1116476526565551757 8125/113540926952062088016336+1712896926116943359375/99348311083054327 014294*7^(1/2), a[18,11] = -257181496023244780272848520935056875/46633 374722631685682590892829656032-24412968277003167712734734927511875/466 33374722631685682590892829656032*7^(1/2), a[18,12] = 31325146179236954 14144270190280539/39424279228932655113708734833726976+8879404488073521 58464070945/150677935947549953424507673856*7^(1/2), a[18,14] = 4584404 187072404233/169253216588093779584-47650637886965575/26445815091889653 06*7^(1/2), a[18,1] = 70926771436976695009804379644612309403/163071325 8316916799374387836496873072640-29849566024332586811787951198583421/16 3071325831691679937438783649687307264*7^(1/2), a[17,11] = 118364493841 744414303786169254455656328/22770202501285002774702584389480484375-353 315848337304788532673730285272096/239686342118789502891606151468215625 *7^(1/2), a[17,8] = -365032134010684654668509433415077086432/103723686 305239653444288515533579762663+5531563793954378477130959983247997120/5 459141384486297549699395554398934877*7^(1/2), a[17,16] = -102651225697 21941440000/49674155541527163507147+2411555846605400000000/49674155541 527163507147*7^(1/2), a[17,1] = 75904016597212274352620201077585695621 748/1555169351879994201063525997635720322265625-1603113211373398762368 89599190517742013/311033870375998840212705199527144064453125*7^(1/2), \+ a[17,15] = -10969058169805055883776/56011535488023799921875+8953379007 044874752/448092283904190399375*7^(1/2), a[17,13] = -43884412335325588 358/2956370631395177734375+12558120070786700736/2956370631395177734375 *7^(1/2), a[17,9] = 24635436365841922229934366224054149750141193382214 11297474786532509646203059368497583694524067255325664382625575544/3823 1649647625848841557333871463334154707064092287367735522942158871531858 2472671110079488595970841204373017578125-11592026816867758054499492684 0509851653431625841698719263191663083353612400657431413037172143283745 436569876116611/764632992952516976831146677429266683094141281845747354 71045884317743063716494534222015897719194168240874603515625*7^(1/2), a [17,6] = 2652112948108236779675852103101319384090464/12420533056451560 429826348580335898898078125-773102852863002395047055595586658963648/13 0742453225805899261329985056167356821875*7^(1/2), a[17,12] = -12718689 83299399472812234477872349348/30078338034769176569907176844579296875+3 752601621753783651501005551584508/226152917554655462931632908605859375 *7^(1/2), a[17,10] = -875118359158215911384486198234977080928/10977265 2071629208992639244024476495017+30439516745616652500404896664815444160 /15681807438804172713234177717782356431*7^(1/2), a[17,7] = 32655056064 704801750397863652882675227617248/155176782436333689589257284317258630 419371875+110378602051450377273915565181129491329216/31035356487266737 917851456863451726083874375*7^(1/2), a[17,14] = 1969695348355905638835 92/2582599130067348931640625-5241128924571054652928/103303965202693957 265625*7^(1/2), a[14,10] = 30448415149825325326308593750/1512742633073 1345559308904251+5887803942383224482632816406250/741243890205835932406 136308299*7^(1/2), c[15] = 9/20, a[10,5] = -18026925980317228116372466 3224981097/38100922558256871086579832832000000, a[10,7] = 318607235173 649312405151265849660869927653414425413/671471671555896530313293807293 5465423910912000000, a[10,8] = 212083202434519082281842245535894/20022 426044775672563822865371173879, a[9,6] = 78012515584389364132309055253 0431036567795592568497182701460674803126770111481625/18311042541273197 2197889874507158786859226102980861859505241443073629143100805376, a[9, 8] = 10332848184452015604056836767286656859124007796970668046446015775 000000/131270355003603364807383424874072791453797202863895016524958273 3679393783, a[9,7] = 6641131229599116421347821358391064699281403281605 77035357155340392950009492511875/1517846559858624813633302310729534917 5279765150089078301139943253016877823170816, a[9,4] = -517229431108566 8458375175655246981230039025336933699114138315270772319372469280000/12 4619381004809145897278630571215298365257079410236252921850936749076487 132995191, c[6] = 93/200, a[9,5] = -1207067925846925480797893644173318 7949484571516120469966534514296406891652614970375/27220311547616572217 10478184531100699497284085048389015085076961673446140398628096, a[8,5] = -1024030607959889/168929280000000, a[8,7] = 6070139212132283/925020 16000000, a[9,1] = -14725142644862158038813847088772642463460444333070 94207829051978044531801133057155/1246894801620032001157059621643986024 803301558393487900440453636168046069686436608, a[8,6] = 15014083535286 89/265697280000000, a[14,9] = -128719957415479235191318173494800977965 419044818935750521702127672316317131710725250664367373074947/959116489 0876824145947562349212814323821053901457549227864771925211857719822468 4430692759024543840-88218508206817762290678760500547255563644475990463 6504531314529608120034066591927204665390933748491981/15978880708200789 0271486388737885486634858757998282770136227100274029549612242324261534 136534890037440*7^(1/2), a[14,8] = 74200703416028798327128906250/42881 511753324799479599510367+28676647199217261041085964843750/210119407591 2915174500376007983*7^(1/2), a[14,1] = 8941065567926479206438689/19875 3096356125278622613280+152838094177334666489948287/3311226585293047141 85273724480*7^(1/2), a[14,11] = -1466584824552592368357500/60247497386 3778980873559-946677979546641857718938375/59042547438650340125608782*7 ^(1/2), a[10,6] = 21127670214172802870128286992003940810655221489/4679 473877997892906145822697976708633673728000, a[5,4] = 3982992/2907025, \+ a[12,1] = -2866556991825663971778295329101033887534912787724034363/868 226711619262703011213925016143612030669233795338240, a[11,5] = -260530 85959256534152588089363841/4377552804565683061011299942400, a[11,4] = \+ -996286030132538159613930889652/16353068885996164905464325675, a[11,6] = 20980822345096760292224086794978105312644533925634933539/3775889992 007550803878727839115494641972212962174156800, a[10,9] = -269840492940 0842518721166485087129798562269848229517793703413951226714583/46954567 4913934315077000442080871141884676035902717550325616728175875000000, a [10,1] = -29055573360337415088538618442231036441314060511/226747598910 89577691327962602370597632000000000, a[11,1] = -2342659845814086836951 207140065609179073838476242943917/135848096135105677702223140013915876 0857532162795520000, a[10,4] = -20462749524591049105403365239069/45425 1913499893469596231268750, c[18] = 3/4, a[11,10] = -311552374371117306 65923206875/392862141594230515010338956291, c[17] = 6/25, a[13,1] = 44 901867737754616851973/1014046409980231013380680, a[12,7] = 16571215593 19846802171283690913610698586256573484808662625/1343148041125514647725 9155104956093505361644432088109056, a[12,8] = 345685379554677052215495 825476969226377187500/74771167436930077221667203179551347546362089, a[ 12,9] = -3205890962717072542791434312152727534008102774023210240571361 570757249056167015230160352087048674542196011/947569549683965814783015 1244512736049846577471272576153724492059731926573060172391034910747383 24033259120, a[6,1] = 5611/114400, a[12,4] = -169570887141714686763870 54358954754000/143690415119654683326368228101570221, a[12,5] = -458349 3974484572912949314673356033540575/45195770365525074715731303427033513 5744, a[11,7] = 890722993756379186418929622095833835264322635782294899 /13921242001395112657501941955594013822830119803764736, a[12,6] = 2346 305388553404258656258473446184419154740172519949575/256726716407895402 892744978301151486254183185289662464, a[11,9] = 3007606697681025178342 3249756545243494667226619587649637187426239268485224392535986488496251 3/46554433375013464555850653366045056037608247796155212857518928103156 80492364106674524398280000, a[11,8] = 16102142614312417838907512192924 6710833125/10997207722131034650667041364346422894371443, a[12,11] = -6 122933601070769591613093993993358877250/105051700151023551319824672130 2027675953\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 19" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order conditions can be adapted to give a method of stage by stage co nstruction for an interpolation scheme that avoids dealing with the we ight polynomials for a given stage (corresponding to an \"approximate \" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_18 := SimpleOrderConditions(7,1 8,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,17,25,27,32,61,63,64]:\ni nterp_order_eqns19 := []:\nfor ct in whch do\n temp_eqn := convert(S O7_18[ct],'interpolation_order_condition'):\n interp_order_eqns19 := [op(interp_order_eqns19),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order \+ conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 584 "interp_order_eqns19 := [add(a[19,i],i=1..18 )=c[19],seq(op(StageOrderConditions(i,19..19,'expanded')),i=2..7),\nad d(a[19,i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*add(a[l,m]*c[m],\n \+ m=2..l-1),l=2..k-1),k=2..j-1),j=2..i-1),i=2..18)=c[19]^6/720, ##17\n \+ add(a[19,i]*add(a[i,j]*c[j]^2*add(a[j,k]*c[k],k=2..j-1),j=2..i-1),i= 2..18)=c[19]^6/60, ##25\n add(a[19,i]*add(a[i,j]*add(a[j,k]*c[k]^3 ,k=2..j-1),j=2..i-1),i=2..18)=c[19]^6/120, ##27\n add(a[19,i]*c[i]* add(a[i,j]*c[j]^4,j=2..i-1),i=2..18)=c[19]^7/35, ##61\n add(a[19,i ]*add(a[i,j]*c[j]^5,j=2..i-1),i=2..18)=c[19]^7/42]: ##63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We specify " } {XPPEDIT 18 0 "c[19] = 7/10;" "6#/&%\"cG6#\"#>*&\"\"(\"\"\"\"#5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[19,17]=0" "6#/&%\"aG6$\"#>\"#<\"\" !" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[19,18] = 0;" "6#/&%\"aG6$\" #>\"#=\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "e18 := `union`(e17,\{c[19]=7/10,seq(a[19,i]=0,i=2..5),a[19,17]=0, a[19,18]=0\}):\neqs_19 := expand(subs(e18,interp_order_eqns19)):\nnops (eqs_19);\nindets(eqs_19);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#>\"#9&F%6$F'\"# :&F%6$F'\"#7&F%6$F'\"#8&F%6$F'\"#;&F%6$F'\"\"'&F%6$F'\"\"(&F%6$F'\"\") &F%6$F'\"\"*&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "e19 := solve(\{op(eqs_19)\}):\ninfolevel[sol ve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e20 := `union`(e18,e19):\nseq(a[19,i]=subs(e20,a[19,i ]),i=1..18):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "64/&%\"aG 6$\"#>\"\"\"$\"I)=ejKwd5P@&f4'*)o>IN?K%!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F '\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"I\\$pA:(GhM@h03a xjID!f\"=!#S/&F%6$F'\"\"($\"I-=Qa))=$*)=8y)pV8>T6\\RDFD/&F%6$F'\"\")$ \"Iy8nAN*49xSF:c\"RX#)[(\\K%!#R/&F%6$F'\"\"*$!I>/J%f.QDTEx&zTZAjv93_FQ /&F%6$F'\"#5$\"IWjx&z.V!['f2cYE2\\Qjl'pFQ/&F%6$F'\"#6$!IsGIVXVum&fR$o' ejI:$y2iFQ/&F%6$F'\"#7$\"I,7d_QL$)QE&>3&)QU-v[ss)F+/&F%6$F'\"#8$\"I\\$ 4n'*fy'zo=Fk!3@\\@R$))=F+/&F%6$F'\"#9$!I_C'[ZxgdGq>p:;7Be6Zi\"F+/&F%6$ F'\"#:$\"I#*f[oQ0kyQUx.Vs5c/%pm\"FD/&F%6$F'\"#;$\"I$>4W\\!*ysNtLL*37`G >*H!*)F+/&F%6$F'\"# " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_18) do\n tt : = convert(SO7_18[ct],'interpolation_order_condition'):\n if expand(s ubs(e20,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do: \nop(recd);\nnops(recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\" \"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#< \"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\" #N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"# _\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 70 "#------------------------------------------------------ ---------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19564 "e20 := \{a[9,3] = 0, a[17,4] = 0, a[17,3] = 0, a[13,7] = 3847 749490868980348119500000/15517045062138271618141237517, a[13,8] = -137 34512432397741476562500000/875132892924995907746928783, a[13,11] = 282 035543183190840068750/12295407629873040425991, a[13,12] = -30681427293 6976936753/1299331183183744997286, c[3] = 341/3200, a[13,6] = 79163867 5191615279648100000/2235604725089973126411512319, a[14,2] = 0, c[2] = \+ 1/20, a[5,3] = -3899844/2907025, c[4] = 1023/6400, a[8,1] = -122110182 1869329/690812928000000, c[10] = 909/1000, a[13,10] = -979836368457773 9445312500000/308722986341456031822630699, a[13,9] = 12274765470313196 8784288120377406350503192342760069863982944435549696163422742153163306 84448207141/4893451474937155176503858341435109348888292806866096544828 96526796523353052166757299452852166040, a[9,2] = 0, a[8,2] = 0, a[11,2 ] = 0, a[11,3] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12, 2] = 0, a[7,2] = 0, a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[10,3] = 0, a[14,3] = 0, a[14,5] = 0, a[13,5] = 0, a[8,3] = 0, c[12] = 1, c[13] = 1, a[6,3] = 0, a[5,2] = 0, a[13,4] = 0, a[7,3] = 0, a[13 ,2] = 0, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100, a[8,4] = -125 /2, a[6,2] = 0, a[13,3] = 0, a[10,2] = 0, a[7,6] = 5611/283500, a[14,6 ] = 3037913416047823635649583750/15649233075629811884880586233-3024656 25814318865951896498250/5367686944941025476514041077919*7^(1/2), a[12, 10] = 40279545832706233433100438588458933210937500/8896460842799482846 916972126377338947215101, a[15,1] = 339349033530268807690405611/740259 3223688213516896000000+784699017603056191374015321/6151316175555109038 55616000000*7^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 183874328 794901398385760606250/760335208044775309288920638333-11478722909055440 7592495836250/37256425194193990155157111278317*7^(1/2), c[7] = 31/200, c[8] = 943/1000, a[17,5] = 0, c[9] = 7067558016280/7837150160667, a[3 ,2] = 116281/1024000, a[7,4] = 8602624/76559175, a[15,11] = -500958158 6726585160535323/182154187109230228533200+164995836726103803189698643/ 13162109004021797158528000*7^(1/2), a[15,8] = 417178092023823016728215 4375/180381659089210106588738944-32173135433579259286326757125/3056994 432985560753767049472*7^(1/2), a[15,6] = 39667151342526762946267729161 /264960560010663481648771830400-154459727654214653639901999/5591570455 455665877423584000*7^(1/2), a[15,7] = 331704901880412220279128473073/1 418701262824070547944341715840-1100134249641546169459352427/1326901352 89901616433045894400*7^(1/2), a[15,14] = 708939/16000000*7^(1/2), a[15 ,13] = -79893/16000000+101277/64000000*7^(1/2), a[15,10] = 66598288816 84523938253413125/487858546317362618188848512-105274812306435785357342 625/16947939574300438348992256*7^(1/2), a[15,9] = -1537277560253172821 5983123367589049442461995162196337284574757119670408788921748236494825 887253876104043/166377350147863276001131183608793717862201955433447282 5241848191108179400377366974818139697364536000000+40573243814860756154 1740662873496460776199401666888744396064357816739210992744431247131193 9993/96852668533426148027403526835032792014972524881166166688187363374 7980768250399197689344000000*7^(1/2), a[16,3] = 0, a[16,4] = 0, a[19,8 ] = 2859387143344524736201037565051296515625/8001541514975630408559399 76973329597686+4208420674863241824630827149071356250/14817669472177093 349184073647654251809*7^(1/2), a[19,7] = 35655064984394480210827873928 75980383385/14187591537036223048160665994720789066914+4184424985720086 9530169750656331246/41975122890639713160238656789114760553*7^(1/2), a[ 19,15] = 365031149291536/65177059476973149*7^(1/2)+544436534587547644/ 3584738271233523195, a[19,1] = 535655479158074959687111905785126128324 39/1228494921165087990600124765218068442000000-32866372517193693834368 0176682875249/2274990594750162945555786602255682300000*7^(1/2), a[19,1 3] = 2381481586979950007/151366176327433100000+511997202259998/4300175 46384753125*7^(1/2), a[19,9] = -45680153313647226875167598116159240355 0264322861223910175490376607996825535317233283995012274112360895978616 38289/1118548835404824834678705996696527262126286675157207558890728364 8766985320812914720477754180636404039807942000000-21605000076188669933 2908318984753896339403089055559398626500021503254015743998006035396648 20101981262758272973/5084312888203749248539572712256942100574030341623 6707222305834767122660549149612365807973548347291090036100000*7^(1/2), a[19,16] = 96563932308593750000/7096307934503880501021*7^(1/2)+376299 572125585937500/7096307934503880501021, a[19,12] = 2226538932772419221 283628295549708587/29700210356617792635885486621390300000+137604393969 352284402586088077/29586008364331473149528307355000*7^(1/2), a[19,10] \+ = 1560143195331346599933913496950810484375/282272533898475108838215198 920082415758+50485505364845502056312650212481250/927917599929240988948 76791229481399*7^(1/2), a[19,14] = 353341535891060455739/1652863443243 1033162500-2350501240657313744/165286344324310331625*7^(1/2), a[19,11] = -114996879666326606709923843682437653/22483948526983134168392037614 298444-430085025149565428288387897266111/10409235429158858411292610006 61965*7^(1/2), a[3,1] = -7161/1024000, c[5] = 39/100, a[14,12] = 11352 128098668146659861/254668911904014019468056-5215842639928607924801/127 334455952007009734028*7^(1/2), a[2,1] = 1/20, a[4,1] = 1023/25600, a[1 9,6] = -941084192745729767895274038102396218/5677957968663570482206330 77958212521055*7^(1/2)+380145805841840287029128269834313988041/2044064 868718885373594279080649565075798, a[6,5] = 923521/5106400, a[7,1] = 2 1173/343200, c[11] = 47/50, a[6,4] = 31744/135025, a[18,2] = 0, a[19,2 ] = 0, a[16,8] = 98379382275525268694725052370138586269885671336545475 52254520645532228606707500790611669969401/3225973101185186391885993265 584557912306501662005116787905345280869508810883745511450895875000+285 6162814976375620056144380757726653475128559212326231546110709141922086 15136563873961039661/9558438818326478198180720786917208629056301220755 90159379361564702076684706294966355821000000*7^(1/2), a[16,7] = 465247 7425778247402497959532986401455742388651319406578472925692245394755132 7371902009709751399763/17874988753503347472491505181518744306400296228 6337680548268494761712979027730922844930286328125000+59179479916745834 2437446085371408705981571275584440562182985089530113491027916891680327 2958541371/14299991002802677977993204145214995445120236982907014443861 47958093703832221847382759442290625000000*7^(1/2), a[16,13] = 76576005 575785192369273567429183684773974122379421535069547976473129717157/554 3799288572840622784044387537403737759671764330612769171296320800781250 000+118909404989046123362367409734124703424015186286628637854661183118 141741311/887007886171654499645447102005984598041547482292898043067407 41132812500000000*7^(1/2), a[16,11] = -9615863326300002173617608151867 7971992754802559724735244662028793265489697951563483807219489943/22662 0748704260115974929574558754437959802651363229475937828922253474675906 06715447437500000000-8480979682230601716688730254053998544182676155292 2862300464496874472373645260113147330694619/20983402657801862590271256 9035883738851669121632619885127619372456920996209321439328125000000*7^ (1/2), a[16,15] = 1263588834736310952013135822216381066950029077743093 90964517191171456273477/2062293335349096711675664512163914190446597896 330987950131722231337890625000-667187169216219383532290229658720215318 230956208501949338288072118597379/412458667069819342335132902432782838 08931957926619759002634444626757812500*7^(1/2), a[16,6] = 495337331448 1393631472009715821845054925887895017645198589189719808023283132738773 826477363486812/160957712785504856586477513483521792882297988430017895 91740513389498327596076419765602058349609375+1200359209636617634217246 4284365465267756691261402416681090968222207996905537209826919329354697 01/3815293932693448452420207727016812868321137503526350125449603173807 0109857366328333278953125000000*7^(1/2), a[16,10] = 159385728532773376 0124571263098653045392989361542527153916765424022637207535538065172875 193509/569017607329528267425311622251345464267306663022026387368840720 992228188711915769497165687500+307412044659630074756695406895036897431 54973507615199658230759885748954946271746253192825123/3371956191582389 7328907355392672323808432987438342304436672042725465374145891304859091 3000000*7^(1/2), a[16,9] = -318405203323081407692841375826141471960585 4448788885892943635710156499067977578602196904297100796120493022555269 8166222935715559315151116468777498783943543083388279233890381/19165976 8391555081369824324462805438958033411539266476775669297424659171219582 7230590922530779821359617221649116046771985003636615158443734095303183 0090690571875000000000000+41420140256469841686369843655407910695214632 6832641129850400941676791362334522616869825127467825507020939475640733 13337424266169065614041755234408498232180327610377394167/3833195367831 1016273964864892561087791606682307853295355133859484931834243916544611 8184506155964271923444329823209354397000727323031688746819060636601813 8114375000000000000*7^(1/2), a[16,14] = 957880618843726168937309082721 0246131323841838285462486649801851406702570303/49994989947856889980016 1093857918591623417671837815260637993268203125000000000-28601158314418 0210983205400664586635494201599866230048023381904611072990151/99989979 89571377996003221877158371832468353436756305212759865364062500000000*7 ^(1/2), a[16,12] = 273475390656489312731136022228123093736719752249770 182523001316211267752774954161575132912540789/299355066553955439929574 8354755428588502466437509622622977423648011521411250435360468750000000 000+263188988017655336747829145557206233152074762297307472568607403164 903392978255666269313019069/221744493743670696244129507759661376926108 62500071278688721656651937195638892113781250000000000*7^(1/2), a[16,1] = 1674617170805713679604128850898316202137053991505903836096367839971 885317604473217871843621390982227/397167318549112569235805287529573107 38091829268688077817226330743141912382660576772065625000000000000-6254 7244342110197309932391817633333695992957712218041139743607093656896553 066548038906825979321/980660045800277948730383425998945944150415537498 47105721546495662078796006569325363125000000000000*7^(1/2), a[16,2] = \+ 0, a[19,17] = 0, a[19,18] = 0, a[18,17] = 0, c[19] = 7/10, c[16] = 371 /500, a[18,5] = 0, a[18,3] = 0, a[18,4] = 0, a[16,5] = 0, a[15,2] = 0, a[15,4] = 0, a[15,5] = 0, a[15,3] = 0, a[17,2] = 0, a[15,12] = -17644 88883660554363266503/153994806895851258937600000, a[19,3] = 0, a[19,4] = 0, a[19,5] = 0, a[18,15] = 1791582608825903125/11471162467947274224 +925013716834375/130354118953946298*7^(1/2), a[18,7] = 400496411534857 632473620487549793366403125/158901025214805698139399459140872837549436 8+1003526315135168217750873893180856009375/794505126074028490696997295 704364187747184*7^(1/2), a[18,9] = -1802703477710111322945264409868310 6369586923840053703055239154013703160260478734328297681436604225417701 89474551/4008879026090892207488482292160353707460611443763431891064370 45971808753897934863581922709834008720786716641280-1154907339589389870 7178189409937016283047719903485547586212857227374962419656921159809142 14471862767372483/2145047368019097976076024555706754619006159475500792 921539071357332167338530337971972404675659525500490752*7^(1/2), a[18,8 ] = 12774383137101436095033568724910537109375/331915796176766891021723 2497074552405216+597207728843802066516925307526298828125/1659578980883 834455108616248537276202608*7^(1/2), a[18,6] = 46846623995426436479777 51441516649198125/25437251699612795760284361892527920943264-2670943790 1843964919449276414071475625/12718625849806397880142180946263960471632 *7^(1/2), a[18,10] = 21271135272151061077797757004198544921875/3512724 866292134687764455808783247840544+172966360250610619535692995992880859 375/250908919020866763411746843484517702896*7^(1/2), a[18,13] = 130603 95422189465/774994822796457472+3321435324453/2201689837489936*7^(1/2), a[18,16] = 11164765265655517578125/113540926952062088016336+171289692 6116943359375/99348311083054327014294*7^(1/2), a[18,11] = -25718149602 3244780272848520935056875/46633374722631685682590892829656032-24412968 277003167712734734927511875/46633374722631685682590892829656032*7^(1/2 ), a[18,12] = 3132514617923695414144270190280539/394242792289326551137 08734833726976+887940448807352158464070945/150677935947549953424507673 856*7^(1/2), a[18,14] = 4584404187072404233/169253216588093779584-4765 0637886965575/2644581509188965306*7^(1/2), a[18,1] = 70926771436976695 009804379644612309403/1630713258316916799374387836496873072640-2984956 6024332586811787951198583421/163071325831691679937438783649687307264*7 ^(1/2), a[17,11] = 118364493841744414303786169254455656328/22770202501 285002774702584389480484375-353315848337304788532673730285272096/23968 6342118789502891606151468215625*7^(1/2), a[17,8] = -365032134010684654 668509433415077086432/103723686305239653444288515533579762663+55315637 93954378477130959983247997120/5459141384486297549699395554398934877*7^ (1/2), a[17,16] = -10265122569721941440000/49674155541527163507147+241 1555846605400000000/49674155541527163507147*7^(1/2), a[17,1] = 7590401 6597212274352620201077585695621748/15551693518799942010635259976357203 22265625-160311321137339876236889599190517742013/311033870375998840212 705199527144064453125*7^(1/2), a[17,15] = -10969058169805055883776/560 11535488023799921875+8953379007044874752/448092283904190399375*7^(1/2) , a[17,13] = -43884412335325588358/2956370631395177734375+125581200707 86700736/2956370631395177734375*7^(1/2), a[17,9] = 2463543636584192222 9934366224054149750141193382214112974747865325096462030593684975836945 24067255325664382625575544/3823164964762584884155733387146333415470706 4092287367735522942158871531858247267111007948859597084120437301757812 5-11592026816867758054499492684050985165343162584169871926319166308335 3612400657431413037172143283745436569876116611/76463299295251697683114 6677429266683094141281845747354710458843177430637164945342220158977191 94168240874603515625*7^(1/2), a[17,6] = 265211294810823677967585210310 1319384090464/12420533056451560429826348580335898898078125-77310285286 3002395047055595586658963648/13074245322580589926132998505616735682187 5*7^(1/2), a[17,12] = -1271868983299399472812234477872349348/300783380 34769176569907176844579296875+3752601621753783651501005551584508/22615 2917554655462931632908605859375*7^(1/2), a[17,10] = -87511835915821591 1384486198234977080928/109772652071629208992639244024476495017+3043951 6745616652500404896664815444160/15681807438804172713234177717782356431 *7^(1/2), a[17,7] = 32655056064704801750397863652882675227617248/15517 6782436333689589257284317258630419371875+11037860205145037727391556518 1129491329216/31035356487266737917851456863451726083874375*7^(1/2), a[ 17,14] = 196969534835590563883592/2582599130067348931640625-5241128924 571054652928/103303965202693957265625*7^(1/2), a[14,10] = 304484151498 25325326308593750/15127426330731345559308904251+5887803942383224482632 816406250/741243890205835932406136308299*7^(1/2), c[15] = 9/20, a[10,5 ] = -180269259803172281163724663224981097/3810092255825687108657983283 2000000, a[10,7] = 318607235173649312405151265849660869927653414425413 /6714716715558965303132938072935465423910912000000, a[10,8] = 21208320 2434519082281842245535894/20022426044775672563822865371173879, a[9,6] \+ = 78012515584389364132309055253043103656779559256849718270146067480312 6770111481625/18311042541273197219788987450715878685922610298086185950 5241443073629143100805376, a[9,8] = 1033284818445201560405683676728665 6859124007796970668046446015775000000/13127035500360336480738342487407 27914537972028638950165249582733679393783, a[9,7] = 664113122959911642 134782135839106469928140328160577035357155340392950009492511875/151784 6559858624813633302310729534917527976515008907830113994325301687782317 0816, a[9,4] = -517229431108566845837517565524698123003902533693369911 4138315270772319372469280000/12461938100480914589727863057121529836525 7079410236252921850936749076487132995191, c[6] = 93/200, a[9,5] = -120 7067925846925480797893644173318794948457151612046996653451429640689165 2614970375/27220311547616572217104781845311006994972840850483890150850 76961673446140398628096, a[8,5] = -1024030607959889/168929280000000, a [8,7] = 6070139212132283/92502016000000, a[9,1] = -1472514264486215803 881384708877264246346044433307094207829051978044531801133057155/124689 4801620032001157059621643986024803301558393487900440453636168046069686 436608, a[8,6] = 1501408353528689/265697280000000, a[14,9] = -12871995 7415479235191318173494800977965419044818935750521702127672316317131710 725250664367373074947/959116489087682414594756234921281432382105390145 75492278647719252118577198224684430692759024543840-8821850820681776229 0678760500547255563644475990463650453131452960812003406659192720466539 0933748491981/15978880708200789027148638873788548663485875799828277013 6227100274029549612242324261534136534890037440*7^(1/2), a[14,8] = 7420 0703416028798327128906250/42881511753324799479599510367+28676647199217 261041085964843750/2101194075912915174500376007983*7^(1/2), a[14,1] = \+ 8941065567926479206438689/198753096356125278622613280+1528380941773346 66489948287/331122658529304714185273724480*7^(1/2), a[14,11] = -146658 4824552592368357500/602474973863778980873559-9466779795466418577189383 75/59042547438650340125608782*7^(1/2), a[10,6] = 211276702141728028701 28286992003940810655221489/4679473877997892906145822697976708633673728 000, a[5,4] = 3982992/2907025, a[12,1] = -2866556991825663971778295329 101033887534912787724034363/868226711619262703011213925016143612030669 233795338240, a[11,5] = -26053085959256534152588089363841/437755280456 5683061011299942400, a[11,4] = -996286030132538159613930889652/1635306 8885996164905464325675, a[11,6] = 209808223450967602922240867949781053 12644533925634933539/3775889992007550803878727839115494641972212962174 156800, a[10,9] = -269840492940084251872116648508712979856226984822951 7793703413951226714583/46954567491393431507700044208087114188467603590 2717550325616728175875000000, a[10,1] = -29055573360337415088538618442 231036441314060511/22674759891089577691327962602370597632000000000, a[ 11,1] = -2342659845814086836951207140065609179073838476242943917/13584 80961351056777022231400139158760857532162795520000, a[10,4] = -2046274 9524591049105403365239069/454251913499893469596231268750, c[18] = 3/4, a[11,10] = -31155237437111730665923206875/392862141594230515010338956 291, c[17] = 6/25, a[13,1] = 44901867737754616851973/10140464099802310 13380680, a[12,7] = 16571215593198468021712836909136106985862565734848 08662625/13431480411255146477259155104956093505361644432088109056, a[1 2,8] = 345685379554677052215495825476969226377187500/74771167436930077 221667203179551347546362089, a[12,9] = -320589096271707254279143431215 2727534008102774023210240571361570757249056167015230160352087048674542 196011/947569549683965814783015124451273604984657747127257615372449205 973192657306017239103491074738324033259120, a[6,1] = 5611/114400, a[12 ,4] = -16957088714171468676387054358954754000/143690415119654683326368 228101570221, a[12,5] = -4583493974484572912949314673356033540575/4519 57703655250747157313034270335135744, a[11,7] = 89072299375637918641892 9622095833835264322635782294899/13921242001395112657501941955594013822 830119803764736, a[12,6] = 2346305388553404258656258473446184419154740 172519949575/256726716407895402892744978301151486254183185289662464, a [11,9] = 3007606697681025178342324975654524349466722661958764963718742 62392684852243925359864884962513/4655443337501346455585065336604505603 760824779615521285751892810315680492364106674524398280000, a[11,8] = 1 61021426143124178389075121929246710833125/1099720772213103465066704136 4346422894371443, a[12,11] = -6122933601070769591613093993993358877250 /1050517001510235513198246721302027675953\}: " }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculatio n for stage 20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order conditions can be adapted to give a method of stage by stage construction for an interpolation sch eme that avoids dealing with the weight polynomials for a given stage \+ (corresponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO 7_19 := SimpleOrderConditions(7,19,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1, 2,4,8,16,17,25,27,32,61,63,64]:\ninterp_order_eqns20 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_19[ct],'interpolation_order_condi tion'):\n interp_order_eqns20 := [op(interp_order_eqns20),temp_eqn]; \nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 590 "interp_orde r_eqns20 := [add(a[20,i],i=1..19)=c[20],seq(op(StageOrderConditions(i, 20..20,'expanded')),i=2..7),\n add(a[20,i]*add(a[i,j]*add(a[j,k]*ad d(a[k,l]*add(a[l,m]*c[m],\n m=2..l-1),l=2..k-1),k=2..j-1),j= 2..i-1),i=2..19)=c[20]^6/720, ##17\n add(a[20,i]*add(a[i,j]*c[j]^2* add(a[j,k]*c[k],k=2..j-1),j=2..i-1),i=2..19)=c[20]^6/60, ##25\n ad d(a[20,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..19)=c[ 20]^6/120, ##27\n add(a[20,i]*c[i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2. .19)=c[20]^7/35, ##61\n add(a[20,i]*add(a[i,j]*c[j]^5,j=2..i-1),i= 2..19)=c[20]^7/42]: ##63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We specify " }{XPPEDIT 18 0 "c[20] = 9/10;" "6#/ &%\"cG6#\"#?*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[20,17] = 0;" "6#/&%\"aG6$\"#?\"#<\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[20,18] = 0;" "6#/&%\"aG6$\"#?\"#=\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[20,19] = 0;" "6#/&%\"aG6$\"#?\"#>\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "e21 := `union`(e20, \{c[20]=9/10,seq(a[20,i]=0,i=2..5),seq(a[20,i]=0,i=17..19)\}):\neqs_20 := expand(subs(e21,interp_order_eqns20)):\nnops(eqs_20);\nindets(eqs_ 20);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#?\"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F' \"\"*&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\"#7&F%6$F'\"#8&F%6$F'\"#9&F%6$F'\"# :&F%6$F'\"#;&F%6$F'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e22 := solve(\{op(eqs_20)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e23 := `u nion`(e21,e22):\nseq(a[20,i]=subs(e23,a[20,i]),i=1..19):\nevalf[40](%) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "65/&%\"aG6$\"#?\"\"\"$\"IZL;!e(QB.M i7EYDa+N.[W!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6 $F'\"\"&F0/&F%6$F'\"\"'$\"I>M2:<9/rU_Jdq6EPN[YE!#S/&F%6$F'\"\"($\"I-S/ d!zKv)fGHes%Gwu38Y#FD/&F%6$F'\"\")$!I/)*Rl$HeXq;rh)\\w!fi!Q&z&!#R/&F%6 $F'\"\"*$\"I.O&oPP,!4[2Toecx]1-35!#Q/&F%6$F'\"#5$!I)ouMmO0upF=2&RCemwN d7FX/&F%6$F'\"#6$\"I4B\"QnJ\\vVCv6Q.7z'y#[_)FQ/&F%6$F'\"#7$!I8W7!4)=]z Tuw)*y>]]1(\\l(F+/&F%6$F'\"#8$!I4POzTqR6>B$z2JHwO>!48F+/&F%6$F'\"#9$\" ItPP-'=j3T54+k![F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#----------------------------------------------------------------- ----" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check which of the (adapte d) simple order conditions are satisfied at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "recd : = []:\nfor ct to nops(SO7_19) do\n tt := convert(SO7_19[ct],'interpo lation_order_condition'):\n if expand(subs(e23,lhs(tt)=rhs(tt))) the n recd := [op(recd),ct] end if: \nend do:\nop(recd);\nnops(recd);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"( \"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C \"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\" #U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"# f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#-------- -------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e23" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21750 "e23 := \{ c[20] = 9/10, a[9,3] = 0, a[17,4] = 0, a[17,3] = 0, a[13,7] = 38477494 90868980348119500000/15517045062138271618141237517, a[13,8] = -1373451 2432397741476562500000/875132892924995907746928783, a[13,11] = 2820355 43183190840068750/12295407629873040425991, a[13,12] = -306814272936976 936753/1299331183183744997286, c[3] = 341/3200, a[13,6] = 791638675191 615279648100000/2235604725089973126411512319, a[14,2] = 0, c[2] = 1/20 , a[5,3] = -3899844/2907025, c[4] = 1023/6400, a[8,1] = -1221101821869 329/690812928000000, c[10] = 909/1000, a[13,10] = -9798363684577739445 312500000/308722986341456031822630699, a[13,9] = 122747654703131968784 2881203774063505031923427600698639829444355496961634227421531633068444 8207141/48934514749371551765038583414351093488882928068660965448289652 6796523353052166757299452852166040, a[9,2] = 0, a[8,2] = 0, a[11,2] = \+ 0, a[11,3] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12,2] = 0, a[7,2] = 0, a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[1 0,3] = 0, a[14,3] = 0, a[14,5] = 0, a[13,5] = 0, a[8,3] = 0, c[12] = 1 , c[13] = 1, a[6,3] = 0, a[5,2] = 0, a[13,4] = 0, a[7,3] = 0, a[13,2] \+ = 0, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100, a[8,4] = -125/2, \+ a[6,2] = 0, a[13,3] = 0, a[10,2] = 0, a[7,6] = 5611/283500, a[14,6] = \+ 3037913416047823635649583750/15649233075629811884880586233-30246562581 4318865951896498250/5367686944941025476514041077919*7^(1/2), a[12,10] \+ = 40279545832706233433100438588458933210937500/88964608427994828469169 72126377338947215101, a[15,1] = 339349033530268807690405611/7402593223 688213516896000000+784699017603056191374015321/61513161755551090385561 6000000*7^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 1838743287949 01398385760606250/760335208044775309288920638333-114787229090554407592 495836250/37256425194193990155157111278317*7^(1/2), c[7] = 31/200, c[8 ] = 943/1000, a[17,5] = 0, c[9] = 7067558016280/7837150160667, a[3,2] \+ = 116281/1024000, a[7,4] = 8602624/76559175, a[15,11] = -5009581586726 585160535323/182154187109230228533200+164995836726103803189698643/1316 2109004021797158528000*7^(1/2), a[15,8] = 4171780920238230167282154375 /180381659089210106588738944-32173135433579259286326757125/30569944329 85560753767049472*7^(1/2), a[15,6] = 39667151342526762946267729161/264 960560010663481648771830400-154459727654214653639901999/55915704554556 65877423584000*7^(1/2), a[15,7] = 331704901880412220279128473073/14187 01262824070547944341715840-1100134249641546169459352427/13269013528990 1616433045894400*7^(1/2), a[15,14] = 708939/16000000*7^(1/2), a[15,13] = -79893/16000000+101277/64000000*7^(1/2), a[15,10] = 665982888168452 3938253413125/487858546317362618188848512-105274812306435785357342625/ 16947939574300438348992256*7^(1/2), a[15,9] = -15372775602531728215983 1233675890494424619951621963372845747571196704087889217482364948258872 53876104043/1663773501478632760011311836087937178622019554334472825241 848191108179400377366974818139697364536000000+405732438148607561541740 6628734964607761994016668887443960643578167392109927444312471311939993 /968526685334261480274035268350327920149725248811661666881873633747980 768250399197689344000000*7^(1/2), a[16,3] = 0, a[16,4] = 0, a[19,8] = \+ 2859387143344524736201037565051296515625/80015415149756304085593997697 3329597686+4208420674863241824630827149071356250/148176694721770933491 84073647654251809*7^(1/2), a[19,7] = 356550649843944802108278739287598 0383385/14187591537036223048160665994720789066914+41844249857200869530 169750656331246/41975122890639713160238656789114760553*7^(1/2), a[19,1 5] = 365031149291536/65177059476973149*7^(1/2)+544436534587547644/3584 738271233523195, a[19,1] = 53565547915807495968711190578512612832439/1 228494921165087990600124765218068442000000-328663725171936938343680176 682875249/2274990594750162945555786602255682300000*7^(1/2), a[19,13] = 2381481586979950007/151366176327433100000+511997202259998/43001754638 4753125*7^(1/2), a[19,9] = -456801533136472268751675981161592403550264 3228612239101754903766079968255353172332839950122741123608959786163828 9/11185488354048248346787059966965272621262866751572075588907283648766 985320812914720477754180636404039807942000000-216050000761886699332908 3189847538963394030890555593986265000215032540157439980060353966482010 1981262758272973/50843128882037492485395727122569421005740303416236707 222305834767122660549149612365807973548347291090036100000*7^(1/2), a[1 9,16] = 96563932308593750000/7096307934503880501021*7^(1/2)+3762995721 25585937500/7096307934503880501021, a[19,12] = 22265389327724192212836 28295549708587/29700210356617792635885486621390300000+1376043939693522 84402586088077/29586008364331473149528307355000*7^(1/2), a[19,10] = 15 60143195331346599933913496950810484375/2822725338984751088382151989200 82415758+50485505364845502056312650212481250/9279175999292409889487679 1229481399*7^(1/2), a[19,14] = 353341535891060455739/16528634432431033 162500-2350501240657313744/165286344324310331625*7^(1/2), a[19,11] = - 114996879666326606709923843682437653/224839485269831341683920376142984 44-430085025149565428288387897266111/104092354291588584112926100066196 5*7^(1/2), a[20,5] = 0, a[20,17] = 0, a[3,1] = -7161/1024000, c[5] = 3 9/100, a[14,12] = 11352128098668146659861/254668911904014019468056-521 5842639928607924801/127334455952007009734028*7^(1/2), a[2,1] = 1/20, a [4,1] = 1023/25600, a[19,6] = -941084192745729767895274038102396218/56 7795796866357048220633077958212521055*7^(1/2)+380145805841840287029128 269834313988041/2044064868718885373594279080649565075798, a[6,5] = 923 521/5106400, a[7,1] = 21173/343200, c[11] = 47/50, a[6,4] = 31744/1350 25, a[18,2] = 0, a[19,2] = 0, a[16,8] = 983793822755252686947250523701 3858626988567133654547552254520645532228606707500790611669969401/32259 7310118518639188599326558455791230650166200511678790534528086950881088 3745511450895875000+28561628149763756200561443807577266534751285592123 2623154611070914192208615136563873961039661/95584388183264781981807207 8691720862905630122075590159379361564702076684706294966355821000000*7^ (1/2), a[16,7] = 46524774257782474024979595329864014557423886513194065 784729256922453947551327371902009709751399763/178749887535033474724915 0518151874430640029622863376805482684947617129790277309228449302863281 25000+5917947991674583424374460853714087059815712755844405621829850895 301134910279168916803272958541371/142999910028026779779932041452149954 4512023698290701444386147958093703832221847382759442290625000000*7^(1/ 2), a[16,13] = 7657600557578519236927356742918368477397412237942153506 9547976473129717157/55437992885728406227840443875374037377596717643306 12769171296320800781250000+1189094049890461233623674097341247034240151 86286628637854661183118141741311/8870078861716544996454471020059845980 4154748229289804306740741132812500000000*7^(1/2), a[16,11] = -96158633 2630000217361760815186779719927548025597247352446620287932654896979515 63483807219489943/2266207487042601159749295745587544379598026513632294 7593782892225347467590606715447437500000000-84809796822306017166887302 540539985441826761552922862300464496874472373645260113147330694619/209 8340265780186259027125690358837388516691216326198851276193724569209962 09321439328125000000*7^(1/2), a[16,15] = 12635888347363109520131358222 1638106695002907774309390964517191171456273477/20622933353490967116756 64512163914190446597896330987950131722231337890625000-6671871692162193 83532290229658720215318230956208501949338288072118597379/4124586670698 1934233513290243278283808931957926619759002634444626757812500*7^(1/2), a[16,6] = 49533733144813936314720097158218450549258878950176451985891 89719808023283132738773826477363486812/1609577127855048565864775134835 2179288229798843001789591740513389498327596076419765602058349609375+12 0035920963661763421724642843654652677566912614024166810909682222079969 0553720982691932935469701/38152939326934484524202077270168128683211375 035263501254496031738070109857366328333278953125000000*7^(1/2), a[16,1 0] = 15938572853277337601245712630986530453929893615425271539167654240 22637207535538065172875193509/5690176073295282674253116222513454642673 06663022026387368840720992228188711915769497165687500+3074120446596300 7475669540689503689743154973507615199658230759885748954946271746253192 825123/337195619158238973289073553926723238084329874383423044366720427 254653741458913048590913000000*7^(1/2), a[16,9] = -3184052033230814076 9284137582614147196058544487888858929436357101564990679775786021969042 9710079612049302255526981662229357155593151511164687774987839435430833 88279233890381/1916597683915550813698243244628054389580334115392664767 7566929742465917121958272305909225307798213596172216491160467719850036 366151584437340953031830090690571875000000000000+414201402564698416863 6984365540791069521463268326411298504009416767913623345226168698251274 6782550702093947564073313337424266169065614041755234408498232180327610 377394167/383319536783110162739648648925610877916066823078532953551338 5948493183424391654461181845061559642719234443298232093543970007273230 316887468190606366018138114375000000000000*7^(1/2), a[16,14] = 9578806 188437261689373090827210246131323841838285462486649801851406702570303/ 4999498994785688998001610938579185916234176718378152606379932682031250 00000000-2860115831441802109832054006645866354942015998662300480233819 04611072990151/9998997989571377996003221877158371832468353436756305212 759865364062500000000*7^(1/2), a[16,12] = 2734753906564893127311360222 28123093736719752249770182523001316211267752774954161575132912540789/2 9935506655395543992957483547554285885024664375096226229774236480115214 11250435360468750000000000+2631889880176553367478291455572062331520747 62297307472568607403164903392978255666269313019069/2217444937436706962 4412950775966137692610862500071278688721656651937195638892113781250000 000000*7^(1/2), a[16,1] = 16746171708057136796041288508983162021370539 91505903836096367839971885317604473217871843621390982227/3971673185491 1256923580528752957310738091829268688077817226330743141912382660576772 065625000000000000-625472443421101973099323918176333336959929577122180 41139743607093656896553066548038906825979321/9806600458002779487303834 2599894594415041553749847105721546495662078796006569325363125000000000 000*7^(1/2), a[16,2] = 0, a[19,17] = 0, a[19,18] = 0, a[18,17] = 0, c[ 19] = 7/10, c[16] = 371/500, a[20,3] = 0, a[20,4] = 0, a[18,5] = 0, a[ 18,3] = 0, a[18,4] = 0, a[16,5] = 0, a[15,2] = 0, a[15,4] = 0, a[15,5] = 0, a[15,3] = 0, a[17,2] = 0, a[15,12] = -1764488883660554363266503/ 153994806895851258937600000, a[20,2] = 0, a[20,18] = 0, a[20,19] = 0, \+ a[19,3] = 0, a[19,4] = 0, a[19,5] = 0, a[18,15] = 1791582608825903125/ 11471162467947274224+925013716834375/130354118953946298*7^(1/2), a[18, 7] = 400496411534857632473620487549793366403125/1589010252148056981393 994591408728375494368+1003526315135168217750873893180856009375/7945051 26074028490696997295704364187747184*7^(1/2), a[18,9] = -18027034777101 1132294526440986831063695869238400537030552391540137031602604787343282 9768143660422541770189474551/40088790260908922074884822921603537074606 1144376343189106437045971808753897934863581922709834008720786716641280 -115490733958938987071781894099370162830477199034855475862128572273749 6241965692115980914214471862767372483/21450473680190979760760245557067 5461900615947550079292153907135733216733853033797197240467565952550049 0752*7^(1/2), a[18,8] = 12774383137101436095033568724910537109375/3319 157961767668910217232497074552405216+597207728843802066516925307526298 828125/1659578980883834455108616248537276202608*7^(1/2), a[18,6] = 468 4662399542643647977751441516649198125/25437251699612795760284361892527 920943264-26709437901843964919449276414071475625/127186258498063978801 42180946263960471632*7^(1/2), a[18,10] = 21271135272151061077797757004 198544921875/3512724866292134687764455808783247840544+1729663602506106 19535692995992880859375/250908919020866763411746843484517702896*7^(1/2 ), a[18,13] = 13060395422189465/774994822796457472+3321435324453/22016 89837489936*7^(1/2), a[18,16] = 11164765265655517578125/11354092695206 2088016336+1712896926116943359375/99348311083054327014294*7^(1/2), a[1 8,11] = -257181496023244780272848520935056875/466333747226316856825908 92829656032-24412968277003167712734734927511875/4663337472263168568259 0892829656032*7^(1/2), a[18,12] = 3132514617923695414144270190280539/3 9424279228932655113708734833726976+887940448807352158464070945/1506779 35947549953424507673856*7^(1/2), a[18,14] = 4584404187072404233/169253 216588093779584-47650637886965575/2644581509188965306*7^(1/2), a[18,1] = 70926771436976695009804379644612309403/1630713258316916799374387836 496873072640-29849566024332586811787951198583421/163071325831691679937 438783649687307264*7^(1/2), a[17,11] = 1183644938417444143037861692544 55656328/22770202501285002774702584389480484375-3533158483373047885326 73730285272096/239686342118789502891606151468215625*7^(1/2), a[17,8] = -365032134010684654668509433415077086432/1037236863052396534442885155 33579762663+5531563793954378477130959983247997120/54591413844862975496 99395554398934877*7^(1/2), a[17,16] = -10265122569721941440000/4967415 5541527163507147+2411555846605400000000/49674155541527163507147*7^(1/2 ), a[17,1] = 75904016597212274352620201077585695621748/155516935187999 4201063525997635720322265625-160311321137339876236889599190517742013/3 11033870375998840212705199527144064453125*7^(1/2), a[17,15] = -1096905 8169805055883776/56011535488023799921875+8953379007044874752/448092283 904190399375*7^(1/2), a[17,13] = -43884412335325588358/295637063139517 7734375+12558120070786700736/2956370631395177734375*7^(1/2), a[17,9] = 246354363658419222299343662240541497501411933822141129747478653250964 6203059368497583694524067255325664382625575544/38231649647625848841557 3338714633341547070640922873677355229421588715318582472671110079488595 970841204373017578125-115920268168677580544994926840509851653431625841 698719263191663083353612400657431413037172143283745436569876116611/764 6329929525169768311466774292666830941412818457473547104588431774306371 6494534222015897719194168240874603515625*7^(1/2), a[17,6] = 2652112948 108236779675852103101319384090464/124205330564515604298263485803358988 98078125-773102852863002395047055595586658963648/130742453225805899261 329985056167356821875*7^(1/2), a[17,12] = -127186898329939947281223447 7872349348/30078338034769176569907176844579296875+37526016217537836515 01005551584508/226152917554655462931632908605859375*7^(1/2), a[17,10] \+ = -875118359158215911384486198234977080928/109772652071629208992639244 024476495017+30439516745616652500404896664815444160/156818074388041727 13234177717782356431*7^(1/2), a[17,7] = 326550560647048017503978636528 82675227617248/155176782436333689589257284317258630419371875+110378602 051450377273915565181129491329216/310353564872667379178514568634517260 83874375*7^(1/2), a[17,14] = 196969534835590563883592/2582599130067348 931640625-5241128924571054652928/103303965202693957265625*7^(1/2), a[2 0,14] = -6490938609056455869/918257468468390731250+86358141932804448/1 8365149369367814625*7^(1/2), a[20,9] = 1085937758822506219350844201821 3234053035919839630515793148446073702850609398845638334564577611465341 2602968969867/11185488354048248346787059966965272621262866751572075588 907283648766985320812914720477754180636404039807942000000+275042822381 9152995621184588625747040165971814747860717993944668956246484752090750 97338014024387949868132668919/1957460461958443460687735494218922708721 0016815251132280587746385342224311422600760836069816113707069663898500 00*7^(1/2), a[20,7] = 3504377652209101253000936599614744190785/1418759 1537036223048160665994720789066914-16368385528267532523268450939050589 404/49656570379626780668562330981522761734199*7^(1/2), a[20,15] = 3013 4866374897324/398304252359280355-147524929494432/79660850471856071*7^( 1/2), a[20,12] = -79729418419338678860004195613446243/1100007790985844 171699462467458900000-42292855673402870431176574643577/275001947746461 04292486561686472500*7^(1/2), a[20,6] = 199257218650726586084803607591 66923779/75706106248847606429417743727761669474+7260902051545711780767 63027631835676/1324856859354833112514810515235829215795*7^(1/2), a[20, 8] = -164384582991190944712321248030516328125/296353389443541866983681 47295308503618-9740976592985247191508217896823012500/10372368630523965 3444288515533579762663*7^(1/2), a[20,13] = -1823744191363273521/151366 176327433100000-1862282607394284/4730193010232284375*7^(1/2), a[20,10] = -417408429277495029867103478642578125/34503426708039983967511942173 338518-313470168511820086574442090135037500/17424230487560191903593530 79753595159*7^(1/2), a[20,11] = 6797946802643594773144205710597803/832 738834332708672903408800529572+995491773909111364600646287590606/72864 64800411200887904827004633755*7^(1/2), a[20,16] = 74002009127050781250 0/5519350615725240389683-24834520992187500000/5519350615725240389683*7 ^(1/2), a[20,1] = 46622685338636643276318136578436913601/1051150769851 560436890462456487774000000+126789671676822527127205819671838659/26541 55693875190103148417702631629350000*7^(1/2), a[14,10] = 30448415149825 325326308593750/15127426330731345559308904251+588780394238322448263281 6406250/741243890205835932406136308299*7^(1/2), c[15] = 9/20, a[10,5] \+ = -180269259803172281163724663224981097/381009225582568710865798328320 00000, a[10,7] = 318607235173649312405151265849660869927653414425413/6 714716715558965303132938072935465423910912000000, a[10,8] = 2120832024 34519082281842245535894/20022426044775672563822865371173879, a[9,6] = \+ 7801251558438936413230905525304310365677955925684971827014606748031267 70111481625/1831104254127319721978898745071587868592261029808618595052 41443073629143100805376, a[9,8] = 103328481844520156040568367672866568 59124007796970668046446015775000000/1312703550036033648073834248740727 914537972028638950165249582733679393783, a[9,7] = 66411312295991164213 4782135839106469928140328160577035357155340392950009492511875/15178465 5985862481363330231072953491752797651500890783011399432530168778231708 16, a[9,4] = -51722943110856684583751756552469812300390253369336991141 38315270772319372469280000/1246193810048091458972786305712152983652570 79410236252921850936749076487132995191, c[6] = 93/200, a[9,5] = -12070 6792584692548079789364417331879494845715161204699665345142964068916526 14970375/2722031154761657221710478184531100699497284085048389015085076 961673446140398628096, a[8,5] = -1024030607959889/168929280000000, a[8 ,7] = 6070139212132283/92502016000000, a[9,1] = -147251426448621580388 1384708877264246346044433307094207829051978044531801133057155/12468948 0162003200115705962164398602480330155839348790044045363616804606968643 6608, a[8,6] = 1501408353528689/265697280000000, a[14,9] = -1287199574 1547923519131817349480097796541904481893575052170212767231631713171072 5250664367373074947/95911648908768241459475623492128143238210539014575 492278647719252118577198224684430692759024543840-882185082068177622906 7876050054725556364447599046365045313145296081200340665919272046653909 33748491981/1597888070820078902714863887378854866348587579982827701362 27100274029549612242324261534136534890037440*7^(1/2), a[14,8] = 742007 03416028798327128906250/42881511753324799479599510367+2867664719921726 1041085964843750/2101194075912915174500376007983*7^(1/2), a[14,1] = 89 41065567926479206438689/198753096356125278622613280+152838094177334666 489948287/331122658529304714185273724480*7^(1/2), a[14,11] = -14665848 24552592368357500/602474973863778980873559-946677979546641857718938375 /59042547438650340125608782*7^(1/2), a[10,6] = 21127670214172802870128 286992003940810655221489/467947387799789290614582269797670863367372800 0, a[5,4] = 3982992/2907025, a[12,1] = -286655699182566397177829532910 1033887534912787724034363/86822671161926270301121392501614361203066923 3795338240, a[11,5] = -26053085959256534152588089363841/43775528045656 83061011299942400, a[11,4] = -996286030132538159613930889652/163530688 85996164905464325675, a[11,6] = 20980822345096760292224086794978105312 644533925634933539/377588999200755080387872783911549464197221296217415 6800, a[10,9] = -26984049294008425187211664850871297985622698482295177 93703413951226714583/4695456749139343150770004420808711418846760359027 17550325616728175875000000, a[10,1] = -2905557336033741508853861844223 1036441314060511/22674759891089577691327962602370597632000000000, a[11 ,1] = -2342659845814086836951207140065609179073838476242943917/1358480 961351056777022231400139158760857532162795520000, a[10,4] = -204627495 24591049105403365239069/454251913499893469596231268750, c[18] = 3/4, a [11,10] = -31155237437111730665923206875/39286214159423051501033895629 1, c[17] = 6/25, a[13,1] = 44901867737754616851973/1014046409980231013 380680, a[12,7] = 1657121559319846802171283690913610698586256573484808 662625/13431480411255146477259155104956093505361644432088109056, a[12, 8] = 345685379554677052215495825476969226377187500/7477116743693007722 1667203179551347546362089, a[12,9] = -32058909627170725427914343121527 2753400810277402321024057136157075724905616701523016035208704867454219 6011/94756954968396581478301512445127360498465774712725761537244920597 3192657306017239103491074738324033259120, a[6,1] = 5611/114400, a[12,4 ] = -16957088714171468676387054358954754000/14369041511965468332636822 8101570221, a[12,5] = -4583493974484572912949314673356033540575/451957 703655250747157313034270335135744, a[11,7] = 8907229937563791864189296 22095833835264322635782294899/1392124200139511265750194195559401382283 0119803764736, a[12,6] = 234630538855340425865625847344618441915474017 2519949575/256726716407895402892744978301151486254183185289662464, a[1 1,9] = 300760669768102517834232497565452434946672266195876496371874262 392684852243925359864884962513/465544333750134645558506533660450560376 0824779615521285751892810315680492364106674524398280000, a[11,8] = 161 021426143124178389075121929246710833125/109972077221310346506670413643 46422894371443, a[12,11] = -6122933601070769591613093993993358877250/1 050517001510235513198246721302027675953\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "calculation of the \+ interpolation coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order condition gives ris e to a group \{list) of equations to be satisfied by the \"d\" coeffi cients of the weight polynomials for a given stage (corresponding to a n \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO8_20 := SimpleOrder Conditions(8,20,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "whch := [1,2,4,8,16,17,25,2 7,32,58,61,63,64,102,117,121,123,125,127,128]:\nordeqns := []:\nfor ct in whch do\n eqn_group := convert(SO8_20[ct],'polynom_order_conditi ons',8):\n ordeqns := [op(ordeqns),op(eqn_group)];\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitut e for all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "eqns := []:\nfor ct to nops(ordeqns) do\n eqns := [op(eqns),e xpand(subs(e23,ordeqns[ct]))];\nend do:\nnops(eqns);\nnops(indets(eqns ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$g\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$g\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Solve the system of equations. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dd := solve(\{op(eqns)\}):\ninfolevel[solve]:=0: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "dd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52753 "dd := \{d[20,1] = 0, d[2,3] = 0, d[4,4] = 0, d[13,1] = 0, d[9,1] = 0, d[5,1] = 0, d[2,4] = 0, d[11,1] = 0, d[18,1] = 0, d[3,5] = 0, d[4,1] \+ = 0, d[16,1] = 0, d[10,1] = 0, d[5,7] = 0, d[5,5] = 0, d[4,3] = 0, d[3 ,2] = 0, d[2,6] = 0, d[1,1] = 1, d[7,1] = 0, d[4,5] = 0, d[3,4] = 0, d [4,2] = 0, d[2,5] = 0, d[8,1] = 0, d[19,1] = 0, d[5,8] = 0, d[15,1] = \+ 0, d[5,4] = 0, d[2,1] = 0, d[6,1] = 0, d[4,6] = 0, d[4,7] = 0, d[3,3] \+ = 0, d[3,1] = 0, d[17,1] = 0, d[14,1] = 0, d[2,2] = 0, d[3,6] = 0, d[2 0,8] = -23927710289771033952904996482934510925284677631437167811827008 3361448339501889677056982554762500000/79829306137901543961820895933235 7254136412938009703985730035142070597499153654586471922302758117+42750 4580977075835660824004733013238231372599158881245034398215771857044278 363949687110156250000/798293061379015439618208959332357254136412938009 703985730035142070597499153654586471922302758117*7^(1/2), d[11,7] = -1 4967326189220801286190271553784511290357139204634832241290867171910637 67563148939253810393855275524537847533487500000/2147386491677497043460 9273228721399433622159250727485949819741849175059700839629328141102464 465572984414631784409-477616551494062582045050844872875193639627909847 1028962361377724386225349558942326293718545468445210452367187500000/58 2862047741034911796537416208152270341172893948317475780821564477608763 30850422462097278117835126671982571986253*7^(1/2), d[11,3] = -13437666 3303738607535240586272499980206428777797414557109600371716057957568705 2086680750553248032684107908009435750000/58286204774103491179653741620 8152270341172893948317475780821564477608763308504224620972781178351266 71982571986253+6963081639396513788976467772099471420178477539255056241 966219484117018082586818906019745014460160295923109020875000/582862047 7410349117965374162081522703411728939483174757808215644776087633085042 2462097278117835126671982571986253*7^(1/2), d[9,7] = -3094191529655506 8178125271923402248837741732810788653801154461340479926521956085466244 6596620463378374745644086335067434565318083756779983249402209656500670 93051196984385238334516687742716550/4059533980377062861648313655269398 0099496990545581206818305590102415727939473327151472426058627140233081 0478670332321916561624802794924882103297120187426195458930663148645235 994285394731-519671305268909261752701664513083692863855451355164435883 5461584225172317705647348093152012038582961953781321074243855948395374 558506542510021050131213037418234576085852871935299506031250/579933425 7681518373783305221813425728499570077940172402615084300345103991353332 4502103465798038771761544354095761760273794517828970703554586185302883 918027922704380449806462284897913533*7^(1/2), d[19,5] = 16426954085382 2734610061838890980584663826749977967514530628660803341085182530405668 56654194175625000/3029213064559428957743170024133220960982280744568809 400396463316880684786350736427454062071408747+162005916590469448192380 9659429006864550479056710487357344968436279429740220223888920346731025 000/131704915850409954684485653223183520042707858459513452191150578994 812382015249409889307046582989*7^(1/2), d[18,4] = 10786296179310264658 2629760388085179331280789739093434322484066117265711411840303525871303 9925248/13846527949641120776971280796500431915436200332787437769013975 52821156768249375366801620155789+1776264125926630556111396905917014519 39007264343696722377377822131294741286378177478221715462144/4615509316 5470402589904269321668106384787334442624792563379918427371892274979178 8933873385263*7^(1/2), d[16,5] = -229637698332254179571285935502130438 71263600494868468299631450412600993055513054735546875000000000000/2904 7306763524285812188013260867926662322375106054628149383758341533298575 427748884296525080254703+193497519295361941053361752531090313216244308 674624627870158379464689016857995369531250000000000000/548062391764609 1662676983634126023898551391529444269462147878932364773316118443185716 32548684051*7^(1/2), d[14,5] = 268597354462934569853805796975077808409 2686033418648485101326547821075182981148189371853375628/17047708829927 5084805391966030785071462278800266876798797709685025860614421947719578 85884271-8595445534603102776493287191224249411688096239610435627616997 94165942063687863682022223642568/1704770882992750848053919660307850714 6227880026687679879770968502586061442194771957885884271*7^(1/2), d[16, 6] = 23986057739895116189798008278833125459727230577030233362242179201 7462607951451972656250000000000000/96824355878414286040626710869559755 54107458368684876049794586113844432858475916294765508360084901-2021115 3064379615909670674064516330119253556832196327474896438935782461489840 23437500000000000000/1826874639215363887558994544708674632850463843148 08982071595964412159110537281439523877516228017*7^(1/2), d[19,3] = 543 4937586096407536385138769400656893031880957738929633746389724543031560 59669873628610826816675000/1009737688186476319247723341377740320327426 914856269800132154438960228262116912142484687357136249+318851208994152 8465473036942609514474876749114283947596156051838097807918017520928157 38891325000/4390163861680331822816188440772784001423595281983781739705 0192998270794005083136629769015527663*7^(1/2), d[2,7] = 0, d[6,7] = -4 2011422184116046265766930861463745753350030342479174927800229226970897 83557026951379030383482052397644920192781800000000/3904471923098139408 3422648935864016181346128763276929008410377424271767669643131192500120 89577525294782649942263341281-1340610228792191939719874301659000105157 2270708625888097827598300479406850477250621129187808995084757706501625 000000000/105978523626949498226432904254488043920796635214608807308542 45300873765510331707037964318528853282942981478414714783477*7^(1/2), d [20,3] = 5658830014886157112704375198302564742656137620017840576732823 8137952328724680704699916169698175000/26609768712633847987273631977745 2418045470979336567995243345047356865833051218195490640767586039+51326 5419957725064901070586674493557926491793530946093838258773041417742216 957984905941950475000/266097687126338479872736319777452418045470979336 567995243345047356865833051218195490640767586039*7^(1/2), d[7,3] = 949 9580631035826355154343126728135546713570424258532854943697057983431100 6370690588450055961166644706314693458830000000/73558331143358901840849 9290589778130439172795136736231559561557895957800389090200208511739940 97196295167196933959136911*7^(1/2)-18332715514363168982290016572312172 6428904778635792606656104995445594531416898473226721828487647668034096 98782414140000000/7355833114335890184084992905897781304391727951367362 3155956155789595780038909020020851173994097196295167196933959136911, d [11,4] = -144354826523565474797296167354377798525641139472188379358410 280341849274965069472869901349318238288040712345875000/588749543172762 5371682196123314669397385584787356742179604258227046553164732365905262 35132503385117898813858447*7^(1/2)+44556123902201815446095850104426752 7661392755324304500678506250901598911068580075165866099635013190887085 80180843750/5887495431727625371682196123314669397385584787356742179604 25822704655316473236590526235132503385117898813858447, d[20,7] = 32002 5509436218930111211692065981238472193011362081486733393830550522705374 338099956643940134000000/266097687126338479872736319777452418045470979 336567995243345047356865833051218195490640767586039-352062596098768335 2500903568389520785434833169543727900283279424003528599939467820952671 87500000/2660976871263384798727363197774524180454709793365679952433450 47356865833051218195490640767586039*7^(1/2), d[15,2] = -48183330317765 4252948010869337507200355424640920311724672460053875223302725772345235 296000000/208360885699113992539923514037626198453896311437293865197200 72614271852873793610170749414109+4469536614263275323583275549719383104 072299681633333344370126137128079715270634801856000000/208360885699113 9925399235140376261984538963114372938651972007261427185287379361017074 9414109*7^(1/2), d[7,6] = -5041536670213565865796224263467253270205909 1450316363182237728243217811643130584447707818537203752817141313085000 00000/7355833114335890184084992905897781304391727951367362315595615578 9595780038909020020851173994097196295167196933959136911*7^(1/2)+105557 3392380305916879849469103611859046946769229120536398216805869327757200 78215429390552961077870781300001731532400000000/7355833114335890184084 9929058977813043917279513673623155956155789595780038909020020851173994 097196295167196933959136911, d[13,3] = 1303325087614746216014337387698 2029300580781998150335412152712106489826833150909414383651709184263/73 1992627549557367192005297165584597788383131710357077824285871011984463 309243318908597667063279-409521122330732098015725934093466876263447159 613092032753335919617039525509357260826909602763538/731992627549557367 1920052971655845977883831317103570778242858710119844633092433189085976 67063279*7^(1/2), d[9,6] = 8418502724072180370086806795416251150371038 5668637806388826155651494672283271632659706283100590742832331418726750 9940658390508145233397181564748030851668809999958264029653235792167396 1743780/57993342576815183737833052218134257284995700779401724026150843 0034510399135333245021034657980387717615443540957617602737945178289707 03554586185302883918027922704380449806462284897913533-4020771127623446 5166466174500040875436437730350565294067789799914519333246705408624675 0161388585218884881137641972924625950133551555306203346771535866425866 530835201156845062879744638075/579933425768151837378330522181342572849 9570077940172402615084300345103991353332450210346579803877176154435409 5761760273794517828970703554586185302883918027922704380449806462284897 913533*7^(1/2), d[17,6] = 17815928424498078431015238924503133373313740 44077086088871695617929599669674371864512561035156250/9549950326867480 9998770923653463478920763473695234958292889389218075182306159419048307 74214476733*7^(1/2)-24580458638991102047268660932213940402231659061678 498526855242323050875531731055106939909071777343750/115604661851553717 3669332233699821060619768365784423179334976816850383785811403493742672 6680682361, d[12,2] = -87854848626211768896280645079181037080404406652 705454599891350377216695609279018191510960374452417701800784160/278205 1025513361810917036557362906919206434804860319650095632352015511767975 280314504067087706779903927638667+252307268834524033100086779322979507 1771687706660455026371704584395836856537958582910777913536937251606942 0830/11406409204604783424759849885187918368746382699927310565392092643 2635982486986492894666750595977976061033185347*7^(1/2), d[9,5] = -8486 5844740885018444483331612632368928858096066847907032934206323145794658 6282908119172223210821605549292039075232113289480948901530641094593887 73855821659907110121408879271966065801075824493/5799334257681518373783 3052218134257284995700779401724026150843003451039913533324502103465798 0387717615443540957617602737945178289707035545861853028839180279227043 804498064622848979135330+256627033721206488495100132257394320618419069 0629597558042393029380209814907510893109831407714408990609590810579238 9128428951299944172900594351280476611288328278485323096477744388729112 623/115986685153630367475666104436268514569991401558803448052301686006 9020798270666490042069315960775435230887081915235205475890356579414071 09172370605767836055845408760899612924569795827066*7^(1/2), d[16,3] = \+ -451960393082219281935761744000891597639574634810432925666333632643881 9881578107874609375000000000000/96824355878414286040626710869559755541 07458368684876049794586113844432858475916294765508360084901+3808312638 3993500024679108844466762852740506290640939967565351170864333633188678 906250000000000000/182687463921536388755899454470867463285046384314808 982071595964412159110537281439523877516228017*7^(1/2), d[10,8] = -7058 2445508865155414595006366420299028381217226889251798202272411325871900 97443815202059421003260561383365429687500000000/3377300585348710346293 6863916037468638760539976690830038065969413959699889819493924983074646 2801367756362117505575927-11852256584637434181315220246611226252956244 2376785278586947164573876683433669402751892196913324419541552734375000 00000/2582641624090190264812819005344041719434629527629298767616809425 77338881510384365308694100236259869460747501621911003*7^(1/2), d[18,5] = -460784299590987262136067384119616920108781196129315031175660923238 358516311147080056699815384530944/137080626701447095692015679885354275 962818383294595633913238357729294520056688161313360395423111-435331938 0683910889160888910466121440246169180850129492601996573344724633016302 1816318153079584768/13708062670144709569201567988535427596281838329459 5633913238357729294520056688161313360395423111*7^(1/2), d[8,5] = 11394 9973463717620757151671863539901033763946265446628747742522435429459879 5006266477884346625002352245393451858468750000000/12445665049196891396 2213262179227446795107464827891589042097505922858997601296206795259007 77011721401554846975875097967-1722874719027930293941119587823780071740 1691719950868917273355145479919411962149722552735352531720869793757157 81250000000/1244566504919689139622132621792274467951074648278915890420 9750592285899760129620679525900777011721401554846975875097967*7^(1/2), d[10,3] = 46684591697538759336024712818101577091964348207530805238990 985757037941624263595698225061805097758210325703761562500000000/146349 6920317774483393930769694956974346290065656602634982858674604920328558 844736749266568005472593610902509190829017-142299140735683805775605039 4984341112910929048171074197537174613647129371439309542991437867517126 0469694454531250000000/86088054136339675493760633511468057314487650920 976625587226980859112960503461455102898033412086623153582500540637001* 7^(1/2), d[15,3] = -67092804234390328470554378935012146142355999054799 1897789170855062707269530021675976512000000/56257439138760777985779348 7901590735825520040880693436032441960585340027592427474610234180943*7^ (1/2)+7232863330964485021841824822627838433803201006018262771611261213 0585208899614096153215392000000/56257439138760777985779348790159073582 5520040880693436032441960585340027592427474610234180943, d[1,5] = -446 7731007255406038572572203061297325490161190519307689543511957255203806 5628078189324013031807096013506241721530189/31798785022584914013552366 2796619173427393633124573137743611772606634543512020268374088062321163 155793437863169212506*7^(1/2)+1448871315490599932453848980378327777086 2949332300900251953690605333582423882932330942922493630617218969540389 310116982869/842667803098500221359137706411040809582593127780118815020 57119740758154030685371119133336515108236285261033739841314090, d[12,5 ] = 127276007909058472644304952121529678151124653145623845543806808739 23180944152136490711660933349820752167298561851414/9239191455729874574 0554784070022138786845699869411215579675950410435145814459059244680067 98274216060943688013107-1480275879641138918320106682915688218537888240 363061548974165970181889815146571149112918554323093614124122784290/710 7070350561441980042675697694010675911207682262401198436611570033472754 95838917266769753713401235457206770239*7^(1/2), d[20,4] = -10640739904 4891741645987309451004876218982397716289632058161837311082648404570475 42047355475000/2687855425518570503765013331085377960055262417541090861 043889367241069020719375712026674420061*7^(1/2)-2440706549165311450257 4202367241799955912662390293353492728816248415633613403311563577292120 0000/29865060283539672264055703678726421778391804639345454011598770747 1229891191041745780741602229, d[1,8] = -366195481452245567125392231394 1382675955900277535195315609334325020847182682690500704683900506941143 398694756609375/158993925112924570067761831398309586713696816562286568 871805886303317271756010134187044031160581577896718931584606253*7^(1/2 )-13866190394671881778436412797656966682110309566180900358165221147195 9345066610703963148168657423990686396670819629710625/84266780309850022 1359137706411040809582593127780118815020571197407581540306853711191333 3651510823628526103373984131409, d[17,2] = -10769493640445832266867478 2877115058970646310740153082050950921077324570606819086947947759643554 6875/32540571484141046369951573985624592817445331777635615418317865955 788580637654320564608563990068868+223633323118736588636288774564885712 564874691677504838747882856147204401667545230496717041015625/353701863 9580548518472997172350499219287536062786479936773681082150932678005904 40919658304239879*7^(1/2), d[6,8] = 7413332887926935700794848609883182 2839001363724780399397075210913505688156090529714399111236457948220741 62423267500000000/3179355708808484946792987127634641317623899056438264 2192562735902621296530995121113892955586559848828944435244144350431+16 2788384924766164108841879487164298483377572890457212616477979362964226 04150947182799728053779745777215037687500000000/3179355708808484946792 9871276346413176238990564382642192562735902621296530995121113892955586 559848828944435244144350431*7^(1/2), d[1,7] = 316096137901180821623537 8439455919273251963533105363097006681225320093004282454748569691658959 2096685742048985028719450/40127038242785724826625605067192419503933006 0847675626200271998765515019193739862472063507214801125167909684475434 829+430818213473230078971049683993103844230105915004140625365804038237 746727374434176553492223589051899223375853718750/757113929109164619370 2944352300456510176038883918408041514566014443679607429054008906858626 694360852224711027838393*7^(1/2), d[8,6] = -41865124060354371804929562 9662992525286083806122205092065446363571919406495294585328341372877728 37305087806431250000000000/4609505573776626443044935636267683214633609 8084404292237813891082540369481961558072318151025969338524277211021759 6221+17995726655219282455071679213064878087198451976431765328554053751 283129395649276910314681832068096717762710937500000000/414855501639896 3798740442072640914893170248827596386301403250197428633253376540226508 633592337240467184948991958365989*7^(1/2), d[6,5] = -65679219757485744 9565897644617332979439835992046924422579029486421094353123816218295084 25559627622455116870528657175600000/3179355708808484946792987127634641 3176238990564382642192562735902621296530995121113892955586559848828944 435244144350431+993041629111759766171503534969068130236865289733391610 54082126783796408324466264874657403285559885764404454211858000000/3179 3557088084849467929871276346413176238990564382642192562735902621296530 995121113892955586559848828944435244144350431*7^(1/2), d[19,8] = -1343 5032978554077707703559009781090454817854260258010172908339321455561067 3975362560138631950000000/13170491585040995468448565322318352004270785 8459513452191150578994812382015249409889307046582989+26557478293843856 3250706054150023527652385217680037747221078973161748707156500783616083 593750000/131704915850409954684485653223183520042707858459513452191150 578994812382015249409889307046582989*7^(1/2), d[7,7] = -65160185914872 5278075582899635723280861177326721223305531137551149417801886842856598 88244047346061617427841875000000000/7355833114335890184084992905897781 3043917279513673623155956155789595780038909020020851173994097196295167 196933959136911*7^(1/2)-3879723457585315003985142664396068948860219340 9621543941088731468744470122984770102696833308423760289925323716324900 0000000/51490831800351231288594950341284469130742095659571536209169309 0527170460272363140145958217958680374066170378537713958377, d[1,2] = 8 3682129785040210539336690618820490703255772930404275071053776387299924 32521009445375032950993744090514852582633/5608251326734552736076255075 7781159334637325066062281788996785292175404499474474140050804642180450 7572200816876918*7^(1/2)-186198759505274026628130590016622492352986866 1302454726016152085606965556763953004754891004748566484244947596408537 /207134021126781390252293741474731807995524615226571493715458510138348 183865654852224577884741154278058025388811684, d[9,3] = -1462087019352 4414543778879585305117667945310550019188730368146839202321070533580460 4395845630016892947623555077228036432836455179927470539393545137449543 9584002886576369821504352450477055233/57993342576815183737833052218134 2572849957007794017240261508430034510399135333245021034657980387717615 4435409576176027379451782897070355458618530288391802792270438044980646 2284897913533+15152379928710011979835128400673306524960131453700802102 3337628070182952374672873531557416949690747453359847476641711494060497 50797852884403973285750128883040962889247117423252061882675977/1159866 8515363036747566610443626851456999140155880344805230168600690207982706 6649004206931596077543523088708191523520547589035657941407109172370605 767836055845408760899612924569795827066*7^(1/2), d[12,7] = 81411535062 6850662250437786627337847238378685052521154801192398602043089802050864 028313920613808874139525123317000/113463754719489687751558506752658766 9312140173834874577294266057672010562633707745110106098033675656607119 580557+259789197729122768842758215942112342645354994507872222649475348 4756833666122280254232679070775264880155418125000/30797304852432915246 8515946900073795956152332898037385265586501368117152714863530815600226 6091405353647896004369*7^(1/2), d[6,3] = 19544503724775365475838843124 7325354073530981032866053125044393847009753499244603952468158991923930 18090591967914000000/1059785236269494982264329042544880439207966352146 0880730854245300873765510331707037964318528853282942981478414714783477 *7^(1/2)-3771785730047000355838270131625587477112176642852397089207500 292839741824229819663361213212984911710967614227570612000000/105978523 6269494982264329042544880439207966352146088073085424530087376551033170 7037964318528853282942981478414714783477, d[17,5] = -17056611976086287 2117643858951385417784438438323071765963818582321672362395967409360521 608154296875/286498509806024429996312770960390436762290421085704874878 66816765422554691847825714492322643430199*7^(1/2)+13534439563956865642 4711548970283254283494763399481190804806454033369725195698897878573838 6201171875000/65894657255385618899151937320889800455326796849712121222 0936785604718757912499991433323420798894577, d[20,5] = 135555845853545 6706182553204413164691872774887601208108684357818986422088376660140136 828265015125000/798293061379015439618208959332357254136412938009703985 730035142070597499153654586471922302758117+260786324369773167054482416 5049266305029259961684254825602955979664086684162384340768862463575000 /798293061379015439618208959332357254136412938009703985730035142070597 499153654586471922302758117*7^(1/2), d[10,2] = -2300687236755043771672 2851825015015605882082046956511306714830540983722176691776756759436521 3964529407219132500000000000/54203589641399054940515954433146554605418 1505798741716660318027631451973540312865462691321483508368004037966366 97371+1611527623299982650765067866491235211161954616508674075891003207 278086489310916135336927822991089467622584218750000000/542035896413990 5494051595443314655460541815057987417166603180276314519735403128654626 9132148350836800403796636697371*7^(1/2), d[7,2] = 24393526059945478414 2016636904314969719554082216471364240416221316347728535676174453999248 5382315876353748727334560000000/73558331143358901840849929058977813043 9172795136736231559561557895957800389090200208511739940971962951671969 33959136911-1708656459134153373180916502740779016005413633221460600695 9701321260493368197548122878534907071273661568398407830000000/73558331 1433589018408499290589778130439172795136736231559561557895957800389090 20020851173994097196295167196933959136911*7^(1/2), d[11,5] = -23399405 5629823393803681889788252816897884129744463666484788482335764239972890 34280201410911532962305115614656440225000/1748586143223104735389612248 6244568110235186818449524273424646934328262899255126738629183435350538 0015947715958759+35378897475197826870250049589844975980775885620724836 937651085700863763344689144512073528675598856949290309412375000/174858 6143223104735389612248624456811023518681844952427342464693432826289925 51267386291834353505380015947715958759*7^(1/2), d[18,3] = 436874070234 3669668744353682143407095215716984369037434943715804816734530910295080 624793121714176/456935422338156985640052266284514253209394610981985446 37746119243098173352229387104453465141037-5039979863922048841771421047 65185417477093968911075953027061824740531452644908329486909214431232/2 6878554255185705037650133310853779600552624175410908610438893672410690 20719375712026674420061*7^(1/2), d[10,7] = 987978596923236660456548537 4465799786787474328275052793635830168258672138444118588372776185771469 92212325930609375000000000/1024447844222442138375751538786469882042403 0459596218444880010722234442299911913157244865976038308155276317564335 803119+165931592184924078538413083452557167541387419327499390021726030 403427356807137163852649075678654187358173828125000000000/146349692031 7774483393930769694956974346290065656602634982858674604920328558844736 749266568005472593610902509190829017*7^(1/2), d[15,6] = 35606922662464 5492358893568740882838486343478991029262666342126951719224198425483744 00000000/5625743913876077798577934879015907358255200408806934360324419 60585340027592427474610234180943*7^(1/2)-38385637355998108878065822374 85200925401535442492951725150099511338269898156600114430400000000/5625 7439138760777985779348790159073582552004088069343603244196058534002759 2427474610234180943, d[17,3] = -19746996116949583593892488898328893974 39758427170699939981599642218630723973789009886167724609375/5617617839 3338123528688778619684399365154984526608798995817287775338342533034952 3813574953792749*7^(1/2)+131701899074727968679494870302386316009377922 661963267877997437055787825042359472167413620648681640625/439297715035 9041259943462488059320030355119789980808081472911904031458386083333276 22215613865929718, d[8,8] = -12861740577554588383902606281457985425784 9553086833990155570286355529244623583743414010008266216253508607621679 687500000000/124456650491968913962213262179227446795107464827891589042 09750592285899760129620679525900777011721401554846975875097967-1661348 4726014847170487148461101253773262972651801851300363786698008797447977 545153540142016311019864995117187500000000/732097944070405376248313306 9366320399712203813405387590711617995462293976546835693838765162948071 41267932175051476351*7^(1/2), d[7,5] = 3712819946551046363328636212605 4982746523499942023678666500728391630837846620661758392138728756017514 111392954270000000/169749994946212850401961374751487260870578337339246 82266759112874522103085902081543273347844791660683500122369375185441*7 ^(1/2)-319232993917796816439781191791502619700566017051554145341627707 014214716289476681076510546592637045271284803525789482000000/220674993 4300767055225497871769334391317518385410208694678684673687873401167270 60062553521982291588885501590801877410733, d[13,2] = -8458636409253903 4261053786183581065397429945856253988015944182110848130660921558953119 603644821/271108380573910135997038998950216517699401159892724843638624 39667110535678120122922540654335677+2728116183930778106886737376926107 646233182217003820883174367061791125349649133068662526911294/271108380 5739101359970389989502165176994011598927248436386243966711053567812012 2922540654335677*7^(1/2), d[3,8] = 0, d[13,5] = 1189032320085712996551 2914610380731513223423751390969130119162897393407317934202984673860024 319224/219597788264867210157601589149675379336514939513107123347285761 3035953389927729956725793001189837-20807462200202811167590405521918384 93406662000304309120863862024600003635089146777768834902274586/2195977 8826486721015760158914967537933651493951310712334728576130359533899277 29956725793001189837*7^(1/2), d[1,6] = 2333311444171014107707205088506 6504203502536356624256269811946710956362754599355002137138829583050861 9380362374075/52997975037641523355920610466103195571232272187428856290 601962101105757252003378062348010386860525965572977194868751*7^(1/2)-4 4375946531423148122760183259877491048252965453363493519588231185165268 5985201155627858088271953697941399150694550899130/28088926769950007378 6379235470346936527531042593372938340190399135860513435617903730444455 0503607876175367791328043803, d[19,2] = -27449659069394216960146177769 0836020193458152002022810673616079581132132479988291861269231875000/53 4252745072209692723663143586105989591231171881624232874155787809644583 1306413452299933106541-30344262257624886191657143504765041183058414518 9238423648366939687870993400463012997576925000/23228380220530856205376 6584167872169387491813861575753423545994699845470926365802273910135067 *7^(1/2), d[10,4] = 50151164421971453497449970342700878317708933617543 41564016647542913188432138913640282465663311644158713445781250000000/1 4782797174927014983777078481767242165114041067238410454372309844494144 732917623603527945131368410036473762719099283*7^(1/2)-1547950664095234 5636586785419076291440730126150786798539677328056528715793106243760576 58775944745973651705009101562500000/1478279717492701498377707848176724 2165114041067238410454372309844494144732917623603527945131368410036473 762719099283, d[16,7] = 4428740350793042132532867112044520948989518201 07648326481576425438446469629711914062500000000000000/1383205083977346 9434375244409937107934439226240978394356849408734063475512108451849665 01194297843-3731749088696384030589120026683221957026136785856042739087 230231865299388818359375000000000000000/260982091316480555365570649244 09637612149483473544140295942280630308444362468777074839645175431*7^(1 /2), d[17,7] = 2302649537878259767671836640906978094778363794043501222 6923687824053171506131098692001342773437500/95499503268674809998770923 6534634789207634736952349582928893892180751823061594190483077421447673 3*7^(1/2)+133030369696984930144779184056947893490468544969932234762945 59367680524575342187668321022822265625000/1156046618515537173669332233 6998210606197683657844231793349768168503837858114034937426726680682361 , d[16,4] = 1338542483623689054136733755944336011622591981093356301957 9165882451606098088412890625000000000000/13971768525023706499368933747 411220135797198223210499350352938115215631830412577625924254487857-112 7883844567595109405255636864737004291579582157130357461724465278968087 27646093750000000000000/2636182740570510660258289386304003799207018532 68122629251942228588974185479482596715551971469*7^(1/2), d[12,8] = -11 0506725398453652158946406525239361414929783819655753305779298948374738 908144532006174722432345744254466085087500/710707035056144198004267569 7694010675911207682262401198436611570033472754958389172667697537134012 35457206770239-3154583115282205050233492622154221303550739219024162703 600772088633298023148483165853967443084250211617293437500/923919145572 9874574055478407002213878684569986941121557967595041043514581445905924 468006798274216060943688013107*7^(1/2), d[13,7] = -2719351414657628595 3478589572465998732662734868375947568138014482617095995000640615022280 09262000/3852592776576617722063185774555708409412542798475563567496241 4263788655963644385205715666687541+28090158401264189733183045479057945 2865854841124775626356504022013089512937513701468546840125000/73199262 7549557367192005297165584597788383131710357077824285871011984463309243 318908597667063279*7^(1/2), d[14,8] = 44030912950946558117546420743656 3188112262157906243069143992429032198658561520525956581125000/17047708 8299275084805391966030785071462278800266876798797709685025860614421947 71957885884271-1409043342461283688776408415366126500459094299185289963 02733246187559757762732414715786750000/1704770882992750848053919660307 8507146227880026687679879770968502586061442194771957885884271*7^(1/2), d[12,3] = 73091280953291415088728477747534155214352475818509004546908 2646379899181902049861958801404636282071021965903539780/30797304852432 9152468515946900073795956152332898037385265586501368117152714863530815 6002266091405353647896004369-37874177248727376409507331100993353308134 4591227416193524471577578984667297625407678960014390649547806941543941 0/30797304852432915246851594690007379595615233289803738526558650136811 71527148635308156002266091405353647896004369*7^(1/2), d[20,6] = -52469 6548252183437092920800220789359485061031047005758160421884401831168494 601244610762920069975000/266097687126338479872736319777452418045470979 336567995243345047356865833051218195490640767586039-272395860067275614 8163556246628234939130722349464118615419177337200444413896022531182810 1250000/26609768712633847987273631977745241804547097933656799524334504 7356865833051218195490640767586039*7^(1/2), d[10,6] = -298670748788790 1021004335613680932370770209178449664427968404348768171186754566871027 5714652114064435745699806250000000000/16261076892419716482154786329943 9663816254451739622514998095408289435592062093859638807396445052510401 211389910092113+128383643324792687052006465711292802772022037582533813 76538116866642322349535069706084962769651301124740992187500000000/1463 4969203177744833939307696949569743462900656566026349828586746049203285 58844736749266568005472593610902509190829017*7^(1/2), d[14,6] = -28055 4617108619480899566370291344261978354334967083819115748587722163286443 43472571539333800/5682569609975836160179732201026169048742626675562559 959923656167528687147398257319295294757+897809263855330877460356232661 5224854689946734573565223242391310021454682858338330596482800/56825696 0997583616017973220102616904874262667556255995992365616752868714739825 7319295294757*7^(1/2), d[12,4] = -484706807009233051677396105115946700 7377962454880881355860175968120404435575838369013119718610838999123854 9116005/62216777479662455044144635737388645647707541999603512174865959 872346899538356268851636409415987986942381737462+785186871216500656550 3524318634403444113209354005930057357742932329054072487979840392849223 5111605737817357410/31108388739831227522072317868694322823853770999801 756087432979936173449769178134425818204707993993471190868731*7^(1/2), \+ d[14,4] = -12055394600384055939522785009745021251857480970152804355049 6246174156031028233120116635471764/63139662333064846224219246678068544 9860291852840284439991517351947631905266473035477254973+38578744713603 9710839708109628704372463579563481219728926521578498242436628004101979 39444984/6313966233306484622421924667806854498602918528402844399915173 51947631905266473035477254973*7^(1/2), d[19,4] = -34741985501142410320 2327433105464777898650089979730892171350134532628956976440721517032390 5900000/14570529411060264347008994825075617897942668324044297260204248 75844485226719930941536345392693-9443214399448667218458046800741542470 31257743430760102589398927628316814717421139295589475000/6335012787417 5062378299977500328773469315949234975205479148907645412401161736127892 884582291*7^(1/2), d[17,4] = -2902371299943364711345183977811024231279 9026542395344895197588777177183642207496988628378304931640625/26624103 9415699470299603787155110304870007259998836853422600721456452023398989 89552861552355510892+6959527963283252321811359063477250593658126731117 07809546541540794183055601306326867048583984375/9646414471583314141289 9922882286342344205528985085816457464029513207254854706483887179537519 967*7^(1/2), d[12,6] = -4676109786647717886742971723992168041862703325 69429233195865087758517314756738814699958597056440087718219667517200/3 4219227613814350274279549655563755106239148099781931696176277929790794 7460959478684000251787933928183099556041+20100261355727555943605407107 7497206823891807179233708267079783906328730510260997956059854961697637 013167779500/307973048524329152468515946900073795956152332898037385265 5865013681171527148635308156002266091405353647896004369*7^(1/2), d[11, 6] = 85969218513659298943510425913829934045506909791783510490507134397 5222098607451881497138855661375989715659069455000000/64762449749003879 0885041573564613633712414326609241639756468404975120848120560249578858 6457537236296886952442917-36953874898454899205085648226164172125031782 2818844183716417453646797092760160451988668280603672846568714581250000 /582862047741034911796537416208152270341172893948317475780821564477608 76330850422462097278117835126671982571986253*7^(1/2), d[8,2] = -786562 8278323187216375157393833043497967974692241554081011840767770926289249 058261030003626061808740201882500000000000/374756550713546865288206149 2900555459051715291414983108765356998580517844061915290432370002111328 335307090326972327+225890229122678674007679660195901527304301986552019 4117607863349754860171406610859436546029673776749003656093750000000/15 3650185792554214768164521208922773821120326948014307459379636941801231 606538526907727170086564461747590703405865407*7^(1/2), d[11,2] = 66222 8506106738222588049543385699753968862145105668737667984760327100127225 7360935521902551105106769745762334000000/21587483249667959696168052452 1537877904138108869747213252156134991706949373520083192952881917907876 5628984147639-46386119481103357968215338054053638806280662604350633282 453700459239020594916447872964594513589539883913390125000/215874832496 6795969616805245215378779041381088697472132521561349917069493735200831 929528819179078765628984147639*7^(1/2), d[14,2] = 11269694422810342387 9735705541471463537895372748015003348764905986059559667548240965153487 2/21046554111021615408073082226022848328676395094676147999717245064921 0635088824345159084991*7^(1/2)-352164421889076565536855984451380138595 5768299188003389727668387685293365134871933721436612/21046554111021615 4080730822260228483286763950946761479997172450649210635088824345159084 991, d[13,4] = 8489969475198088704957243665590473383417596718155218530 998977561323617439023414113185359695938/739386492474300370901015451682 4086846347304360710677553780665363757418821305488069783814818821*7^(1/ 2)-2494621341773228587516531943492565820219439042006125377605464845340 19737190065886775622127864696/7393864924743003709010154516824086846347 304360710677553780665363757418821305488069783814818821, d[7,8] = 79123 0828966309266234636378129092555331429611018628299573524169252864473719 73775444150010628920217678305236562500000000/2206749934300767055225497 8717693343913175183854102086946786846736878734011672706006255352198229 1588885501590801877410733*7^(1/2)+277172370319838098740900487523036354 0195244393128764863194741833862014005005255138194368202794748817266157 550512500000000/169749994946212850401961374751487260870578337339246822 66759112874522103085902081543273347844791660683500122369375185441, d[1 5,5] = -34089352462880768422384919170434563723801665849751174399703326 06848957143739402116275264000000/1687723174162823339573380463704772207 476560122642080308097325881756020082777282423830702542829*7^(1/2)+3674 9638089908746641808901214380067002001309766870386815945072790188427608 0441265744110624000000/16877231741628233395733804637047722074765601226 42080308097325881756020082777282423830702542829, d[19,7] = 59191605923 5001949404169330749043320023949856637270217708569193235032157440377021 936236346981000000/144248241169496617035389048768248617189632416408038 542876022062708604037445273163212098193876607-312440921104045368530242 4166470865031204531972706326437895046743079396554782362160189218750000 0/62716626595433311754516977725325485734622789742625453424357418568958 27715011876661395573646809*7^(1/2), d[14,3] = 528638663362100452483911 731161997256153086844879498678758830462235489967058230035627678951324/ 5682569609975836160179732201026169048742626675562559959923656167528687 147398257319295294757-169170870930575688701446615475589440546319228835 625261193031957238753745647998638967091298344/568256960997583616017973 2201026169048742626675562559959923656167528687147398257319295294757*7^ (1/2), d[18,2] = -8586514296522614043439696583579360982200091242700339 6981321030044716108335899421461537503891456/16923534160672480949631565 41794497234108868929562909060657263675670302716749236559424202412631+5 7077412622417403258846898988439799537838755667879253829046235062835313 901975412244193002268672/169235341606724809496315654179449723410886892 9562909060657263675670302716749236559424202412631*7^(1/2), d[7,4] = 54 7082545030831472812327543789716124468884349915025452869565606415926121 3773031453022902852669935530058091737693417500000/66871210130326274400 7726627808889209490157086487942028690510507178143454899172909280465218 1281563299560654266723557901-17724613131819965524100775802731090396673 4011705400278677357912113460032798051446480601605927828862969262518354 70000000/6687121013032627440077266278088892094901570864879420286905105 071781434548991729092804652181281563299560654266723557901*7^(1/2), d[1 5,4] = -14994779402019586224362315192397876465756130917602452189559753 57117457691678721515896992000000/5682569609975836160179732201026169048 742626675562559959923656167528687147398257319295294757+139093157567435 3720963559722144961232881807923987704561369879451900320030007616003411 2000000/56825696099758361601797322010261690487426266755625599599236561 67528687147398257319295294757*7^(1/2), d[6,6] = 2413048989616421697254 1247101179712384530933807686540286915947592984767568248783970878109584 73225724103882230684880000000/1177539151410549980293698936160978265786 6292801623200812060272556526406122590785597738131698725869936646087127 46087053-1037249285591215935074691316826449224218791687970254427683346 748505663821459782705200509731050248272110548754300000000/105978523626 9494982264329042544880439207966352146088073085424530087376551033170703 7964318528853282942981478414714783477*7^(1/2), d[16,8] = -376442929817 4085812652937045237842806641090470915010775093399616226794991852551269 531250000000000000/290473067635242858121880132608679266623223751060546 28149383758341533298575427748884296525080254703+3171986725391926426000 7520226807386634722162679776363282241456970855044804956054687500000000 000000/548062391764609166267698363412602389855139152944426946214787893 236477331611844318571632548684051*7^(1/2), d[13,6] = 22728711259621610 7352981340050722826455833991232150021375038377877755709310101598701561 4588514600/38525927765766177220631857745557084094125427984755635674962 414263788655963644385205715666687541+217337568430352645135599106163682 61667449568850454068462097511188898468600993917245052138373100/7319926 2754955736719200529716558459778838313171035707782428587101198446330924 3318908597667063279*7^(1/2), d[8,4] = 70297634730270183723312500512365 4011660340119599282815183073124824865051694742654608836153221377970159 4573906250000000/41904596125242058573135778511524392860305543713094811 125285355529582154074510507338471046387244853203888373656145111*7^(1/2 )-21697855198227982661844457970411424912205181842400370958118026528474 40299333279198804342827689133890684550806757812500000/4190459612524205 8573135778511524392860305543713094811125285355529582154074510507338471 046387244853203888373656145111, d[15,7] = 4602076415015728298582450112 97300566729025914500961011570235393032133413845823188000000000000/5625 7439138760777985779348790159073582552004088069343603244196058534002759 2427474610234180943*7^(1/2)-496121605413564922722416463485901153578485 00918483497186208819385834359835849706058000000000000/5625743913876077 7985779348790159073582552004088069343603244196058534002759242747461023 4180943, d[11,8] = 264113342787125024403196816880935149765401730876623 0265920747732424017262908489939350633314948288234148147346406250000/17 4858614322310473538961224862445681102351868184495242734246469343282628 992551267386291834353505380015947715958759+579962955385647421054704597 3456341637052624619571963740010244379611845067321572824785229662354540 612692160156250000/174858614322310473538961224862445681102351868184495 242734246469343282628992551267386291834353505380015947715958759*7^(1/2 ), d[14,7] = -36260751841955989037979405318305203726892177709925899811 8582000379457718815369844905419750000/56825696099758361601797322010261 69048742626675562559959923656167528687147398257319295294757+1160388634 9681159789923363420662218239074894228584740871989796744857862403989728 2707118500000/56825696099758361601797322010261690487426266755625599599 23656167528687147398257319295294757*7^(1/2), d[18,8] = 137740076837761 6470861576644949555140359988154130520934337276658267678113931700634253 21210809600000/1370806267014470956920156798853542759628183832945956339 13238357729294520056688161313360395423111-4197856306073296897714675437 48821778196625350581601949201622698449894930439334345170871104000000/8 0635662765557115112950399932561338801657872526232725831316681017232070 62158127136080023260183*7^(1/2), d[16,2] = 430119262869020251911592053 9217638745658620076885480546789070243858192113043762109375000000000000 /512298179250869238310194237405078071645897268184384976179607730891239 83378179451295055599788809-3624274714941928170508153369914745164663784 0464233887082015180011875787664203906250000000000000/96660033820918724 2094706108311468059709240128649782973923788171492905346758102854623690 562053*7^(1/2), d[17,8] = -1644749669913042691194169029219270067698831 281459643730494549130289512250437935620857238769531250/168528535180014 3705860663358590531980954649535798263969874518633260150275991048571440 724861378247*7^(1/2)-7306028195661488086910686656694613484093429258762 188987462777552520672361471579454785674633789062500/286498509806024429 9963127709603904367622904210857048748786681676542255469184782571449232 2643430199, d[20,2] = -24237381536481244982344125669257838320324339679 0833268226506545243894558046328705219953841625000/98554698935680918471 3838221397971918686929553098399982382759434655058640930437761076447287 3557-34192319333112380719488775456199025868143099742608685367551209765 92226976261211147709234925000/9855469893568091847138382213979719186869 295530983999823827594346550586409304377610764472873557*7^(1/2), d[8,3] = 6543848806923424220776003786522526335370644211586548535470604943808 5131684006065869252228738014656404908749817812500000000/41485550163989 6379874044207264091489317024882759638630140325019742863325337654022650 8633592337240467184948991958365989-19946282669931233631664234339585854 0902097110695345170860271652199229462864214245317015228259191409380720 57656250000000/2440326480234684587494377689788773466570734604468462530 23720599848743132551561189794625505431602380422644058350492117*7^(1/2) , d[1,4] = -3562564185862677576522566600130553326062803925057047320291 312404495638129538317219378526838639219014269096935822706775/324258894 8911977763767725661995347030620848976546873747072905040529414296515069 5957338465229478878801447247230338168+26042099368029811813642011298015 1411760214423501702925221121225033953141763297971043054979315110092042 54623605591/1529523089109423473475342293394031618217381592710789503336 27596251387466816748565836502194478673956610600222784614*7^(1/2), d[10 ,5] = 8129325939607217651345089720828030338238500111054705443197990708 76657668061637397430454090887303441381391859579093750000000/4390490760 9533234501817923090848709230388701969698079049485760238147609856765342 10247799704016417780832707527572487051-9454763159086438575224087390774 3128285462504232901349577933753008409589706013262092195040294182094637 616180781250000000/337730058534871034629368639160374686387605399766908 300380659694139596998898194939249830746462801367756362117505575927*7^( 1/2), d[9,2] = -272540576705668924507680882550557716555063263752313278 0702540156923723171676091340412757386409610757230736706274745041764424 057396459639206295519699214415047918286959073366980723955339077/115986 6851536303674756661044362685145699914015588034480523016860069020798270 6664900420693159607754352308870819152352054758903565794140710917237060 5767836055845408760899612924569795827066*7^(1/2)+474501016589432935418 9318103721279561655942465649358750274883007920993262626721719056371075 8325183694780581014784244811444840891505684278856906387272774709911345 54656606742615807527996952/1414471770166223993605684200442298958170626 8482780908299061181220353912174032518171244747755619212624766915633112 6244570230531290172447694112647080204678116884644830365381615329019301 3, d[1,3] = -879314449190835291710565832299307114362812304082195155496 6183330589258588612599353340098256839675184587240228686537/10599595007 5283046711841220932206391142464544374857712581203924202211514504006756 124696020773721051931145954389737502*7^(1/2)+2338565078845547751586570 4897429779739544494102135779165555274224146736651670325640559279931978 5276790901085272527908599/56177853539900014757275847094069387305506208 5186745876680380798271721026871235807460888910100721575235073558265608 7606, d[13,8] = 554010499685026831220719158954166334390956091635139103 32211292601316365576343023665474609701775000/2195977882648672101576015 8914967537933651493951310712334728576130359533899277299567257930011898 37-3410947805867794467600798379599893356228237356515132605757548838730 37265709838066068949734437500/2195977882648672101576015891496753793365 149395131071233472857613035953389927729956725793001189837*7^(1/2), d[6 ,2] = 1858793013573990927163041768414884892033827943419339301404291059 5102046894920974422195339071158831563053354334624000000/39251305047018 3326764566312053659421928876426720773360402009085217546870753026186591 271056624195664554869570915362351-130200065420297681185594192177122090 2128698931221746252061016346942559993318350580324066720009602631668455 43782000000/3925130504701833267645663120536594219288764267207733604020 09085217546870753026186591271056624195664554869570915362351*7^(1/2), d [18,6] = 2404353913223379539529601904531172994916596759556941941351274 21148504963542471415597102746778777600/4569354223381569856400522662845 1425320939461098198544637746119243098173352229387104453465141037+45471 1795073859519960453643388723750142584579749991231375197706960926188651 886962689087579852800/456935422338156985640052266284514253209394610981 98544637746119243098173352229387104453465141037*7^(1/2), d[6,4] = 1250 6349554212754700070171702073334628141897207792885551083340167194836188 1521128274627499479901218640922952458488500000/10704901376459545275397 2630560088933253329934560210916473275205059330964750825323615801197261 144272151328064795098823-405186035550152091860934808933416191782720309 8975088418687413310336895926488244227730085723390674417169213051140000 00/1070490137645954527539726305600889332533299345602109164732752050593 30964750825323615801197261144272151328064795098823*7^(1/2), d[15,8] = \+ -558823564680481293399297513718150688170960039036881228335285834396162 002527071014000000000000/168772317416282333957338046370477220747656012 2642080308097325881756020082777282423830702542829*7^(1/2)+602433378002 1859775915057056614514007738746540101567515468213782565600837210321449 9000000000000/16877231741628233395733804637047722074765601226420803080 97325881756020082777282423830702542829, d[9,8] = 957896211621177814978 1266278342342332660674426808331010622465369464286389952582714666210942 1679740794547821685808145091638388151963348425330920837627914013398198 8799043594645079529377568125/57993342576815183737833052218134257284995 7007794017240261508430034510399135333245021034657980387717615443540957 6176027379451782897070355458618530288391802792270438044980646228489791 3533+21034314737074898689990305468386720901632244459613798595286392126 6256974764276202184722819534895024650510196329195584645530288970225264 8158818044100729086574047328415702352926192657203125/57993342576815183 7378330522181342572849957007794017240261508430034510399135333245021034 6579803877176154435409576176027379451782897070355458618530288391802792 2704380449806462284897913533*7^(1/2), d[9,4] = -2827178195480552434898 4579194838194759610900891165281838259738365891889463460387380778136754 1337436595763955966666303343931743241181343133481177194498409117271699 2897138709624704860865193/10544244104875487952333282221478955869999218 3235275861865728805460820072570060590003824483269161403202807916537748 6550432627596890376428265203369143343964144049170553632844768816325700 6*7^(1/2)+349051306437255673020204067738789871093523923352832143225741 8629217412044176339647602870027123787897110544596268268117568604304121 566518768991960141832220637361541400357206295989898840915773/421769764 1950195180933312888591582347999687329411034474629152218432802902802423 6001529793307664561281123166615099462017305103875615057130608134765733 758565761966822145313790752653028024, d[8,7] = 72887596118366544475467 7047611896785317417137253036154237862282799540830017979690801375276905 39992954519418765625000000000/1528415006041723504799110237288758118536 4074627986686373590921780000227775597779781873913234926675405418233128 26766417+2325887861642078603868200784554175528256816171252259182050930 13772123164271685632149561988228354278109931640625000000000/4148555016 3989637987404420726409148931702488275963863014032501974286332533765402 26508633592337240467184948991958365989*7^(1/2), d[18,7] = -17039914297 1935852852273740444208586276743235346347327264017458490281812941560720 918010689421312000/456935422338156985640052266284514253209394610981985 44637746119243098173352229387104453465141037+5876998828502615656800545 6124835048947527549081424272888227177782985290261506808323921954560000 00/4569354223381569856400522662845142532093946109819854463774611924309 8173352229387104453465141037*7^(1/2), d[19,6] = -667303888915594215696 5682243728399231408151217259336797801169465880685437507392782565272790 512475000/100973768818647631924772334137774032032742691485626980013215 4438960228262116912142484687357136249-16921800286995097159597929285606 205009003745164177463987639573160518011740701273459584808750000/439016 3861680331822816188440772784001423595281983781739705019299827079400508 3136629769015527663*7^(1/2), d[12,1] = 0, d[5,2] = 0, d[5,3] = 0, d[3, 7] = 0, d[4,8] = 0, d[5,6] = 0, d[2,8] = 0\}: " }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "subs(dd,matrix([seq([seq(d[j,i],j=1..11)],i=1..8)])) :\nevalf[8](%);\nsubs(dd,matrix([seq([seq(d[j,i],j=12..20)],i=1..8)])) :\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7-$\" \"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+7-$!)7\")\\*)!\"(F+F+F+F+$\")!fyk%!\"' $\")#eZD$F2$!)n'*f?!\"%$\")xW#H$F7$!)%oe;%F7$\"))*z5IF77-$\")(R39%F2F+ F+F+F+$!)n@5N!\"&$!)45eCFC$\")]vb:!\"$$!)`c'[#FH$\"):?YJFH$!)b&QF#FH7- $!)U<%4\"FCF+F+F+F+$\")$o#e6F7$\")+,6\")FC$!)N`L^FH$\")Q$\\?)FH$!)R:Q5 !\"#$\")T0.vFH7-$\"))ocr\"FCF+F+F+F+$!)!Rv0#F7$!)P$3W\"F7$\")2<>\"*FH$ !)$=vX\"Ffn$\")*oT%=Ffn$!)p$GL\"Ffn7-$!)7sz:FCF+F+F+F+$\"):(*[?F7$\")T $[V\"F7$!)#)>\"3*FH$\")#\\9X\"Ffn$!)(*[O=Ffn$\")qGF8Ffn7-$\")-W#*yF2F+ F+F+F+$!)!H$z5F7$!)>AevFC$\"))oOy%FH$!)YuXwFH$\")$3Sn*FH$!)**p\"*pFH7- $!)Yg^;F2F+F+F+F+$\")fDXBFC$\")LJU;FC$!)aVR5FH$\")5Lh;FH$!)!\\?5#FH$\" )Z@>:FHQ(pprint46\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7 +$\"\"!F)F(F(F(F(F(F(F(F(7+$!)JR*4$!\"'$!)Iy`G!\"($!(mbc#F-$!)-ubAF-$! )eQC:F-$!)tFUJF-$\")3_\\QF-$!)+e$[&F-$!)K2^DF-7+$\")SwSB!\"&$\")t\\K;F -$\")EPE9F-$\")!>TD\"FB$\"(%4v%)FB$\")p+0HFB$!)\\$\\+%FB$\")8oubFB$\") ;jx@FB7+$!)G$Qs(FB$!)+6qIF-$!(2v#HFB$!)<(Rd#FB$!)-WR(z\"Fjn$!)'RQU#Fjn$!)()=x#)FB7+$\")f0s8Fjn$\")Kn2HF0$\"(kdT#FB$\")% GS7#FB$\")vPN9FB$\")'*>Q?Fjn$!)Fjn7+$\")>V(>(FB$!)k'p&pF-$!(vQy*F-$!)2M-')F-$!(&H8eFB$\")Y6 d6Fjn$!)H)))Q$Fjn$\")PF!4%Fjn$\")7;*>\"Fjn7+$!)z\"Rc\"FB$\")iu\"[#F-$ \"(R,'RF-$\"))**=[$F-$\"(,IN#FB$!):$fd#FB$\")3uq')FB$!)4v95Fjn$!)1>$)H FBQ(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions are \+ satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "nm := NULL:\n for ct to nops(SO8_20) do\n eqn_group := convert(SO8_20[ct],'polynom _order_conditions',8):\n tt := expand(subs(\{op(e23),op(dd)\},eqn_gr oup));\n tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op( i,tt),i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([% ]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\s\"\"\"\"\"#\"\"$\"\"%\"\"&\" \"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A \"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\" #S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"# d\"#e\"#f\"#g\"#h\"#i\"#j\"#k\"#l\"#m\"#n\"#o\"#p\"#q\"#r\"#s\"#t\"#u \"#v\"#w\"#x\"#y\"#z\"#!)\"#\")\"##)\"#$)\"#%)\"#&)\"#')\"#()\"#))\"#* )\"#!*\"#\"*\"##*\"#$*\"#%*\"#&*\"#'*\"#(*\"#)*\"#**\"$+\"\"$,\"\"$-\" \"$.\"\"$/\"\"$0\"\"$1\"\"$2\"\"$3\"\"$4\"\"$5\"\"$6\"\"$7\"\"$8\"\"$9 \"\"$:\"\"$;\"\"$<\"\"$=\"\"$>\"\"$?\"\"$@\"\"$A\"\"$B\"\"$C\"\"$D\"\" $E\"\"$F\"\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$G\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "principle error graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 96 "The interpolation scheme amounts to having a Runge -Kutta method for each value of the parameter " }{TEXT 268 1 "u" } {TEXT -1 8 " where " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "The nodes and linking coefficients are ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "e_u := map(_U->lhs(_U)=rhs(_U)/u,e23):" }}} {PARA 0 "" 0 "" {TEXT -1 43 " ... and the weight polynomials (of degre e " }{XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 33 " and re-usi ng the weight symbol " }{XPPEDIT 18 0 "b[j]" "6#&%\"bG6#%\"jG" }{TEXT -1 10 ") are ... " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "pols := [seq(b[j]=add(simplify(subs(dd,d[j,i]))*u^(i-1),i=1..8),j=1..20)]:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "The wh ole interpolation scheme (Runge-Kutta scheme with a parameter), includ ing the weights, is given by the set of equations:" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "eu := `union`(e_u,\{op(pols)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can calcula te the principal error norm, that is, the root mean square of the resi dues of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "errterms8_20 := PrincipalErrorTerms(8,20,'expanded'): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "sm := 0:\nfor ct to nops(errterms8_20) do\n sm := s m+expand(subs(eu,errterms8_20[ct]))^2;\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Because the step has widt h " }{XPPEDIT 18 0 "u*h" "6#*&%\"uG\"\"\"%\"hGF%" }{TEXT -1 17 " we \+ multiply by " }{XPPEDIT 18 0 "u^9;" "6#*$%\"uG\"\"*" }{TEXT -1 45 " in order to provide appropriate weighting. 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}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "abreviated c alculation for stages 15 to 20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 172 "If the only criterion in the selection o f the nodes is to obtain a reasonable principal error curve for an ord er 8 interpolant (without considering an order 7 interpolant) " }} {PARA 0 "" 0 "" {TEXT -1 120 "then one can improve slightly on the sit uation given by Verner's nodes as indicated by the calculations in thi s section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e5 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6888 "e5 := \{a[13,5] = 0, a[14,10] = 30448415149825325326308593750/15 127426330731345559308904251+5887803942383224482632816406250/7412438902 05835932406136308299*7^(1/2), a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), a [12,3] = 0, a[4,2] = 0, c[13] = 1, a[11,3] = 0, a[12,2] = 0, a[11,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[10,2] = \+ 0, a[8,2] = 0, a[8,3] = 0, a[9,2] = 0, a[14,2] = 0, a[6,3] = 0, a[9,3] = 0, a[14,5] = 0, a[6,2] = 0, a[7,2] = 0, a[7,3] = 0, a[5,2] = 0, a[1 3,2] = 0, a[14,4] = 0, a[13,4] = 0, a[13,3] = 0, a[14,11] = -146658482 4552592368357500/602474973863778980873559-946677979546641857718938375/ 59042547438650340125608782*7^(1/2), a[14,8] = 742007034160287983271289 06250/42881511753324799479599510367+28676647199217261041085964843750/2 101194075912915174500376007983*7^(1/2), a[14,1] = 89410655679264792064 38689/198753096356125278622613280+152838094177334666489948287/33112265 8529304714185273724480*7^(1/2), a[14,9] = -128719957415479235191318173 4948009779654190448189357505217021276723163171317107252506643673730749 47/9591164890876824145947562349212814323821053901457549227864771925211 8577198224684430692759024543840-88218508206817762290678760500547255563 6444759904636504531314529608120034066591927204665390933748491981/15978 8807082007890271486388737885486634858757998282770136227100274029549612 242324261534136534890037440*7^(1/2), a[9,1] = -14725142644862158038813 84708877264246346044433307094207829051978044531801133057155/1246894801 6200320011570596216439860248033015583934879004404536361680460696864366 08, a[8,7] = 6070139212132283/92502016000000, a[8,6] = 150140835352868 9/265697280000000, a[8,5] = -1024030607959889/168929280000000, c[6] = \+ 93/200, a[9,5] = -1207067925846925480797893644173318794948457151612046 9966534514296406891652614970375/27220311547616572217104781845311006994 97284085048389015085076961673446140398628096, a[9,4] = -51722943110856 68458375175655246981230039025336933699114138315270772319372469280000/1 2461938100480914589727863057121529836525707941023625292185093674907648 7132995191, a[9,8] = 1033284818445201560405683676728665685912400779697 0668046446015775000000/13127035500360336480738342487407279145379720286 38950165249582733679393783, a[9,7] = 664113122959911642134782135839106 469928140328160577035357155340392950009492511875/151784655985862481363 33023107295349175279765150089078301139943253016877823170816, a[9,6] = \+ 7801251558438936413230905525304310365677955925684971827014606748031267 70111481625/1831104254127319721978898745071587868592261029808618595052 41443073629143100805376, a[10,7] = 31860723517364931240515126584966086 9927653414425413/6714716715558965303132938072935465423910912000000, a[ 10,8] = 212083202434519082281842245535894/2002242604477567256382286537 1173879, a[10,6] = 21127670214172802870128286992003940810655221489/467 9473877997892906145822697976708633673728000, a[10,5] = -18026925980317 2281163724663224981097/38100922558256871086579832832000000, a[10,4] = \+ -20462749524591049105403365239069/454251913499893469596231268750, a[10 ,1] = -29055573360337415088538618442231036441314060511/226747598910895 77691327962602370597632000000000, a[11,1] = -2342659845814086836951207 140065609179073838476242943917/135848096135105677702223140013915876085 7532162795520000, a[10,9] = -26984049294008425187211664850871297985622 69848229517793703413951226714583/4695456749139343150770004420808711418 84676035902717550325616728175875000000, a[11,6] = 20980822345096760292 224086794978105312644533925634933539/377588999200755080387872783911549 4641972212962174156800, a[11,5] = -26053085959256534152588089363841/43 77552804565683061011299942400, a[11,4] = -9962860301325381596139308896 52/16353068885996164905464325675, a[5,4] = 3982992/2907025, a[12,1] = \+ -2866556991825663971778295329101033887534912787724034363/8682267116192 62703011213925016143612030669233795338240, a[11,10] = -311552374371117 30665923206875/392862141594230515010338956291, a[11,9] = 3007606697681 0251783423249756545243494667226619587649637187426239268485224392535986 4884962513/46554433375013464555850653366045056037608247796155212857518 92810315680492364106674524398280000, a[11,8] = 16102142614312417838907 5121929246710833125/10997207722131034650667041364346422894371443, a[11 ,7] = 890722993756379186418929622095833835264322635782294899/139212420 01395112657501941955594013822830119803764736, a[12,6] = 23463053885534 04258656258473446184419154740172519949575/2567267164078954028927449783 01151486254183185289662464, a[12,5] = -4583493974484572912949314673356 033540575/451957703655250747157313034270335135744, a[6,1] = 5611/11440 0, a[12,4] = -16957088714171468676387054358954754000/14369041511965468 3326368228101570221, a[12,9] = -32058909627170725427914343121527275340 08102774023210240571361570757249056167015230160352087048674542196011/9 4756954968396581478301512445127360498465774712725761537244920597319265 7306017239103491074738324033259120, a[12,8] = 345685379554677052215495 825476969226377187500/74771167436930077221667203179551347546362089, a[ 12,7] = 1657121559319846802171283690913610698586256573484808662625/134 31480411255146477259155104956093505361644432088109056, a[13,1] = 44901 867737754616851973/1014046409980231013380680, a[12,11] = -612293360107 0769591613093993993358877250/1050517001510235513198246721302027675953, a[12,10] = 40279545832706233433100438588458933210937500/8896460842799 482846916972126377338947215101, c[3] = 341/3200, a[13,6] = 79163867519 1615279648100000/2235604725089973126411512319, a[13,8] = -137345124323 97741476562500000/875132892924995907746928783, a[13,7] = 3847749490868 980348119500000/15517045062138271618141237517, a[13,12] = -30681427293 6976936753/1299331183183744997286, c[2] = 1/20, a[13,11] = 28203554318 3190840068750/12295407629873040425991, a[13,10] = -9798363684577739445 312500000/308722986341456031822630699, a[13,9] = 122747654703131968784 2881203774063505031923427600698639829444355496961634227421531633068444 8207141/48934514749371551765038583414351093488882928068660965448289652 6796523353052166757299452852166040, a[8,1] = -1221101821869329/6908129 28000000, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = -3899844/290702 5, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100, a[8,4] = -125/2, a[ 7,6] = 5611/283500, a[14,6] = 3037913416047823635649583750/15649233075 629811884880586233-302465625814318865951896498250/53676869449410254765 14041077919*7^(1/2), a[7,5] = -26782109/689364000, a[14,7] = 183874328 794901398385760606250/760335208044775309288920638333-11478722909055440 7592495836250/37256425194193990155157111278317*7^(1/2), c[7] = 31/200, c[8] = 943/1000, c[9] = 7067558016280/7837150160667, a[3,2] = 116281/ 1024000, a[7,4] = 8602624/76559175, a[4,1] = 1023/25600, a[2,1] = 1/20 , a[3,1] = -7161/1024000, c[5] = 39/100, a[14,12] = 113521280986681466 59861/254668911904014019468056-5215842639928607924801/1273344559520070 09734028*7^(1/2), c[11] = 47/50, a[6,5] = 923521/5106400, a[7,1] = 211 73/343200, a[6,4] = 31744/135025\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "set up order relations e tc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1975 "SO7_14 := SimpleOrderConditions(7,14,'expanded'):\n SO7_15 := SimpleOrderConditions(7,15,'expanded'):\nSO7_16 := SimpleOrd erConditions(7,16,'expanded'):\nSO7_17 := SimpleOrderConditions(7,17,' expanded'):\nSO7_18 := SimpleOrderConditions(7,18,'expanded'):\nSO7_19 := SimpleOrderConditions(7,19,'expanded'):\nSO8_20 := SimpleOrderCond itions(8,20,'expanded'):\nerrterms8_20 := PrincipalErrorTerms(8,20,'ex panded'):\n\nwhch := [1,2,4,8,16,17,25,27,32,64]:\ninterp_order_eqns15 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_14[ct],'interpo lation_order_condition'):\n interp_order_eqns15 := [op(interp_order_ eqns15),temp_eqn];\nend do:\nwhch := [1,2,4,8,16,17,25,27,32,63,64]:\n interp_order_eqns16 := []:\nfor ct in whch do\n temp_eqn := convert( SO7_15[ct],'interpolation_order_condition'):\n interp_order_eqns16 : = [op(interp_order_eqns16),temp_eqn];\nend do:\nwhch := [1,2,4,8,16,17 ,25,27,32,61,63,64]:\ninterp_order_eqns17 := []:\nfor ct in whch do\n \+ temp_eqn := convert(SO7_16[ct],'interpolation_order_condition'):\n \+ interp_order_eqns17 := [op(interp_order_eqns17),temp_eqn];\nend do:\n whch := [1,2,4,8,16,17,25,27,32,61,63,64]:\ninterp_order_eqns18 := []: \nfor ct in whch do\n temp_eqn := convert(SO7_17[ct],'interpolation_ order_condition'):\n interp_order_eqns18 := [op(interp_order_eqns18) ,temp_eqn];\nend do:\nwhch := [1,2,4,8,16,17,25,27,32,61,63,64]:\ninte rp_order_eqns19 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_ 18[ct],'interpolation_order_condition'):\n interp_order_eqns19 := [o p(interp_order_eqns19),temp_eqn];\nend do:\nwhch := [1,2,4,8,16,17,25, 27,32,61,63,64]:\ninterp_order_eqns20 := []:\nfor ct in whch do\n te mp_eqn := convert(SO7_19[ct],'interpolation_order_condition'):\n int erp_order_eqns20 := [op(interp_order_eqns20),temp_eqn];\nend do:\nwhch := [1,2,4,8,16,17,25,27,32,58,61,63,64,102,117,121,123,125,127,128]: \nordeqns := []:\nfor ct in whch do\n eqn_group := convert(SO8_20[ct ],'polynom_order_conditions',8):\n ordeqns := [op(ordeqns),op(eqn_gr oup)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1722 "calc_coeffs := proc()\n local e qns,pols,e_u,eu,ct,eqs_15,eqs_16,eqs_17,eqs_18,eqs_19,eqs_20;\n glob al dd,sm,e6,e7,e8,e9,e10,e11,e12,e13,e14,e15,e16,e17,e18,e19,e20,e21,e 22,e23;\n\n e6 := `union`(e5,\{c[15]=c_15,seq(a[15,i]=0,i=2..5)\}): \n eqs_15 := expand(subs(e6,interp_order_eqns15)):\n e7 := solve( \{op(eqs_15)\}):\n e8 := `union`(e6,e7):\n e9 := `union`(e8,\{c[16 ]=c_16,seq(a[16,i]=0,i=2..5)\}):\n eqs_16 := expand(subs(e9,interp_o rder_eqns16)):\n e10 := solve(\{op(eqs_16)\}):\n e11 := `union`(e9 ,e10):\n e12 := `union`(e11,\{c[17]=c_17,seq(a[17,i]=0,i=2..5)\}):\n eqs_17 := expand(subs(e12,interp_order_eqns17)):\n e13 := solve( \{op(eqs_17)\}):\n e14 := `union`(e12,e13):\n e15 := `union`(e14, \{c[18]=c_18,seq(a[18,i]=0,i=2..5),a[18,17]=0\}):\n eqs_18 := expand (subs(e15,interp_order_eqns18)):\n e16 := solve(\{op(eqs_18)\}):\n \+ e17 := `union`(e15,e16):\n e18 := `union`(e17,\{c[19]=c_19,seq(a[19 ,i]=0,i=2..5),a[19,17]=0,a[19,18]=0\}):\n eqs_19 := expand(subs(e18, interp_order_eqns19)):\n e19 := solve(\{op(eqs_19)\}):\n e20 := `u nion`(e18,e19):\n e21 := `union`(e20,\{c[20]=c_20,seq(a[20,i]=0,i=2. .5),seq(a[20,i]=0,i=17..19)\}):\n eqs_20 := expand(subs(e21,interp_o rder_eqns20)):\n e22 := solve(\{op(eqs_20)\}):\n e23 := `union`(e2 1,e22):\n eqns := []:\n for ct to nops(ordeqns) do\n eqns := \+ [op(eqns),expand(subs(e23,ordeqns[ct]))];\n end do:\n dd := solve( \{op(eqns)\}):\n e_u := map(_U->lhs(_U)=rhs(_U)/u,e23):\n pols := \+ [seq(b[j]=add(simplify(subs(dd,d[j,i]))*u^(i-1),i=1..8),j=1..20)]:\n \+ eu := `union`(e_u,\{op(pols)\}):\n sm := 0:\n for ct to 286 do\n \+ sm := sm+expand(subs(eu,errterms8_20[ct]))^2;\n end do:\n ret urn(c[15]=c_15,c[16]=c_16,c[17]=c_17,c[18]=c_18,c[19]=c_19,c[20]=c_20) ;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 17 "Sample comparison" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The following error curve is for the published interpolation scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "c_15 := 69/400:\n c_16 := 3923/5000:\nc_17 := 37/100:\nc_18 := 1/2:\nc_19 := 7/10:\nc_20 := 9/10:\ncalc_coeffs();\nssmA := sqrt(sm)*u^9:plot(ssmA,u=0..1,color =blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(/&%\"cG6#\"#:#\"#p\"$+%/&F% 6#\"#;#\"%BR\"%+]/&F%6#\"#<#\"#P\"$+\"/&F%6#\"#=#\"\"\"\"\"#/&F%6#\"#> #\"\"(\"#5/&F%6#\"#?#\"\"*FF" }}{PARA 13 "" 1 "" {GLPLOT2D 405 267 267 {PLOTDATA 2 "6&-%'CURVESG6#7]q7$$\"3`*****\\n5;\"o!#@$\"3#e%oXN8&z B#!#F7$$\"3#******\\8ABO\"!#?$\"3[e4IFn)H*))F-7$$\"33+++-K[V?F1$\"3-a, g)*3w()>!#E7$$\"3#)******pUkCFF1$\"3.Pp$31H0^$F97$$\"3s*****\\Smp3%F1$ 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1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#>\"\"',&*(\"E=iR-\"QSF&*ywHdu#>%3T*\"\"\"\"Hb5_7#ez2L1A[qNmoz&z n&!\"\"\"\"(#F,\"\"#F.#\"HT!))RJM)p#G\"HqGS=%e!e9!Q\"I)zv]c\\13zUft`)) =(o[1W?F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"(,&#\"I& Q$Q!)f(GR(y#3@![%R%)\\1bc$\"J9p1*y?Z*fmg\"[IAOq`\"f(=9\"\"\"*(\"DY7Lc1 vp,`p3?d)\\U%=%F-\"G`0w9\"*yc'Q-;8(R1*G7v>%!\"\"F(#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"),&#\"IDc^'H^]cP5?OZ_WL9(Qf G\"H'o(fHL(p(*Rf&3/jv\\^T:+)\"\"\"*(\"F]iNr!\\r#3jC=Cj[n?%3UF-\"G4=Daw ktS=\\L4x@Zpw\"[\"!\"\"\"\"(#F-\"\"#F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"*,&#\"\\r*GQ;'yf*3O7TF7]*RGLsJNb#o*zgw.\\v,\"RA hGKk-b.Cfh6)fn^(oAZOJ`,oX\"\\r+++Uz!)RSSO1=axZ?Z\"H\"3K&)pw[OG2*)ev?d^ n'GE@EFlp'*fqyY$[#[SN)[&=6!\"\"*(\"iqtHFeFE\")>5?[mRNg+)*Ru:SD.:-+li)R fb0*3.%Rj*Qv%)*=$3HL*p')=w++0;#\"\"\"\"iq++5O+4\"HZ$[N(z!eO7'\\\"\\0mA rwMeIAsqOiT..u05UpD7FdR&[#\\P?))GJ%3&F-\"\"(#F0\"\"#F-" }}{PARA 11 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B" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "nod es and linking coefficients: ee" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21749 "ee := \{c[2] = 1/20, c[3] = 341/3200, c[4] = 1023/6400, c[5] = 39/100, c[6] = 93/200, c[7] = 31 /200, c[8] = 943/1000, c[9] = 7067558016280/7837150160667, c[10] = 909 /1000, c[11] = 47/50, c[12] = 1, c[13] = 1, c[14] = 1/2-1/14*7^(1/2), \+ c[15] = 9/20, c[16] = 371/500, c[17] = 6/25, c[18] = 3/4, c[19] = 7/10 , c[20] = 9/10,\na[2,1] = 1/20, a[3,1] = -7161/1024000, a[3,2] = 11628 1/1024000, a[4,1] = 1023/25600, a[4,2] = 0, a[4,3] = 3069/25600, a[5,1 ] = 4202367/11628100, a[5,2] = 0, a[5,3] = -3899844/2907025, a[5,4] = \+ 3982992/2907025, a[6,1] = 5611/114400, a[6,2] = 0, a[6,3] = 0, a[6,4] \+ = 31744/135025, a[6,5] = 923521/5106400, a[7,1] = 21173/343200, a[7,2] = 0, a[7,3] = 0, a[7,4] = 8602624/76559175, a[7,5] = -26782109/689364 000, a[7,6] = 5611/283500, a[8,1] = -1221101821869329/690812928000000, a[8,2] = 0, a[8,3] = 0, a[8,4] = -125/2, a[8,5] = -1024030607959889/1 68929280000000, a[8,6] = 1501408353528689/265697280000000, a[8,7] = 60 70139212132283/92502016000000, a[9,1] = -14725142644862158038813847088 77264246346044433307094207829051978044531801133057155/1246894801620032 001157059621643986024803301558393487900440453636168046069686436608, a[ 9,2] = 0, a[9,3] = 0, a[9,4] = -51722943110856684583751756552469812300 39025336933699114138315270772319372469280000/1246193810048091458972786 30571215298365257079410236252921850936749076487132995191, a[9,5] = -12 0706792584692548079789364417331879494845715161204699665345142964068916 52614970375/2722031154761657221710478184531100699497284085048389015085 076961673446140398628096, a[9,6] = 78012515584389364132309055253043103 6567795592568497182701460674803126770111481625/18311042541273197219788 9874507158786859226102980861859505241443073629143100805376, a[9,7] = 6 6411312295991164213478213583910646992814032816057703535715534039295000 9492511875/15178465598586248136333023107295349175279765150089078301139 943253016877823170816, a[9,8] = 10332848184452015604056836767286656859 124007796970668046446015775000000/131270355003603364807383424874072791 4537972028638950165249582733679393783, a[10,1] = -29055573360337415088 538618442231036441314060511/226747598910895776913279626023705976320000 00000, a[10,2] = 0, a[10,3] = 0, a[10,4] = -20462749524591049105403365 239069/454251913499893469596231268750, a[10,5] = -18026925980317228116 3724663224981097/38100922558256871086579832832000000, a[10,6] = 211276 70214172802870128286992003940810655221489/4679473877997892906145822697 976708633673728000, a[10,7] = 3186072351736493124051512658496608699276 53414425413/6714716715558965303132938072935465423910912000000, a[10,8] = 212083202434519082281842245535894/200224260447756725638228653711738 79, a[10,9] = -2698404929400842518721166485087129798562269848229517793 703413951226714583/469545674913934315077000442080871141884676035902717 550325616728175875000000, a[11,1] = -234265984581408683695120714006560 9179073838476242943917/13584809613510567770222314001391587608575321627 95520000, a[11,2] = 0, a[11,3] = 0, a[11,4] = -99628603013253815961393 0889652/16353068885996164905464325675, a[11,5] = -26053085959256534152 588089363841/4377552804565683061011299942400, a[11,6] = 20980822345096 760292224086794978105312644533925634933539/377588999200755080387872783 9115494641972212962174156800, a[11,7] = 890722993756379186418929622095 833835264322635782294899/139212420013951126575019419555940138228301198 03764736, a[11,8] = 161021426143124178389075121929246710833125/1099720 7722131034650667041364346422894371443, a[11,9] = 300760669768102517834 2324975654524349466722661958764963718742623926848522439253598648849625 13/4655443337501346455585065336604505603760824779615521285751892810315 680492364106674524398280000, a[11,10] = -31155237437111730665923206875 /392862141594230515010338956291, a[12,1] = -28665569918256639717782953 29101033887534912787724034363/8682267116192627030112139250161436120306 69233795338240, a[12,2] = 0, a[12,3] = 0, a[12,4] = -16957088714171468 676387054358954754000/143690415119654683326368228101570221, a[12,5] = \+ -4583493974484572912949314673356033540575/4519577036552507471573130342 70335135744, a[12,6] = 23463053885534042586562584734461844191547401725 19949575/256726716407895402892744978301151486254183185289662464, a[12, 7] = 1657121559319846802171283690913610698586256573484808662625/134314 80411255146477259155104956093505361644432088109056, a[12,8] = 34568537 9554677052215495825476969226377187500/74771167436930077221667203179551 347546362089, a[12,9] = -320589096271707254279143431215272753400810277 4023210240571361570757249056167015230160352087048674542196011/94756954 9683965814783015124451273604984657747127257615372449205973192657306017 239103491074738324033259120, a[12,10] = 402795458327062334331004385884 58933210937500/8896460842799482846916972126377338947215101, a[12,11] = -6122933601070769591613093993993358877250/105051700151023551319824672 1302027675953, a[13,1] = 44901867737754616851973/101404640998023101338 0680, a[13,2] = 0, a[13,3] = 0, a[13,4] = 0, a[13,5] = 0, a[13,6] = 79 1638675191615279648100000/2235604725089973126411512319, a[13,7] = 3847 749490868980348119500000/15517045062138271618141237517, a[13,8] = -137 34512432397741476562500000/875132892924995907746928783, a[13,9] = 1227 4765470313196878428812037740635050319234276006986398294443554969616342 274215316330684448207141/489345147493715517650385834143510934888829280 686609654482896526796523353052166757299452852166040, a[13,10] = -97983 63684577739445312500000/308722986341456031822630699, a[13,11] = 282035 543183190840068750/12295407629873040425991, a[13,12] = -30681427293697 6936753/1299331183183744997286, a[14,1] = 8941065567926479206438689/19 8753096356125278622613280+152838094177334666489948287/3311226585293047 14185273724480*7^(1/2), a[14,2] = 0, a[14,3] = 0, a[14,4] = 0, a[14,5] = 0, a[14,6] = 3037913416047823635649583750/1564923307562981188488058 6233-302465625814318865951896498250/5367686944941025476514041077919*7^ (1/2), a[14,7] = 183874328794901398385760606250/7603352080447753092889 20638333-114787229090554407592495836250/372564251941939901551571112783 17*7^(1/2), a[14,8] = 74200703416028798327128906250/428815117533247994 79599510367+28676647199217261041085964843750/2101194075912915174500376 007983*7^(1/2), a[14,9] = -1287199574154792351913181734948009779654190 44818935750521702127672316317131710725250664367373074947/9591164890876 8241459475623492128143238210539014575492278647719252118577198224684430 692759024543840-882185082068177622906787605005472555636444759904636504 531314529608120034066591927204665390933748491981/159788807082007890271 4863887378854866348587579982827701362271002740295496122423242615341365 34890037440*7^(1/2), a[14,10] = 30448415149825325326308593750/15127426 330731345559308904251+5887803942383224482632816406250/7412438902058359 32406136308299*7^(1/2), a[14,11] = -1466584824552592368357500/60247497 3863778980873559-946677979546641857718938375/5904254743865034012560878 2*7^(1/2), a[14,12] = 11352128098668146659861/254668911904014019468056 -5215842639928607924801/127334455952007009734028*7^(1/2), a[14,13] = 3 /392-3/392*7^(1/2), a[15,1] = 339349033530268807690405611/740259322368 8213516896000000+784699017603056191374015321/6151316175555109038556160 00000*7^(1/2), a[15,2] = 0, a[15,3] = 0, a[15,4] = 0, a[15,5] = 0, a[1 5,6] = 39667151342526762946267729161/264960560010663481648771830400-15 4459727654214653639901999/5591570455455665877423584000*7^(1/2), a[15,7 ] = 331704901880412220279128473073/1418701262824070547944341715840-110 0134249641546169459352427/132690135289901616433045894400*7^(1/2), a[15 ,8] = 4171780920238230167282154375/180381659089210106588738944-3217313 5433579259286326757125/3056994432985560753767049472*7^(1/2), a[15,9] = -15372775602531728215983123367589049442461995162196337284574757119670 408788921748236494825887253876104043/166377350147863276001131183608793 7178622019554334472825241848191108179400377366974818139697364536000000 +405732438148607561541740662873496460776199401666888744396064357816739 2109927444312471311939993/96852668533426148027403526835032792014972524 8811661666881873633747980768250399197689344000000*7^(1/2), a[15,10] = \+ 6659828881684523938253413125/487858546317362618188848512-1052748123064 35785357342625/16947939574300438348992256*7^(1/2), a[15,11] = -5009581 586726585160535323/182154187109230228533200+16499583672610380318969864 3/13162109004021797158528000*7^(1/2), a[15,12] = -17644888836605543632 66503/153994806895851258937600000, a[15,13] = -79893/16000000+101277/6 4000000*7^(1/2), a[15,14] = 708939/16000000*7^(1/2), a[16,1] = 1674617 1708057136796041288508983162021370539915059038360963678399718853176044 73217871843621390982227/3971673185491125692358052875295731073809182926 8688077817226330743141912382660576772065625000000000000-62547244342110 1973099323918176333336959929577122180411397436070936568965530665480389 06825979321/9806600458002779487303834259989459441504155374984710572154 6495662078796006569325363125000000000000*7^(1/2), a[16,2] = 0, a[16,3] = 0, a[16,4] = 0, a[16,5] = 0, a[16,6] = 4953373314481393631472009715 821845054925887895017645198589189719808023283132738773826477363486812/ 1609577127855048565864775134835217928822979884300178959174051338949832 7596076419765602058349609375+12003592096366176342172464284365465267756 69126140241668109096822220799690553720982691932935469701/3815293932693 4484524202077270168128683211375035263501254496031738070109857366328333 278953125000000*7^(1/2), a[16,7] = 46524774257782474024979595329864014 557423886513194065784729256922453947551327371902009709751399763/178749 8875350334747249150518151874430640029622863376805482684947617129790277 30922844930286328125000+5917947991674583424374460853714087059815712755 844405621829850895301134910279168916803272958541371/142999910028026779 7799320414521499544512023698290701444386147958093703832221847382759442 290625000000*7^(1/2), a[16,8] = 98379382275525268694725052370138586269 88567133654547552254520645532228606707500790611669969401/3225973101185 1863918859932655845579123065016620051167879053452808695088108837455114 50895875000+2856162814976375620056144380757726653475128559212326231546 11070914192208615136563873961039661/9558438818326478198180720786917208 62905630122075590159379361564702076684706294966355821000000*7^(1/2), a [16,9] = -318405203323081407692841375826141471960585444878888589294363 5710156499067977578602196904297100796120493022555269816622293571555931 5151116468777498783943543083388279233890381/19165976839155508136982432 4462805438958033411539266476775669297424659171219582723059092253077982 1359617221649116046771985003636615158443734095303183009069057187500000 0000000+41420140256469841686369843655407910695214632683264112985040094 1676791362334522616869825127467825507020939475640733133374242661690656 14041755234408498232180327610377394167/3833195367831101627396486489256 1087791606682307853295355133859484931834243916544611818450615596427192 3444329823209354397000727323031688746819060636601813811437500000000000 0*7^(1/2), a[16,10] = 159385728532773376012457126309865304539298936154 2527153916765424022637207535538065172875193509/56901760732952826742531 1622251345464267306663022026387368840720992228188711915769497165687500 +307412044659630074756695406895036897431549735076151996582307598857489 54946271746253192825123/3371956191582389732890735539267232380843298743 83423044366720427254653741458913048590913000000*7^(1/2), a[16,11] = -9 6158633263000021736176081518677971992754802559724735244662028793265489 697951563483807219489943/226620748704260115974929574558754437959802651 36322947593782892225347467590606715447437500000000-8480979682230601716 6887302540539985441826761552922862300464496874472373645260113147330694 619/209834026578018625902712569035883738851669121632619885127619372456 920996209321439328125000000*7^(1/2), a[16,12] = 2734753906564893127311 3602222812309373671975224977018252300131621126775277495416157513291254 0789/29935506655395543992957483547554285885024664375096226229774236480 11521411250435360468750000000000+2631889880176553367478291455572062331 52074762297307472568607403164903392978255666269313019069/2217444937436 7069624412950775966137692610862500071278688721656651937195638892113781 250000000000*7^(1/2), a[16,13] = 7657600557578519236927356742918368477 3974122379421535069547976473129717157/55437992885728406227840443875374 03737759671764330612769171296320800781250000+1189094049890461233623674 09734124703424015186286628637854661183118141741311/8870078861716544996 4544710200598459804154748229289804306740741132812500000000*7^(1/2), a[ 16,14] = 9578806188437261689373090827210246131323841838285462486649801 851406702570303/499949899478568899800161093857918591623417671837815260 637993268203125000000000-286011583144180210983205400664586635494201599 866230048023381904611072990151/999899798957137799600322187715837183246 8353436756305212759865364062500000000*7^(1/2), a[16,15] = 126358883473 631095201313582221638106695002907774309390964517191171456273477/206229 3335349096711675664512163914190446597896330987950131722231337890625000 -667187169216219383532290229658720215318230956208501949338288072118597 379/412458667069819342335132902432782838089319579266197590026344446267 57812500*7^(1/2), a[17,1] = 75904016597212274352620201077585695621748/ 1555169351879994201063525997635720322265625-16031132113733987623688959 9190517742013/311033870375998840212705199527144064453125*7^(1/2), a[17 ,2] = 0, a[17,3] = 0, a[17,4] = 0, a[17,5] = 0, a[17,6] = 265211294810 8236779675852103101319384090464/12420533056451560429826348580335898898 078125-773102852863002395047055595586658963648/13074245322580589926132 9985056167356821875*7^(1/2), a[17,7] = 3265505606470480175039786365288 2675227617248/155176782436333689589257284317258630419371875+1103786020 51450377273915565181129491329216/3103535648726673791785145686345172608 3874375*7^(1/2), a[17,8] = -365032134010684654668509433415077086432/10 3723686305239653444288515533579762663+55315637939543784771309599832479 97120/5459141384486297549699395554398934877*7^(1/2), a[17,9] = 2463543 6365841922229934366224054149750141193382214112974747865325096462030593 68497583694524067255325664382625575544/3823164964762584884155733387146 3334154707064092287367735522942158871531858247267111007948859597084120 4373017578125-11592026816867758054499492684050985165343162584169871926 3191663083353612400657431413037172143283745436569876116611/76463299295 2516976831146677429266683094141281845747354710458843177430637164945342 22015897719194168240874603515625*7^(1/2), a[17,10] = -8751183591582159 11384486198234977080928/109772652071629208992639244024476495017+304395 16745616652500404896664815444160/1568180743880417271323417771778235643 1*7^(1/2), a[17,11] = 118364493841744414303786169254455656328/22770202 501285002774702584389480484375-353315848337304788532673730285272096/23 9686342118789502891606151468215625*7^(1/2), a[17,12] = -12718689832993 99472812234477872349348/30078338034769176569907176844579296875+3752601 621753783651501005551584508/226152917554655462931632908605859375*7^(1/ 2), a[17,13] = -43884412335325588358/2956370631395177734375+1255812007 0786700736/2956370631395177734375*7^(1/2), a[17,14] = 1969695348355905 63883592/2582599130067348931640625-5241128924571054652928/103303965202 693957265625*7^(1/2), a[17,15] = -10969058169805055883776/560115354880 23799921875+8953379007044874752/448092283904190399375*7^(1/2), a[17,16 ] = -10265122569721941440000/49674155541527163507147+24115558466054000 00000/49674155541527163507147*7^(1/2), a[18,1] = 709267714369766950098 04379644612309403/1630713258316916799374387836496873072640-29849566024 332586811787951198583421/163071325831691679937438783649687307264*7^(1/ 2), a[18,2] = 0, a[18,3] = 0, a[18,4] = 0, a[18,5] = 0, a[18,6] = 4684 662399542643647977751441516649198125/254372516996127957602843618925279 20943264-26709437901843964919449276414071475625/1271862584980639788014 2180946263960471632*7^(1/2), a[18,7] = 4004964115348576324736204875497 93366403125/1589010252148056981393994591408728375494368+10035263151351 68217750873893180856009375/794505126074028490696997295704364187747184* 7^(1/2), a[18,8] = 12774383137101436095033568724910537109375/331915796 1767668910217232497074552405216+59720772884380206651692530752629882812 5/1659578980883834455108616248537276202608*7^(1/2), a[18,9] = -1802703 4777101113229452644098683106369586923840053703055239154013703160260478 73432829768143660422541770189474551/4008879026090892207488482292160353 7074606114437634318910643704597180875389793486358192270983400872078671 6641280-11549073395893898707178189409937016283047719903485547586212857 22737496241965692115980914214471862767372483/2145047368019097976076024 5557067546190061594755007929215390713573321673385303379719724046756595 25500490752*7^(1/2), a[18,10] = 21271135272151061077797757004198544921 875/3512724866292134687764455808783247840544+1729663602506106195356929 95992880859375/250908919020866763411746843484517702896*7^(1/2), a[18,1 1] = -257181496023244780272848520935056875/466333747226316856825908928 29656032-24412968277003167712734734927511875/4663337472263168568259089 2829656032*7^(1/2), a[18,12] = 3132514617923695414144270190280539/3942 4279228932655113708734833726976+887940448807352158464070945/1506779359 47549953424507673856*7^(1/2), a[18,13] = 13060395422189465/77499482279 6457472+3321435324453/2201689837489936*7^(1/2), a[18,14] = 45844041870 72404233/169253216588093779584-47650637886965575/2644581509188965306*7 ^(1/2), a[18,15] = 1791582608825903125/11471162467947274224+9250137168 34375/130354118953946298*7^(1/2), a[18,16] = 11164765265655517578125/1 13540926952062088016336+1712896926116943359375/99348311083054327014294 *7^(1/2), a[18,17] = 0, a[19,1] = 535655479158074959687111905785126128 32439/1228494921165087990600124765218068442000000-32866372517193693834 3680176682875249/2274990594750162945555786602255682300000*7^(1/2), a[1 9,2] = 0, a[19,3] = 0, a[19,4] = 0, a[19,5] = 0, a[19,6] = -9410841927 45729767895274038102396218/567795796866357048220633077958212521055*7^( 1/2)+380145805841840287029128269834313988041/2044064868718885373594279 080649565075798, a[19,7] = 3565506498439448021082787392875980383385/14 187591537036223048160665994720789066914+418442498572008695301697506563 31246/41975122890639713160238656789114760553*7^(1/2), a[19,8] = 285938 7143344524736201037565051296515625/80015415149756304085593997697332959 7686+4208420674863241824630827149071356250/148176694721770933491840736 47654251809*7^(1/2), a[19,9] = -45680153313647226875167598116159240355 0264322861223910175490376607996825535317233283995012274112360895978616 38289/1118548835404824834678705996696527262126286675157207558890728364 8766985320812914720477754180636404039807942000000-21605000076188669933 2908318984753896339403089055559398626500021503254015743998006035396648 20101981262758272973/5084312888203749248539572712256942100574030341623 6707222305834767122660549149612365807973548347291090036100000*7^(1/2), a[19,10] = 1560143195331346599933913496950810484375/28227253389847510 8838215198920082415758+50485505364845502056312650212481250/92791759992 924098894876791229481399*7^(1/2), a[19,11] = -114996879666326606709923 843682437653/22483948526983134168392037614298444-430085025149565428288 387897266111/1040923542915885841129261000661965*7^(1/2), a[19,12] = 22 26538932772419221283628295549708587/2970021035661779263588548662139030 0000+137604393969352284402586088077/29586008364331473149528307355000*7 ^(1/2), a[19,13] = 2381481586979950007/151366176327433100000+511997202 259998/430017546384753125*7^(1/2), a[19,14] = 353341535891060455739/16 528634432431033162500-2350501240657313744/165286344324310331625*7^(1/2 ), a[19,15] = 365031149291536/65177059476973149*7^(1/2)+54443653458754 7644/3584738271233523195, a[19,16] = 96563932308593750000/709630793450 3880501021*7^(1/2)+376299572125585937500/7096307934503880501021, a[19, 17] = 0, a[19,18] = 0, a[20,1] = 4662268533863664327631813657843691360 1/1051150769851560436890462456487774000000+126789671676822527127205819 671838659/2654155693875190103148417702631629350000*7^(1/2), a[20,2] = \+ 0, a[20,3] = 0, a[20,4] = 0, a[20,5] = 0, a[20,6] = 199257218650726586 08480360759166923779/75706106248847606429417743727761669474+7260902051 54571178076763027631835676/1324856859354833112514810515235829215795*7^ (1/2), a[20,7] = 3504377652209101253000936599614744190785/141875915370 36223048160665994720789066914-16368385528267532523268450939050589404/4 9656570379626780668562330981522761734199*7^(1/2), a[20,8] = -164384582 991190944712321248030516328125/29635338944354186698368147295308503618- 9740976592985247191508217896823012500/10372368630523965344428851553357 9762663*7^(1/2), a[20,9] = 1085937758822506219350844201821323405303591 9839630515793148446073702850609398845638334564577611465341260296896986 7/11185488354048248346787059966965272621262866751572075588907283648766 985320812914720477754180636404039807942000000+275042822381915299562118 4588625747040165971814747860717993944668956246484752090750973380140243 87949868132668919/1957460461958443460687735494218922708721001681525113 228058774638534222431142260076083606981611370706966389850000*7^(1/2), \+ a[20,10] = -417408429277495029867103478642578125/345034267080399839675 11942173338518-313470168511820086574442090135037500/174242304875601919 0359353079753595159*7^(1/2), a[20,11] = 679794680264359477314420571059 7803/832738834332708672903408800529572+9954917739091113646006462875906 06/7286464800411200887904827004633755*7^(1/2), a[20,12] = -79729418419 338678860004195613446243/1100007790985844171699462467458900000-4229285 5673402870431176574643577/27500194774646104292486561686472500*7^(1/2), a[20,13] = -1823744191363273521/151366176327433100000-186228260739428 4/4730193010232284375*7^(1/2), a[20,14] = -6490938609056455869/9182574 68468390731250+86358141932804448/18365149369367814625*7^(1/2), a[20,15 ] = 30134866374897324/398304252359280355-147524929494432/7966085047185 6071*7^(1/2), a[20,16] = 740020091270507812500/5519350615725240389683- 24834520992187500000/5519350615725240389683*7^(1/2), a[20,17] = 0, a[2 0,18] = 0, a[20,19] = 0\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "interpolation coefficients: dd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52753 "dd := \{d[1,1] = 1, d[2,1] = 0, d[3,1] = 0, d[4,1] = 0, d[5,1] = 0, d [6,1] = 0, d[7,1] = 0, d[8,1] = 0, d[9,1] = 0, d[10,1] = 0, d[11,1] = \+ 0, d[12,1] = 0, d[13,1] = 0, d[14,1] = 0, d[15,1] = 0, d[16,1] = 0, d[ 17,1] = 0, d[18,1] = 0, d[19,1] = 0, d[20,1] = 0, d[1,2] = 83682129785 0402105393366906188204907032557729304042750710537763872999243252100944 5375032950993744090514852582633/56082513267345527360762550757781159334 6373250660622817889967852921754044994744741400508046421804507572200816 876918*7^(1/2)-1861987595052740266281305900166224923529868661302454726 016152085606965556763953004754891004748566484244947596408537/207134021 1267813902522937414747318079955246152265714937154585101383481838656548 52224577884741154278058025388811684, d[2,2] = 0, d[3,2] = 0, d[4,2] = \+ 0, d[5,2] = 0, d[6,2] = 1858793013573990927163041768414884892033827943 4193393014042910595102046894920974422195339071158831563053354334624000 000/392513050470183326764566312053659421928876426720773360402009085217 546870753026186591271056624195664554869570915362351-130200065420297681 1855941921771220902128698931221746252061016346942559993318350580324066 72000960263166845543782000000/3925130504701833267645663120536594219288 7642672077336040200908521754687075302618659127105662419566455486957091 5362351*7^(1/2), d[7,2] = 24393526059945478414201663690431496971955408 2216471364240416221316347728535676174453999248538231587635374872733456 0000000/73558331143358901840849929058977813043917279513673623155956155 789595780038909020020851173994097196295167196933959136911-170865645913 4153373180916502740779016005413633221460600695970132126049336819754812 2878534907071273661568398407830000000/73558331143358901840849929058977 8130439172795136736231559561557895957800389090200208511739940971962951 67196933959136911*7^(1/2), d[8,2] = -786562827832318721637515739383304 3497967974692241554081011840767770926289249058261030003626061808740201 882500000000000/374756550713546865288206149290055545905171529141498310 8765356998580517844061915290432370002111328335307090326972327+22589022 9122678674007679660195901527304301986552019411760786334975486017140661 0859436546029673776749003656093750000000/15365018579255421476816452120 8922773821120326948014307459379636941801231606538526907727170086564461 747590703405865407*7^(1/2), d[9,2] = -27254057670566892450768088255055 7716555063263752313278070254015692372317167609134041275738640961075723 0736706274745041764424057396459639206295519699214415047918286959073366 980723955339077/115986685153630367475666104436268514569991401558803448 0523016860069020798270666490042069315960775435230887081915235205475890 35657941407109172370605767836055845408760899612924569795827066*7^(1/2) +474501016589432935418931810372127956165594246564935875027488300792099 3262626721719056371075832518369478058101478424481144484089150568427885 690638727277470991134554656606742615807527996952/141447177016622399360 5684200442298958170626848278090829906118122035391217403251817124474775 5619212624766915633112624457023053129017244769411264708020467811688464 48303653816153290193013, d[10,2] = -2300687236755043771672285182501501 5605882082046956511306714830540983722176691776756759436521396452940721 9132500000000000/54203589641399054940515954433146554605418150579874171 666031802763145197354031286546269132148350836800403796636697371+161152 7623299982650765067866491235211161954616508674075891003207278086489310 916135336927822991089467622584218750000000/542035896413990549405159544 3314655460541815057987417166603180276314519735403128654626913214835083 6800403796636697371*7^(1/2), d[11,2] = 6622285061067382225880495433856 9975396886214510566873766798476032710012722573609355219025511051067697 45762334000000/2158748324966795969616805245215378779041381088697472132 521561349917069493735200831929528819179078765628984147639-463861194811 0335796821533805405363880628066260435063328245370045923902059491644787 2964594513589539883913390125000/21587483249667959696168052452153787790 4138108869747213252156134991706949373520083192952881917907876562898414 7639*7^(1/2), d[12,2] = -878548486262117688962806450791810370804044066 52705454599891350377216695609279018191510960374452417701800784160/2782 0510255133618109170365573629069192064348048603196500956323520155117679 75280314504067087706779903927638667+2523072688345240331000867793229795 0717716877066604550263717045843958368565379585829107779135369372516069 420830/114064092046047834247598498851879183687463826999273105653920926 432635982486986492894666750595977976061033185347*7^(1/2), d[13,2] = -8 4586364092539034261053786183581065397429945856253988015944182110848130 660921558953119603644821/271108380573910135997038998950216517699401159 89272484363862439667110535678120122922540654335677+2728116183930778106 8867373769261076462331822170038208831743670617911253496491330686625269 11294/2711083805739101359970389989502165176994011598927248436386243966 7110535678120122922540654335677*7^(1/2), d[14,2] = 1126969442281034238 7973570554147146353789537274801500334876490598605955966754824096515348 72/2104655411102161540807308222602284832867639509467614799971724506492 10635088824345159084991*7^(1/2)-35216442188907656553685598445138013859 55768299188003389727668387685293365134871933721436612/2104655411102161 5408073082226022848328676395094676147999717245064921063508882434515908 4991, d[15,2] = -48183330317765425294801086933750720035542464092031172 4672460053875223302725772345235296000000/20836088569911399253992351403 762619845389631143729386519720072614271852873793610170749414109+446953 6614263275323583275549719383104072299681633333344370126137128079715270 634801856000000/208360885699113992539923514037626198453896311437293865 19720072614271852873793610170749414109*7^(1/2), d[16,2] = 430119262869 0202519115920539217638745658620076885480546789070243858192113043762109 375000000000000/512298179250869238310194237405078071645897268184384976 17960773089123983378179451295055599788809-3624274714941928170508153369 9147451646637840464233887082015180011875787664203906250000000000000/96 6600338209187242094706108311468059709240128649782973923788171492905346 758102854623690562053*7^(1/2), d[17,2] = -1076949364044583226686747828 7711505897064631074015308205095092107732457060681908694794775964355468 75/3254057148414104636995157398562459281744533177763561541831786595578 8580637654320564608563990068868+22363332311873658863628877456488571256 4874691677504838747882856147204401667545230496717041015625/35370186395 8054851847299717235049921928753606278647993677368108215093267800590440 919658304239879*7^(1/2), d[18,2] = -8586514296522614043439696583579360 9822000912427003396981321030044716108335899421461537503891456/16923534 1606724809496315654179449723410886892956290906065726367567030271674923 6559424202412631+57077412622417403258846898988439799537838755667879253 829046235062835313901975412244193002268672/169235341606724809496315654 1794497234108868929562909060657263675670302716749236559424202412631*7^ (1/2), d[19,2] = -2744965906939421696014617776908360201934581520020228 10673616079581132132479988291861269231875000/5342527450722096927236631 435861059895912311718816242328741557878096445831306413452299933106541- 3034426225762488619165714350476504118305841451892384236483669396878709 93400463012997576925000/2322838022053085620537665841678721693874918138 61575753423545994699845470926365802273910135067*7^(1/2), d[20,2] = -24 2373815364812449823441256692578383203243396790833268226506545243894558 046328705219953841625000/985546989356809184713838221397971918686929553 0983999823827594346550586409304377610764472873557-34192319333112380719 4887754561990258681430997426086853675512097659222697626121114770923492 5000/98554698935680918471383822139797191868692955309839998238275943465 50586409304377610764472873557*7^(1/2), d[1,3] = -879314449190835291710 5658322993071143628123040821951554966183330589258588612599353340098256 839675184587240228686537/105995950075283046711841220932206391142464544 3748577125812039242022115145040067561246960207737210519311459543897375 02*7^(1/2)+23385650788455477515865704897429779739544494102135779165555 2742241467366516703256405592799319785276790901085272527908599/56177853 5399000147572758470940693873055062085186745876680380798271721026871235 8074608889101007215752350735582656087606, d[2,3] = 0, d[3,3] = 0, d[4, 3] = 0, d[5,3] = 0, d[6,3] = 19544503724775365475838843124732535407353 0981032866053125044393847009753499244603952468158991923930180905919679 14000000/1059785236269494982264329042544880439207966352146088073085424 5300873765510331707037964318528853282942981478414714783477*7^(1/2)-377 1785730047000355838270131625587477112176642852397089207500292839741824 229819663361213212984911710967614227570612000000/105978523626949498226 4329042544880439207966352146088073085424530087376551033170703796431852 8853282942981478414714783477, d[7,3] = 9499580631035826355154343126728 1355467135704242585328549436970579834311006370690588450055961166644706 314693458830000000/735583311433589018408499290589778130439172795136736 23155956155789595780038909020020851173994097196295167196933959136911*7 ^(1/2)-183327155143631689822900165723121726428904778635792606656104995 44559453141689847322672182848764766803409698782414140000000/7355833114 3358901840849929058977813043917279513673623155956155789595780038909020 020851173994097196295167196933959136911, d[8,3] = 65438488069234242207 7600378652252633537064421158654853547060494380851316840060658692522287 38014656404908749817812500000000/4148555016398963798740442072640914893 1702488275963863014032501974286332533765402265086335923372404671849489 91958365989-1994628266993123363166423433958585409020971106953451708602 7165219922946286421424531701522825919140938072057656250000000/24403264 8023468458749437768978877346657073460446846253023720599848743132551561 189794625505431602380422644058350492117*7^(1/2), d[9,3] = -14620870193 5244145437788795853051176679453105500191887303681468392023210705335804 6043958456300168929476235550772280364328364551799274705393935451374495 439584002886576369821504352450477055233/579933425768151837378330522181 3425728499570077940172402615084300345103991353332450210346579803877176 1544354095761760273794517828970703554586185302883918027922704380449806 462284897913533+151523799287100119798351284006733065249601314537008021 0233376280701829523746728735315574169496907474533598474766417114940604 9750797852884403973285750128883040962889247117423252061882675977/11598 6685153630367475666104436268514569991401558803448052301686006902079827 0666490042069315960775435230887081915235205475890356579414071091723706 05767836055845408760899612924569795827066*7^(1/2), d[10,3] = 466845916 9753875933602471281810157709196434820753080523899098575703794162426359 5698225061805097758210325703761562500000000/14634969203177744833939307 6969495697434629006565660263498285867460492032855884473674926656800547 2593610902509190829017-14229914073568380577560503949843411129109290481 7107419753717461364712937143930954299143786751712604696944545312500000 00/8608805413633967549376063351146805731448765092097662558722698085911 2960503461455102898033412086623153582500540637001*7^(1/2), d[11,3] = - 1343766633037386075352405862724999802064287777974145571096003717160579 575687052086680750553248032684107908009435750000/582862047741034911796 5374162081522703411728939483174757808215644776087633085042246209727811 7835126671982571986253+69630816393965137889764677720994714201784775392 55056241966219484117018082586818906019745014460160295923109020875000/5 8286204774103491179653741620815227034117289394831747578082156447760876 330850422462097278117835126671982571986253*7^(1/2), d[12,3] = 73091280 9532914150887284777475341552143524758185090045469082646379899181902049 861958801404636282071021965903539780/307973048524329152468515946900073 7959561523328980373852655865013681171527148635308156002266091405353647 896004369-378741772487273764095073311009933533081344591227416193524471 5775789846672976254076789600143906495478069415439410/30797304852432915 2468515946900073795956152332898037385265586501368117152714863530815600 2266091405353647896004369*7^(1/2), d[13,3] = 1303325087614746216014337 3876982029300580781998150335412152712106489826833150909414383651709184 263/731992627549557367192005297165584597788383131710357077824285871011 984463309243318908597667063279-409521122330732098015725934093466876263 447159613092032753335919617039525509357260826909602763538/731992627549 5573671920052971655845977883831317103570778242858710119844633092433189 08597667063279*7^(1/2), d[14,3] = 528638663362100452483911731161997256 153086844879498678758830462235489967058230035627678951324/568256960997 5836160179732201026169048742626675562559959923656167528687147398257319 295294757-169170870930575688701446615475589440546319228835625261193031 957238753745647998638967091298344/568256960997583616017973220102616904 8742626675562559959923656167528687147398257319295294757*7^(1/2), d[15, 3] = -6709280423439032847055437893501214614235599905479918977891708550 62707269530021675976512000000/5625743913876077798577934879015907358255 20040880693436032441960585340027592427474610234180943*7^(1/2)+72328633 3096448502184182482262783843380320100601826277161126121305852088996140 96153215392000000/5625743913876077798577934879015907358255200408806934 36032441960585340027592427474610234180943, d[16,3] = -4519603930822192 8193576174400089159763957463481043292566633363264388198815781078746093 75000000000000/9682435587841428604062671086955975554107458368684876049 794586113844432858475916294765508360084901+380831263839935000246791088 4446676285274050629064093996756535117086433363318867890625000000000000 0/18268746392153638875589945447086746328504638431480898207159596441215 9110537281439523877516228017*7^(1/2), d[17,3] = -197469961169495835938 9248889832889397439758427170699939981599642218630723973789009886167724 609375/561761783933381235286887786196843993651549845266087989958172877 753383425330349523813574953792749*7^(1/2)+1317018990747279686794948703 0238631600937792266196326787799743705578782504235947216741362064868164 0625/43929771503590412599434624880593200303551197899808080814729119040 3145838608333327622215613865929718, d[18,3] = 436874070234366966874435 3682143407095215716984369037434943715804816734530910295080624793121714 176/456935422338156985640052266284514253209394610981985446377461192430 98173352229387104453465141037-5039979863922048841771421047651854174770 93968911075953027061824740531452644908329486909214431232/2687855425518 5705037650133310853779600552624175410908610438893672410690207193757120 26674420061*7^(1/2), d[19,3] = 543493758609640753638513876940065689303 188095773892963374638972454303156059669873628610826816675000/100973768 8186476319247723341377740320327426914856269800132154438960228262116912 142484687357136249+318851208994152846547303694260951447487674911428394 759615605183809780791801752092815738891325000/439016386168033182281618 8440772784001423595281983781739705019299827079400508313662976901552766 3*7^(1/2), d[20,3] = 5658830014886157112704375198302564742656137620017 8405767328238137952328724680704699916169698175000/26609768712633847987 2736319777452418045470979336567995243345047356865833051218195490640767 586039+513265419957725064901070586674493557926491793530946093838258773 041417742216957984905941950475000/266097687126338479872736319777452418 045470979336567995243345047356865833051218195490640767586039*7^(1/2), \+ d[1,4] = -356256418586267757652256660013055332606280392505704732029131 2404495638129538317219378526838639219014269096935822706775/32425889489 1197776376772566199534703062084897654687374707290504052941429651506959 57338465229478878801447247230338168+2604209936802981181364201129801514 1176021442350170292522112122503395314176329797104305497931511009204254 623605591/152952308910942347347534229339403161821738159271078950333627 596251387466816748565836502194478673956610600222784614*7^(1/2), d[2,4] = 0, d[3,4] = 0, d[4,4] = 0, d[5,4] = 0, d[6,4] = 1250634955421275470 0070171702073334628141897207792885551083340167194836188152112827462749 9479901218640922952458488500000/10704901376459545275397263056008893325 3329934560210916473275205059330964750825323615801197261144272151328064 795098823-405186035550152091860934808933416191782720309897508841868741 331033689592648824422773008572339067441716921305114000000/107049013764 5954527539726305600889332533299345602109164732752050593309647508253236 15801197261144272151328064795098823*7^(1/2), d[7,4] = 5470825450308314 7281232754378971612446888434991502545286956560641592612137730314530229 02852669935530058091737693417500000/6687121013032627440077266278088892 0949015708648794202869051050717814345489917290928046521812815632995606 54266723557901-1772461313181996552410077580273109039667340117054002786 7735791211346003279805144648060160592782886296926251835470000000/66871 2101303262744007726627808889209490157086487942028690510507178143454899 1729092804652181281563299560654266723557901*7^(1/2), d[8,4] = 70297634 7302701837233125005123654011660340119599282815183073124824865051694742 6546088361532213779701594573906250000000/41904596125242058573135778511 5243928603055437130948111252853555295821540745105073384710463872448532 03888373656145111*7^(1/2)-21697855198227982661844457970411424912205181 8424003709581180265284744029933327919880434282768913389068455080675781 2500000/41904596125242058573135778511524392860305543713094811125285355 529582154074510507338471046387244853203888373656145111, d[9,4] = -2827 1781954805524348984579194838194759610900891165281838259738365891889463 4603873807781367541337436595763955966666303343931743241181343133481177 1944984091172716992897138709624704860865193/10544244104875487952333282 2214789558699992183235275861865728805460820072570060590003824483269161 4032028079165377486550432627596890376428265203369143343964144049170553 6328447688163257006*7^(1/2)+349051306437255673020204067738789871093523 9233528321432257418629217412044176339647602870027123787897110544596268 2681175686043041215665187689919601418322206373615414003572062959898988 40915773/4217697641950195180933312888591582347999687329411034474629152 2184328029028024236001529793307664561281123166615099462017305103875615 057130608134765733758565761966822145313790752653028024, d[10,4] = 5015 1164421971453497449970342700878317708933617543415640166475429131884321 38913640282465663311644158713445781250000000/1478279717492701498377707 8481767242165114041067238410454372309844494144732917623603527945131368 410036473762719099283*7^(1/2)-1547950664095234563658678541907629144073 0126150786798539677328056528715793106243760576587759447459736517050091 01562500000/1478279717492701498377707848176724216511404106723841045437 2309844494144732917623603527945131368410036473762719099283, d[11,4] = \+ -144354826523565474797296167354377798525641139472188379358410280341849 274965069472869901349318238288040712345875000/588749543172762537168219 6123314669397385584787356742179604258227046553164732365905262351325033 85117898813858447*7^(1/2)+44556123902201815446095850104426752766139275 5324304500678506250901598911068580075165866099635013190887085801808437 50/5887495431727625371682196123314669397385584787356742179604258227046 55316473236590526235132503385117898813858447, d[12,4] = -4847068070092 3305167739610511594670073779624548808813558601759681204044355758383690 131197186108389991238549116005/622167774796624550441446357373886456477 0754199960351217486595987234689953835626885163640941598798694238173746 2+78518687121650065655035243186344034441132093540059300573577429323290 540724879798403928492235111605737817357410/311083887398312275220723178 6869432282385377099980175608743297993617344976917813442581820470799399 3471190868731*7^(1/2), d[13,4] = 8489969475198088704957243665590473383 417596718155218530998977561323617439023414113185359695938/739386492474 3003709010154516824086846347304360710677553780665363757418821305488069 783814818821*7^(1/2)-2494621341773228587516531943492565820219439042006 12537760546484534019737190065886775622127864696/7393864924743003709010 1545168240868463473043607106775537806653637574188213054880697838148188 21, d[14,4] = -1205539460038405593952278500974502125185748097015280435 50496246174156031028233120116635471764/6313966233306484622421924667806 85449860291852840284439991517351947631905266473035477254973+3857874471 3603971083970810962870437246357956348121972892652157849824243662800410 197939444984/631396623330648462242192466780685449860291852840284439991 517351947631905266473035477254973*7^(1/2), d[15,4] = -1499477940201958 6224362315192397876465756130917602452189559753571174576916787215158969 92000000/5682569609975836160179732201026169048742626675562559959923656 167528687147398257319295294757+139093157567435372096355972214496123288 18079239877045613698794519003200300076160034112000000/5682569609975836 1601797322010261690487426266755625599599236561675286871473982573192952 94757*7^(1/2), d[16,4] = 133854248362368905413673375594433601162259198 10933563019579165882451606098088412890625000000000000/1397176852502370 6499368933747411220135797198223210499350352938115215631830412577625924 254487857-112788384456759510940525563686473700429157958215713035746172 446527896808727646093750000000000000/263618274057051066025828938630400 379920701853268122629251942228588974185479482596715551971469*7^(1/2), \+ d[17,4] = -29023712999433647113451839778110242312799026542395344895197 588777177183642207496988628378304931640625/266241039415699470299603787 1551103048700072599988368534226007214564520233989898955286155235551089 2+69595279632832523218113590634772505936581267311170780954654154079418 3055601306326867048583984375/96464144715833141412899922882286342344205 528985085816457464029513207254854706483887179537519967*7^(1/2), d[18,4 ] = 107862961793102646582629760388085179331280789739093434322484066117 2657114118403035258713039925248/13846527949641120776971280796500431915 43620033278743776901397552821156768249375366801620155789+1776264125926 6305561113969059170145193900726434369672237737782213129474128637817747 8221715462144/46155093165470402589904269321668106384787334442624792563 3799184273718922749791788933873385263*7^(1/2), d[19,4] = -347419855011 4241032023274331054647778986500899797308921713501345326289569764407215 170323905900000/145705294110602643470089948250756178979426683240442972 6020424875844485226719930941536345392693-94432143994486672184580468007 4154247031257743430760102589398927628316814717421139295589475000/63350 1278741750623782999775003287734693159492349752054791489076454124011617 36127892884582291*7^(1/2), d[20,4] = -10640739904489174164598730945100 487621898239771628963205816183731108264840457047542047355475000/268785 5425518570503765013331085377960055262417541090861043889367241069020719 375712026674420061*7^(1/2)-2440706549165311450257420236724179995591266 23902933534927288162484156336134033115635772921200000/2986506028353967 2264055703678726421778391804639345454011598770747122989119104174578074 1602229, d[1,5] = -446773100725540603857257220306129732549016119051930 76895435119572552038065628078189324013031807096013506241721530189/3179 8785022584914013552366279661917342739363312457313774361177260663454351 2020268374088062321163155793437863169212506*7^(1/2)+144887131549059993 2453848980378327777086294933230090025195369060533358242388293233094292 2493630617218969540389310116982869/84266780309850022135913770641104080 9582593127780118815020571197407581540306853711191333365151082362852610 33739841314090, d[2,5] = 0, d[3,5] = 0, d[4,5] = 0, d[5,5] = 0, d[6,5] = -656792197574857449565897644617332979439835992046924422579029486421 09435312381621829508425559627622455116870528657175600000/3179355708808 4849467929871276346413176238990564382642192562735902621296530995121113 892955586559848828944435244144350431+993041629111759766171503534969068 1302368652897333916105408212678379640832446626487465740328555988576440 4454211858000000/31793557088084849467929871276346413176238990564382642 192562735902621296530995121113892955586559848828944435244144350431*7^( 1/2), d[7,5] = 3712819946551046363328636212605498274652349994202367866 6500728391630837846620661758392138728756017514111392954270000000/16974 9994946212850401961374751487260870578337339246822667591128745221030859 02081543273347844791660683500122369375185441*7^(1/2)-31923299391779681 6439781191791502619700566017051554145341627707014214716289476681076510 546592637045271284803525789482000000/220674993430076705522549787176933 4391317518385410208694678684673687873401167270600625535219822915888855 01590801877410733, d[8,5] = 113949973463717620757151671863539901033763 9462654466287477425224354294598795006266477884346625002352245393451858 468750000000/124456650491968913962213262179227446795107464827891589042 09750592285899760129620679525900777011721401554846975875097967-1722874 7190279302939411195878237800717401691719950868917273355145479919411962 14972255273535253172086979375715781250000000/1244566504919689139622132 6217922744679510746482789158904209750592285899760129620679525900777011 721401554846975875097967*7^(1/2), d[9,5] = -84865844740885018444483331 6126323689288580960668479070329342063231457946586282908119172223210821 6055492920390752321132894809489015306410945938877385582165990711012140 8879271966065801075824493/57993342576815183737833052218134257284995700 7794017240261508430034510399135333245021034657980387717615443540957617 6027379451782897070355458618530288391802792270438044980646228489791353 30+2566270337212064884951001322573943206184190690629597558042393029380 2098149075108931098314077144089906095908105792389128428951299944172900 594351280476611288328278485323096477744388729112623/115986685153630367 4756661044362685145699914015588034480523016860069020798270666490042069 3159607754352308870819152352054758903565794140710917237060576783605584 5408760899612924569795827066*7^(1/2), d[10,5] = 8129325939607217651345 0897208280303382385001110547054431979907087665766806163739743045409088 7303441381391859579093750000000/43904907609533234501817923090848709230 3887019696980790494857602381476098567653421024779970401641778083270752 7572487051-94547631590864385752240873907743128285462504232901349577933 753008409589706013262092195040294182094637616180781250000000/337730058 5348710346293686391603746863876053997669083003806596941395969988981949 39249830746462801367756362117505575927*7^(1/2), d[11,5] = -23399405562 9823393803681889788252816897884129744463666484788482335764239972890342 80201410911532962305115614656440225000/1748586143223104735389612248624 4568110235186818449524273424646934328262899255126738629183435350538001 5947715958759+35378897475197826870250049589844975980775885620724836937 651085700863763344689144512073528675598856949290309412375000/174858614 3223104735389612248624456811023518681844952427342464693432826289925512 67386291834353505380015947715958759*7^(1/2), d[12,5] = 127276007909058 4726443049521215296781511246531456238455438068087392318094415213649071 1660933349820752167298561851414/92391914557298745740554784070022138786 8456998694112155796759504104351458144590592446800679827421606094368801 3107-14802758796411389183201066829156882185378882403630615489741659701 81889815146571149112918554323093614124122784290/7107070350561441980042 6756976940106759112076822624011984366115700334727549583891726676975371 3401235457206770239*7^(1/2), d[13,5] = 1189032320085712996551291461038 0731513223423751390969130119162897393407317934202984673860024319224/21 9597788264867210157601589149675379336514939513107123347285761303595338 9927729956725793001189837-20807462200202811167590405521918384934066620 00304309120863862024600003635089146777768834902274586/2195977882648672 1015760158914967537933651493951310712334728576130359533899277299567257 93001189837*7^(1/2), d[14,5] = 268597354462934569853805796975077808409 2686033418648485101326547821075182981148189371853375628/17047708829927 5084805391966030785071462278800266876798797709685025860614421947719578 85884271-8595445534603102776493287191224249411688096239610435627616997 94165942063687863682022223642568/1704770882992750848053919660307850714 6227880026687679879770968502586061442194771957885884271*7^(1/2), d[15, 5] = -3408935246288076842238491917043456372380166584975117439970332606 848957143739402116275264000000/168772317416282333957338046370477220747 6560122642080308097325881756020082777282423830702542829*7^(1/2)+367496 3808990874664180890121438006700200130976687038681594507279018842760804 41265744110624000000/1687723174162823339573380463704772207476560122642 080308097325881756020082777282423830702542829, d[16,5] = -229637698332 2541795712859355021304387126360049486846829963145041260099305551305473 5546875000000000000/29047306763524285812188013260867926662322375106054 628149383758341533298575427748884296525080254703+193497519295361941053 3617525310903132162443086746246278701583794646890168579953695312500000 00000000/5480623917646091662676983634126023898551391529444269462147878 93236477331611844318571632548684051*7^(1/2), d[17,5] = -17056611976086 2872117643858951385417784438438323071765963818582321672362395967409360 521608154296875/286498509806024429996312770960390436762290421085704874 87866816765422554691847825714492322643430199*7^(1/2)+13534439563956865 6424711548970283254283494763399481190804806454033369725195698897878573 8386201171875000/65894657255385618899151937320889800455326796849712121 2220936785604718757912499991433323420798894577, d[18,5] = -46078429959 0987262136067384119616920108781196129315031175660923238358516311147080 056699815384530944/137080626701447095692015679885354275962818383294595 633913238357729294520056688161313360395423111-435331938068391088916088 8910466121440246169180850129492601996573344724633016302181631815307958 4768/13708062670144709569201567988535427596281838329459563391323835772 9294520056688161313360395423111*7^(1/2), d[19,5] = 1642695408538227346 1006183889098058466382674997796751453062866080334108518253040566856654 194175625000/302921306455942895774317002413322096098228074456880940039 6463316880684786350736427454062071408747+16200591659046944819238096594 29006864550479056710487357344968436279429740220223888920346731025000/1 3170491585040995468448565322318352004270785845951345219115057899481238 2015249409889307046582989*7^(1/2), d[20,5] = 1355558458535456706182553 2044131646918727748876012081086843578189864220883766601401368282650151 25000/7982930613790154396182089593323572541364129380097039857300351420 70597499153654586471922302758117+2607863243697731670544824165049266305 029259961684254825602955979664086684162384340768862463575000/798293061 3790154396182089593323572541364129380097039857300351420705974991536545 86471922302758117*7^(1/2), d[1,6] = 2333311444171014107707205088506650 4203502536356624256269811946710956362754599355002137138829583050861938 0362374075/52997975037641523355920610466103195571232272187428856290601 962101105757252003378062348010386860525965572977194868751*7^(1/2)-4437 5946531423148122760183259877491048252965453363493519588231185165268598 5201155627858088271953697941399150694550899130/28088926769950007378637 9235470346936527531042593372938340190399135860513435617903730444455050 3607876175367791328043803, d[2,6] = 0, d[3,6] = 0, d[4,6] = 0, d[5,6] \+ = 0, d[6,6] = 24130489896164216972541247101179712384530933807686540286 91594759298476756824878397087810958473225724103882230684880000000/1177 5391514105499802936989361609782657866292801623200812060272556526406122 59078559773813169872586993664608712746087053-1037249285591215935074691 3168264492242187916879702544276833467485056638214597827052005097310502 48272110548754300000000/1059785236269494982264329042544880439207966352 1460880730854245300873765510331707037964318528853282942981478414714783 477*7^(1/2), d[7,6] = -50415366702135658657962242634672532702059091450 3163631822377282432178116431305844477078185372037528171413130850000000 0/73558331143358901840849929058977813043917279513673623155956155789595 780038909020020851173994097196295167196933959136911*7^(1/2)+1055573392 3803059168798494691036118590469467692291205363982168058693277572007821 5429390552961077870781300001731532400000000/73558331143358901840849929 0589778130439172795136736231559561557895957800389090200208511739940971 96295167196933959136911, d[8,6] = -41865124060354371804929562966299252 5286083806122205092065446363571919406495294585328341372877728373050878 06431250000000000/4609505573776626443044935636267683214633609808440429 22378138910825403694819615580723181510259693385242772110217596221+1799 5726655219282455071679213064878087198451976431765328554053751283129395 649276910314681832068096717762710937500000000/414855501639896379874044 2072640914893170248827596386301403250197428633253376540226508633592337 240467184948991958365989*7^(1/2), d[9,6] = 841850272407218037008680679 5416251150371038566863780638882615565149467228327163265970628310059074 2832331418726750994065839050814523339718156474803085166880999995826402 96532357921673961743780/5799334257681518373783305221813425728499570077 9401724026150843003451039913533324502103465798038771761544354095761760 273794517828970703554586185302883918027922704380449806462284897913533- 4020771127623446516646617450004087543643773035056529406778979991451933 3246705408624675016138858521888488113764197292462595013355155530620334 6771535866425866530835201156845062879744638075/57993342576815183737833 0522181342572849957007794017240261508430034510399135333245021034657980 3877176154435409576176027379451782897070355458618530288391802792270438 0449806462284897913533*7^(1/2), d[10,6] = -298670748788790102100433561 3680932370770209178449664427968404348768171186754566871027571465211406 4435745699806250000000000/16261076892419716482154786329943966381625445 1739622514998095408289435592062093859638807396445052510401211389910092 113+128383643324792687052006465711292802772022037582533813765381168666 42322349535069706084962769651301124740992187500000000/1463496920317774 4833939307696949569743462900656566026349828586746049203285588447367492 66568005472593610902509190829017*7^(1/2), d[11,6] = 859692185136592989 4351042591382993404550690979178351049050713439752220986074518814971388 55661375989715659069455000000/6476244974900387908850415735646136337124 1432660924163975646840497512084812056024957885864575372362968869524429 17-3695387489845489920508564822616417212503178228188441837164174536467 97092760160451988668280603672846568714581250000/5828620477410349117965 3741620815227034117289394831747578082156447760876330850422462097278117 835126671982571986253*7^(1/2), d[12,6] = -4676109786647717886742971723 9921680418627033256942923319586508775851731475673881469995859705644008 7718219667517200/34219227613814350274279549655563755106239148099781931 6961762779297907947460959478684000251787933928183099556041+20100261355 7275559436054071077497206823891807179233708267079783906328730510260997 956059854961697637013167779500/307973048524329152468515946900073795956 1523328980373852655865013681171527148635308156002266091405353647896004 369*7^(1/2), d[13,6] = 22728711259621610735298134005072282645583399123 21500213750383778777557093101015987015614588514600/3852592776576617722 0631857745557084094125427984755635674962414263788655963644385205715666 687541+217337568430352645135599106163682616674495688504540684620975111 88898468600993917245052138373100/7319926275495573671920052971655845977 88383131710357077824285871011984463309243318908597667063279*7^(1/2), d [14,6] = -280554617108619480899566370291344261978354334967083819115748 58772216328644343472571539333800/5682569609975836160179732201026169048 742626675562559959923656167528687147398257319295294757+897809263855330 8774603562326615224854689946734573565223242391310021454682858338330596 482800/568256960997583616017973220102616904874262667556255995992365616 7528687147398257319295294757*7^(1/2), d[15,6] = 3560692266246454923588 9356874088283848634347899102926266634212695171922419842548374400000000 /562574391387607779857793487901590735825520040880693436032441960585340 027592427474610234180943*7^(1/2)-3838563735599810887806582237485200925 401535442492951725150099511338269898156600114430400000000/562574391387 6077798577934879015907358255200408806934360324419605853400275924274746 10234180943, d[16,6] = 23986057739895116189798008278833125459727230577 0302333622421792017462607951451972656250000000000000/96824355878414286 0406267108695597555410745836868487604979458611384443285847591629476550 8360084901-20211153064379615909670674064516330119253556832196327474896 43893578246148984023437500000000000000/1826874639215363887558994544708 67463285046384314808982071595964412159110537281439523877516228017*7^(1 /2), d[17,6] = 1781592842449807843101523892450313337331374044077086088 871695617929599669674371864512561035156250/954995032686748099987709236 5346347892076347369523495829288938921807518230615941904830774214476733 *7^(1/2)-2458045863899110204726866093221394040223165906167849852685524 2323050875531731055106939909071777343750/11560466185155371736693322336 998210606197683657844231793349768168503837858114034937426726680682361, d[18,6] = 24043539132233795395296019045311729949165967595569419413512 7421148504963542471415597102746778777600/45693542233815698564005226628 451425320939461098198544637746119243098173352229387104453465141037+454 7117950738595199604536433887237501425845797499912313751977069609261886 51886962689087579852800/4569354223381569856400522662845142532093946109 8198544637746119243098173352229387104453465141037*7^(1/2), d[19,6] = - 6673038889155942156965682243728399231408151217259336797801169465880685 437507392782565272790512475000/100973768818647631924772334137774032032 7426914856269800132154438960228262116912142484687357136249-16921800286 9950971595979292856062050090037451641774639876395731605180117407012734 59584808750000/4390163861680331822816188440772784001423595281983781739 7050192998270794005083136629769015527663*7^(1/2), d[20,6] = -524696548 2521834370929208002207893594850610310470057581604218844018311684946012 44610762920069975000/2660976871263384798727363197774524180454709793365 67995243345047356865833051218195490640767586039-2723958600672756148163 5562466282349391307223494641186154191773372004444138960225311828101250 000/266097687126338479872736319777452418045470979336567995243345047356 865833051218195490640767586039*7^(1/2), d[1,7] = 316096137901180821623 5378439455919273251963533105363097006681225320093004282454748569691658 9592096685742048985028719450/40127038242785724826625605067192419503933 0060847675626200271998765515019193739862472063507214801125167909684475 434829+430818213473230078971049683993103844230105915004140625365804038 237746727374434176553492223589051899223375853718750/757113929109164619 3702944352300456510176038883918408041514566014443679607429054008906858 626694360852224711027838393*7^(1/2), d[2,7] = 0, d[3,7] = 0, d[4,7] = \+ 0, d[5,7] = 0, d[6,7] = -420114221841160462657669308614637457533500303 4247917492780022922697089783557026951379030383482052397644920192781800 000000/390447192309813940834226489358640161813461287632769290084103774 2427176766964313119250012089577525294782649942263341281-13406102287921 9193971987430165900010515722707086258880978275983004794068504772506211 29187808995084757706501625000000000/1059785236269494982264329042544880 4392079663521460880730854245300873765510331707037964318528853282942981 478414714783477*7^(1/2), d[7,7] = -65160185914872527807558289963572328 0861177326721223305531137551149417801886842856598882440473460616174278 41875000000000/7355833114335890184084992905897781304391727951367362315 5956155789595780038909020020851173994097196295167196933959136911*7^(1/ 2)-3879723457585315003985142664396068948860219340962154394108873146874 44701229847701026968333084237602899253237163249000000000/5149083180035 1231288594950341284469130742095659571536209169309052717046027236314014 5958217958680374066170378537713958377, d[8,7] = 7288759611836654447546 7704761189678531741713725303615423786228279954083001797969080137527690 539992954519418765625000000000/152841500604172350479911023728875811853 6407462798668637359092178000022777559777978187391323492667540541823312 826766417+232588786164207860386820078455417552825681617125225918205093 013772123164271685632149561988228354278109931640625000000000/414855501 6398963798740442072640914893170248827596386301403250197428633253376540 226508633592337240467184948991958365989*7^(1/2), d[9,7] = -30941915296 5550681781252719234022488377417328107886538011544613404799265219560854 6624465966204633783747456440863350674345653180837567799832494022096565 0067093051196984385238334516687742716550/40595339803770628616483136552 6939800994969905455812068183055901024157279394733271514724260586271402 3308104786703323219165616248027949248821032971201874261954589306631486 45235994285394731-5196713052689092617527016645130836928638554513551644 3588354615842251723177056473480931520120385829619537813210742438559483 95374558506542510021050131213037418234576085852871935299506031250/5799 3342576815183737833052218134257284995700779401724026150843003451039913 5333245021034657980387717615443540957617602737945178289707035545861853 02883918027922704380449806462284897913533*7^(1/2), d[10,7] = 987978596 9232366604565485374465799786787474328275052793635830168258672138444118 58837277618577146992212325930609375000000000/1024447844222442138375751 5387864698820424030459596218444880010722234442299911913157244865976038 308155276317564335803119+165931592184924078538413083452557167541387419 3274993900217260304034273568071371638526490756786541873581738281250000 00000/1463496920317774483393930769694956974346290065656602634982858674 604920328558844736749266568005472593610902509190829017*7^(1/2), d[11,7 ] = -14967326189220801286190271553784511290357139204634832241290867171 91063767563148939253810393855275524537847533487500000/2147386491677497 0434609273228721399433622159250727485949819741849175059700839629328141 102464465572984414631784409-477616551494062582045050844872875193639627 9098471028962361377724386225349558942326293718545468445210452367187500 000/582862047741034911796537416208152270341172893948317475780821564477 60876330850422462097278117835126671982571986253*7^(1/2), d[12,7] = 814 1153506268506622504377866273378472383786850525211548011923986020430898 02050864028313920613808874139525123317000/1134637547194896877515585067 5265876693121401738348745772942660576720105626337077451101060980336756 56607119580557+2597891977291227688427582159421123426453549945078722226 494753484756833666122280254232679070775264880155418125000/307973048524 3291524685159469000737959561523328980373852655865013681171527148635308 156002266091405353647896004369*7^(1/2), d[13,7] = -2719351414657628595 3478589572465998732662734868375947568138014482617095995000640615022280 09262000/3852592776576617722063185774555708409412542798475563567496241 4263788655963644385205715666687541+28090158401264189733183045479057945 2865854841124775626356504022013089512937513701468546840125000/73199262 7549557367192005297165584597788383131710357077824285871011984463309243 318908597667063279*7^(1/2), d[14,7] = -3626075184195598903797940531830 52037268921777099258998118582000379457718815369844905419750000/5682569 6099758361601797322010261690487426266755625599599236561675286871473982 57319295294757+1160388634968115978992336342066221823907489422858474087 19897967448578624039897282707118500000/5682569609975836160179732201026 169048742626675562559959923656167528687147398257319295294757*7^(1/2), \+ d[15,7] = 460207641501572829858245011297300566729025914500961011570235 393032133413845823188000000000000/562574391387607779857793487901590735 825520040880693436032441960585340027592427474610234180943*7^(1/2)-4961 2160541356492272241646348590115357848500918483497186208819385834359835 849706058000000000000/562574391387607779857793487901590735825520040880 693436032441960585340027592427474610234180943, d[16,7] = 4428740350793 0421325328671120445209489895182010764832648157642543844646962971191406 2500000000000000/13832050839773469434375244409937107934439226240978394 35684940873406347551210845184966501194297843-3731749088696384030589120 0266832219570261367858560427390872302318652993888183593750000000000000 00/2609820913164805553655706492440963761214948347354414029594228063030 8444362468777074839645175431*7^(1/2), d[17,7] = 2302649537878259767671 8366409069780947783637940435012226923687824053171506131098692001342773 437500/954995032686748099987709236534634789207634736952349582928893892 1807518230615941904830774214476733*7^(1/2)+133030369696984930144779184 0569478934904685449699322347629455936768052457534218766832102282226562 5000/11560466185155371736693322336998210606197683657844231793349768168 503837858114034937426726680682361, d[18,7] = -170399142971935852852273 7404442085862767432353463473272640174584902818129415607209180106894213 12000/4569354223381569856400522662845142532093946109819854463774611924 3098173352229387104453465141037+58769988285026156568005456124835048947 52754908142427288822717778298529026150680832392195456000000/4569354223 3815698564005226628451425320939461098198544637746119243098173352229387 104453465141037*7^(1/2), d[19,7] = 59191605923500194940416933074904332 0023949856637270217708569193235032157440377021936236346981000000/14424 8241169496617035389048768248617189632416408038542876022062708604037445 273163212098193876607-312440921104045368530242416647086503120453197270 63264378950467430793965547823621601892187500000/6271662659543331175451 6977725325485734622789742625453424357418568958277150118766613955736468 09*7^(1/2), d[20,7] = 320025509436218930111211692065981238472193011362 081486733393830550522705374338099956643940134000000/266097687126338479 8727363197774524180454709793365679952433450473568658330512181954906407 67586039-3520625960987683352500903568389520785434833169543727900283279 42400352859993946782095267187500000/2660976871263384798727363197774524 18045470979336567995243345047356865833051218195490640767586039*7^(1/2) , d[1,8] = -3661954814522455671253922313941382675955900277535195315609 334325020847182682690500704683900506941143398694756609375/158993925112 9245700677618313983095867136968165622865688718058863033172717560101341 87044031160581577896718931584606253*7^(1/2)-13866190394671881778436412 7976569666821103095661809003581652211471959345066610703963148168657423 990686396670819629710625/842667803098500221359137706411040809582593127 7801188150205711974075815403068537111913333651510823628526103373984131 409, d[2,8] = 0, d[3,8] = 0, d[4,8] = 0, d[5,8] = 0, d[6,8] = 74133328 8792693570079484860988318228390013637247803993970752109135056881560905 2971439911123645794822074162423267500000000/31793557088084849467929871 2763464131762389905643826421925627359026212965309951211138929555865598 48828944435244144350431+1627883849247661641088418794871642984833775728 9045721261647797936296422604150947182799728053779745777215037687500000 000/317935570880848494679298712763464131762389905643826421925627359026 21296530995121113892955586559848828944435244144350431*7^(1/2), d[7,8] \+ = 79123082896630926623463637812909255533142961101862829957352416925286 447371973775444150010628920217678305236562500000000/220674993430076705 5225497871769334391317518385410208694678684673687873401167270600625535 21982291588885501590801877410733*7^(1/2)+27717237031983809874090048752 3036354019524439312876486319474183386201400500525513819436820279474881 7266157550512500000000/16974999494621285040196137475148726087057833733 9246822667591128745221030859020815432733478447916606835001223693751854 41, d[8,8] = -12861740577554588383902606281457985425784955308683399015 5570286355529244623583743414010008266216253508607621679687500000000/12 4456650491968913962213262179227446795107464827891589042097505922858997 60129620679525900777011721401554846975875097967-1661348472601484717048 7148461101253773262972651801851300363786698008797447977545153540142016 311019864995117187500000000/732097944070405376248313306936632039971220 3813405387590711617995462293976546835693838765162948071412679321750514 76351*7^(1/2), d[9,8] = 9578962116211778149781266278342342332660674426 8083310106224653694642863899525827146662109421679740794547821685808145 0916383881519633484253309208376279140133981988799043594645079529377568 125/579933425768151837378330522181342572849957007794017240261508430034 5103991353332450210346579803877176154435409576176027379451782897070355 4586185302883918027922704380449806462284897913533+21034314737074898689 9903054683867209016322444596137985952863921266256974764276202184722819 5348950246505101963291955846455302889702252648158818044100729086574047 328415702352926192657203125/579933425768151837378330522181342572849957 0077940172402615084300345103991353332450210346579803877176154435409576 1760273794517828970703554586185302883918027922704380449806462284897913 533*7^(1/2), d[10,8] = -7058244550886515541459500636642029902838121722 6889251798202272411325871900974438152020594210032605613833654296875000 00000/3377300585348710346293686391603746863876053997669083003806596941 39596998898194939249830746462801367756362117505575927-1185225658463743 4181315220246611226252956244237678527858694716457387668343366940275189 219691332441954155273437500000000/258264162409019026481281900534404171 9434629527629298767616809425773388815103843653086941002362598694607475 01621911003*7^(1/2), d[11,8] = 264113342787125024403196816880935149765 4017308766230265920747732424017262908489939350633314948288234148147346 406250000/174858614322310473538961224862445681102351868184495242734246 469343282628992551267386291834353505380015947715958759+579962955385647 4210547045973456341637052624619571963740010244379611845067321572824785 229662354540612692160156250000/174858614322310473538961224862445681102 3518681844952427342464693432826289925512673862918343535053800159477159 58759*7^(1/2), d[12,8] = -11050672539845365215894640652523936141492978 3819655753305779298948374738908144532006174722432345744254466085087500 /710707035056144198004267569769401067591120768226240119843661157003347 275495838917266769753713401235457206770239-315458311528220505023349262 2154221303550739219024162703600772088633298023148483165853967443084250 211617293437500/923919145572987457405547840700221387868456998694112155 7967595041043514581445905924468006798274216060943688013107*7^(1/2), d[ 13,8] = 55401049968502683122071915895416633439095609163513910332211292 601316365576343023665474609701775000/219597788264867210157601589149675 3793365149395131071233472857613035953389927729956725793001189837-34109 4780586779446760079837959989335622823735651513260575754883873037265709 838066068949734437500/219597788264867210157601589149675379336514939513 1071233472857613035953389927729956725793001189837*7^(1/2), d[14,8] = 4 4030912950946558117546420743656318811226215790624306914399242903219865 8561520525956581125000/17047708829927508480539196603078507146227880026 687679879770968502586061442194771957885884271-140904334246128368877640 841536612650045909429918528996302733246187559757762732414715786750000/ 1704770882992750848053919660307850714622788002668767987977096850258606 1442194771957885884271*7^(1/2), d[15,8] = -558823564680481293399297513 718150688170960039036881228335285834396162002527071014000000000000/168 7723174162823339573380463704772207476560122642080308097325881756020082 777282423830702542829*7^(1/2)+6024333780021859775915057056614514007738 7465401015675154682137825656008372103214499000000000000/16877231741628 2333957338046370477220747656012264208030809732588175602008277728242383 0702542829, d[16,8] = -37644292981740858126529370452378428066410904709 15010775093399616226794991852551269531250000000000000/2904730676352428 5812188013260867926662322375106054628149383758341533298575427748884296 525080254703+317198672539192642600075202268073866347221626797763632822 41456970855044804956054687500000000000000/5480623917646091662676983634 12602389855139152944426946214787893236477331611844318571632548684051*7 ^(1/2), d[17,8] = -164474966991304269119416902921927006769883128145964 3730494549130289512250437935620857238769531250/16852853518001437058606 6335859053198095464953579826396987451863326015027599104857144072486137 8247*7^(1/2)-730602819566148808691068665669461348409342925876218898746 2777552520672361471579454785674633789062500/28649850980602442999631277 0960390436762290421085704874878668167654225546918478257144923226434301 99, d[18,8] = 13774007683776164708615766449495551403599881541305209343 3727665826767811393170063425321210809600000/13708062670144709569201567 9885354275962818383294595633913238357729294520056688161313360395423111 -419785630607329689771467543748821778196625350581601949201622698449894 930439334345170871104000000/806356627655571151129503999325613388016578 7252623272583131668101723207062158127136080023260183*7^(1/2), d[19,8] \+ = -1343503297855407770770355900978109045481785426025801017290833932145 55610673975362560138631950000000/1317049158504099546844856532231835200 42707858459513452191150578994812382015249409889307046582989+2655747829 3843856325070605415002352765238521768003774722107897316174870715650078 3616083593750000/13170491585040995468448565322318352004270785845951345 2191150578994812382015249409889307046582989*7^(1/2), d[20,8] = -239277 1028977103395290499648293451092528467763143716781182700833614483395018 89677056982554762500000/7982930613790154396182089593323572541364129380 09703985730035142070597499153654586471922302758117+4275045809770758356 6082400473301323823137259915888124503439821577185704427836394968711015 6250000/79829306137901543961820895933235725413641293800970398573003514 2070597499153654586471922302758117*7^(1/2)\}: " }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Suppose that we are gi ven the initial value problem: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=f(x,y)" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$%\"xG% \"yG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(x[k])=y[k]" "6#/-%\"yG6#&%\" xG6#%\"kG&F%6#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "When a Runge-Kutta step of width " }{TEXT 272 1 "h" }{TEXT -1 69 " has bee n made using the basic scheme, we wish to obtain the result " } {XPPEDIT 18 0 "y[k](u)" "6#-&%\"yG6#%\"kG6#%\"uG" }{TEXT -1 21 " of a \+ step of width " }{XPPEDIT 18 0 "h*u" "6#*&%\"hG\"\"\"%\"uGF%" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 107 "ee2 \+ := map(_U->lhs(_U)=rhs(_U)/u,ee):\nsubs(ee2,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(8-i)],i=2..8)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7)7*,$*&\"\"\"F**&\"#?F*%\"uGF*!\"\"F*F(%!GF/F/F/F/F/7*,$* (\"$T$F*\"%+KF.F-F.F*,$*(\"%hrF*\"(+S-\"F.F-F.F.,$*(\"'\"G;\"F*F8F.F-F .F*F/F/F/F/F/7*,$*(\"%B5F*\"%+kF.F-F.F*,$*(F?F*\"&+c#F.F-F.F*\"\"!,$*( \"%pIF*FCF.F-F.F*F/F/F/F/7*,$*(\"#RF*\"$+\"F.F-F.F*,$*(\"(nB?%F*\")+\" G;\"F.F-F.F*FD,$*(\"(W)**QF*\"(Dq!HF.F-F.F.,$*(\"(#*H)RF*FTF.F-F.F*F/F /F/7*,$*(\"#$*F*\"$+#F.F-F.F*,$*(\"%6cF*\"'+W6F.F-F.F*FDFD,$*(\"&W<$F* \"'D]8F.F-F.F*,$*(\"'@N#*F*\"(+k5&F.F-F.F*F/F/7*,$*(\"#JF*FfnF.F-F.F*, $*(\"&t6#F*\"'+KMF.F-F.F*FDFD,$*(\"(CEg)F*\")v\"fl(F.F-F.F*,$*(\")4@yE F*\"*+SO*oF.F-F.F.,$*(FinF*\"'+NGF.F-F.F*F/7*,$*(\"$V*F*\"%+5F.F-F.F*, $*(\"1H$p=#=5@7F*\"0+++GH\"3pF.F-F.F.FDFD,$*(\"$D\"F*\"\"#F.F-F.F.,$*( \"1*))fzgIS-\"F*\"0+++!GH*o\"F.F-F.F.,$*(\"1*oGNN39]\"F*\"0+++!G(pl#F. F-F.F*,$*(\"1$GK@@R,2'F*\"/+++;?]#*F.F-F.F*Q(pprint36\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The new polynomials (of degree " } {XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 55 " ) are obtained a s follows (re-using the weight symbol " }{XPPEDIT 18 0 "b[j]" "6#&%\"b G6#%\"jG" }{TEXT -1 3 "). " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "[seq(b[j]=add(d[j,i]*u^(i-1),i=1..8),j=1..20)]:\npols := eval(subs (dd,%)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The first few non-zero polynomials with rough approximations for t he coefficients are . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct to 12 do\n if rhs(pols[ct])<>0 then print(evalf[6](pols[c t])) end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\" \",2$F'\"\"!F'*&$\"'\")\\*)!\"&F'%\"uGF'!\"\"*&$\"'%39%!\"%F')F/\"\"#F 'F'*&$\"'=%4\"!\"$F')F/\"\"$F'F0*&$\"'n:F-)F.\" \"&F-F-*&$\"'AevF2F-)F.\"\"'F-F5*&$\"'KU;F2F-)F.F'F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"),0*&$\"''*f?!\"#\"\"\"%\"uGF-!\"\"*& $\"'vb:F/F-)F.\"\"#F-F-*&$\"'aL^F/F-)F.\"\"$F-F/*&$\"'<>\"*F/F-)F.\"\" %F-F-*&$\"'?\"3*F/F-)F.\"\"&F-F/*&$\"'n$y%F/F-)F.\"\"'F-F-*&$\"'VR5F/F -)F.\"\"(F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*,0*&$\" 'X#H$!\"#\"\"\"%\"uGF-F-*&$\"'d'[#!\"\"F-)F.\"\"#F-F2*&$\"'$\\?)F2F-)F .\"\"$F-F-*&$\"'_d9\"\"!F-)F.\"\"%F-F2*&$\"'X^9F=F-)F.\"\"&F-F-*&$\"'v XwF2F-)F.\"\"'F-F2*&$\"'Lh;F2F-)F.\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5,0*&$\"'(e;%!\"#\"\"\"%\"uGF-!\"\"*&$\"'?Y JF/F-)F.\"\"#F-F-*&$\"':Q5\"\"!F-)F.\"\"$F-F/*&$\"':F2F-)F.\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7,0*&$\"'S*4$!\"%\"\"\"%\"uGF-!\"\"*&$\"'wSB!\"$F-)F .\"\"#F-F-*&$\"'$Qs(F3F-)F.\"\"$F-F/*&$\"'1s8!\"#F-)F.\"\"%F-F-*&$\"'N m8F>F-)F.\"\"&F-F/*&$\"'V(>(F3F-)F.\"\"'F-F-*&$\"'\"Rc\"F3F-)F.\"\"(F- F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The whole scheme, includi ng the weights, is given by the set of equations:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ee3 := `union`(ee2,\{op(pols)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "We can now che ck that this scheme satisfies the order conditions (and row sum condit ions) for a 20 stage, order 8 Runge-Kutta scheme. " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 88 "RK8_20eqs := [op(RowSumConditions(20,'expand ed')),op(OrderConditions(8,20,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "simplify(sub s(ee3,RK8_20eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7gx\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$>#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 43 "#----- -------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 96 "Stage by stage construction of the interpolation scheme C .. [8 stage scheme] .. (longer method)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "In this section an alternative inte rpolation scheme is constructed - one which requires 8 stages - in an \+ effort to improve the principal error curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Start with linking coefficient s using the weights of the 12 stage scheme as the linking coefficients for the first new stage." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5614 "e1 := \{c[ 2] = 1/20,\nc[3] = 341/3200,\nc[4] = 1023/6400,\nc[5] = 39/100,\nc[6] \+ = 93/200,\nc[7] = 31/200,\nc[8] = 943/1000,\nc[9] = 7067558016280/7837 150160667,\nc[10] = 909/1000,\nc[11] = 47/50,\nc[12] = 1,\nc[13] = 1, \na[2,1] = 1/20,\na[3,1] = -7161/1024000,\na[3,2] = 116281/1024000,\na [4,1] = 1023/25600,\na[4,2] = 0,\na[4,3] = 3069/25600,\na[5,1] = 42023 67/11628100,\na[5,2] = 0,\na[5,3] = -3899844/2907025,\na[5,4] = 398299 2/2907025,\na[6,1] = 5611/114400,\na[6,2] = 0,\na[6,3] = 0,\na[6,4] = \+ 31744/135025,\na[6,5] = 923521/5106400,\na[7,1] = 21173/343200,\na[7,2 ] = 0,\na[7,3] = 0,\na[7,4] = 8602624/76559175,\na[7,5] = -26782109/68 9364000,\na[7,6] = 5611/283500,\na[8,1] = -1221101821869329/6908129280 00000,\na[8,2] = 0,\na[8,3] = 0,\na[8,4] = -125/2,\na[8,5] = -10240306 07959889/168929280000000,\na[8,6] = 1501408353528689/265697280000000, \na[8,7] = 6070139212132283/92502016000000,\na[9,1] = -147251426448621 5803881384708877264246346044433307094207829051978044531801133057155/\n 124689480162003200115705962164398602480330155839348790044045 3636168046069686436608,\na[9,2] = 0,\na[9,3] = 0,\na[9,4] = -517229431 1085668458375175655246981230039025336933699114138315270772319372469280 000/\n 124619381004809145897278630571215298365257079410236252 921850936749076487132995191,\na[9,5] = -120706792584692548079789364417 33187949484571516120469966534514296406891652614970375/\n 2722 0311547616572217104781845311006994972840850483890150850769616734461403 98628096,\na[9,6] = 78012515584389364132309055253043103656779559256849 7182701460674803126770111481625/\n 18311042541273197219788987 4507158786859226102980861859505241443073629143100805376,\na[9,7] = 664 1131229599116421347821358391064699281403281605770353571553403929500094 92511875/\n 1517846559858624813633302310729534917527976515008 9078301139943253016877823170816,\na[9,8] = 103328481844520156040568367 67286656859124007796970668046446015775000000/\n 1312703550036 033648073834248740727914537972028638950165249582733679393783,\na[10,1] = -29055573360337415088538618442231036441314060511/\n 226747 59891089577691327962602370597632000000000,\na[10,2] = 0,\na[10,3] = 0, \na[10,4] = -20462749524591049105403365239069/454251913499893469596231 268750,\na[10,5] = -180269259803172281163724663224981097/3810092255825 6871086579832832000000,\na[10,6] = 21127670214172802870128286992003940 810655221489/\n 46794738779978929061458226979767086336737280 00,\na[10,7] = 318607235173649312405151265849660869927653414425413/\n \+ 6714716715558965303132938072935465423910912000000,\na[10,8] \+ = 212083202434519082281842245535894/2002242604477567256382286537117387 9,\na[10,9] = -2698404929400842518721166485087129798562269848229517793 703413951226714583/\n 46954567491393431507700044208087114188 4676035902717550325616728175875000000,\na[11,1] = -2342659845814086836 951207140065609179073838476242943917/\n 13584809613510567770 22231400139158760857532162795520000,\na[11,2] = 0,\na[11,3] = 0,\na[11 ,4] = -996286030132538159613930889652/16353068885996164905464325675,\n a[11,5] = -26053085959256534152588089363841/43775528045656830610112999 42400,\na[11,6] = 2098082234509676029222408679497810531264453392563493 3539/\n 3775889992007550803878727839115494641972212962174156 800,\na[11,7] = 890722993756379186418929622095833835264322635782294899 /\n 13921242001395112657501941955594013822830119803764736,\n a[11,8] = 161021426143124178389075121929246710833125/10997207722131034 650667041364346422894371443,\na[11,9] = 300760669768102517834232497565 452434946672266195876496371874262392684852243925359864884962513/\n \+ 465544333750134645558506533660450560376082477961552128575189281 0315680492364106674524398280000,\na[11,10] = -311552374371117306659232 06875/392862141594230515010338956291,\na[12,1] = -28665569918256639717 78295329101033887534912787724034363/\n 868226711619262703011 213925016143612030669233795338240,\na[12,2] = 0,\na[12,3] = 0,\na[12,4 ] = -16957088714171468676387054358954754000/14369041511965468332636822 8101570221,\na[12,5] = -4583493974484572912949314673356033540575/45195 7703655250747157313034270335135744,\na[12,6] = 23463053885534042586562 58473446184419154740172519949575/\n 256726716407895402892744 978301151486254183185289662464,\na[12,7] = 165712155931984680217128369 0913610698586256573484808662625/\n 1343148041125514647725915 5104956093505361644432088109056,\na[12,8] = 34568537955467705221549582 5476969226377187500/74771167436930077221667203179551347546362089,\na[1 2,9] = \n -3205890962717072542791434312152727534008102774023210240571 361570757249056167015230160352087048674542196011/\n 94756954968396581 4783015124451273604984657747127257615372449205973192657306017239103491 074738324033259120,\na[12,10] = 40279545832706233433100438588458933210 937500/8896460842799482846916972126377338947215101,\na[12,11] = -61229 33601070769591613093993993358877250/1050517001510235513198246721302027 675953,\na[13,1] = 44901867737754616851973/1014046409980231013380680, \na[13,2] = 0,\na[13,3] = 0,\na[13,4] = 0,\na[13,5] = 0,\na[13,6] = 79 1638675191615279648100000/2235604725089973126411512319,\na[13,7] = 384 7749490868980348119500000/15517045062138271618141237517,\na[13,8] = -1 3734512432397741476562500000/875132892924995907746928783,\na[13,9] = 1 2274765470313196878428812037740635050319234276006986398294443554969616 342274215316330684448207141/\n 489345147493715517650385834143510 934888829280686609654482896526796523353052166757299452852166040,\na[13 ,10] = -9798363684577739445312500000/308722986341456031822630699,\na[1 3,11] = 282035543183190840068750/12295407629873040425991,\na[13,12] = \+ -306814272936976936753/1299331183183744997286\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "su bs(e1,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..13)])):\n evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7/$\"+++++] !#6F(%!GF+F+F+F+F+F+F+F+F+F+7/$\"++]il5!#5$!+iS;$*p!#7$\"+TmbN6F/F+F+F +F+F+F+F+F+F+F+7/$\"++vV)f\"F/$\"+]P4'*RF*$\"\"!F;$\"+D\"G))>\"F/F+F+F +F+F+F+F+F+F+7/$\"+++++RF/$\"+Gc(Rh$F/F:$!+nS_T8!\"*$\"+/l7q8FEF+F+F+F +F+F+F+F+7/$\"++++]YF/$\"+!G?Z!\\F*F:F:$\"+U?(4N#F/$\"+Ifb3=F/F+F+F+F+ F+F+F+7/$\"++++]:F/$\"+W!*GphF*F:F:$\"+JolB6F/$!+rg/&)QF*$\"+8()=z>F*F +F+F+F+F+F+7/$\"++++I%*F/$!+S-jnB 3l&FE$\"+U'p@c'F]oF+F+F+F+F+7/$\"+$QWV%FE$\"+*=3/E%FE$\"+ASOvVF]o$\"+!\\D9(yF2F+F+F+F+7/$\"++++!4*F/$ !+**fS\"G\"FEF:F:$!+'*Rr/XF]o$!+p?OJZFE$\"+,$)R'F]o$\"+DG?k9F*$\"+s(3/Y'F*$!+pJKIzF*F+F+7 /$\"\"\"F;$!+oEi,LFEF:F:$!+CF6!=\"!\"($!+RA995F]o$\"+K8JR\"*FE$\"+G%fP B\"F^s$\"+zVCBYFE$!+QxF$Q$FE$\"++@fFXFE$!+'[&\\GeFEF+7/Fhr$\"+>%*)zU%F *F:F:F:F:$\"+#R\\5a$F/$\"+b@pzCF/$!+/-Up:F]o$\"+(\\1%3DF]o$!+zn$Q<$F]o $\"+F$GQH#F]o$!+LYKhBF/Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 14" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order cond ition gives rise to a group \{list) of equations to be satisfied by t he \"d\" coefficients of the weight polynomials for a given stage (cor responding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO6_13 \+ := SimpleOrderConditions(6,13,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "whch := [1,2,3,6, 7,8,12,15,16,24,27,31]:\nordeqns1 := []:\nfor ct in whch do\n eqn_gr oup := convert(SO6_13[ct],'polynom_order_conditions',6):\n ordeqns1 \+ := [op(ordeqns1),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known coeffi cients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns1 := []:\nfo r ct to nops(ordeqns1) do\n eqns1 := [op(eqns1),expand(subs(e1,ordeq ns1[ct]))];\nend do:\nnops(eqns1);\nnops(indets(eqns1));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#y" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Solve th e system of equations to give solutions in terms of " }{XPPEDIT 18 0 "d[1,i];" "6#&%\"dG6$\"\"\"%\"iG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "i=1 " "6#/%\"iG\"\"\"" }{TEXT -1 11 " . . . 6.. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:= 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "d1 := solve(\{op(eqns 1)\},indets(eqns1) minus \{seq(d[1,i],i=1..6)\}):\ninfolevel[solve]:=0 :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Set \+ " }{XPPEDIT 18 0 "d[1,1]=1" "6#/&%\"dG6$\"\"\"F'F'" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "d[1,i]=0" "6#/&%\"dG6$\"\"\"%\"iG\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 9 " . . . 5 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "dd := \{d[1,1]=1,seq(d[1,i]=0,i=2..5)\}:\ndd_1 := `un ion`(dd,simplify(subs(dd,d1))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Take " }{XPPEDIT 18 0 "c[14]=1/2-sqrt(7)/ 14" "6#/&%\"cG6#\"#9,&*&\"\"\"F*\"\"#!\"\"F**&-%%sqrtG6#\"\"(F*F'F,F, " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "We choose " } {XPPEDIT 18 0 "d[1,6]" "6#&%\"dG6$\"\"\"\"\"'" }{TEXT -1 67 " to mini mze the 2-norm of the linking coefficients for this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "sb \+ := \{seq(a[14,j]=add(expand(subs(\{op(dd_1),c[14]=1/2-7^(1/2)/14\},d[j ,i]*c[14]^i)),i=1..6),j=1..13)\}:\nsm := subs(sb,add(a[14,j]^2,j=1..13 )):\nplot(sm,d[1,6]=-293.76..-293.72);" }}{PARA 13 "" 1 "" {GLPLOT2D 420 359 359 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!3\"***********fPH!#:$\"3_ 22ca_;iO!#<7$$!3*HLQ6G\"fPHF*$\"3.WQe@%*HVNF-7$$!3O;M!\\p$ePHF*$\"3k1H Z\"=(oVMF-7$$!31L))Qj^dPHF*$\"3<*G25&3!fL$F-7$$!3SL=KvlcPHF*$\"3*)yFKe 8$>B$F-7$$!3];C2G!ev$HF*$\"3RF>TQ*pH8$F-7$$!3D$3yO5]v$HF*$\"3!\\b\"=IH CXIF-7$$!3v\\nU)*=aPHF*$\"3![wDBo&[eHF-7$$!3E$3WDTLv$HF*$\"3%ylwzbEJ(G F-7$$!3#)\\d(Q&\\_PHF*$\"31@[@#feCz#F-7$$!3\"pm&4`i^PHF*$\"3+Gt2^J39FF -7$$!35LQW*e3v$HF*$\"3e:+?C_\"*[EF-7$$!3*)**p)>'**\\PHF*$\"3w[Os!Q$))z DF-7$$!3<+]5*H\"\\PHF*$\"3G*)zEnW=:DF-7$$!3++I\"3&H[PHF*$\"3g\"obi)>@d CF-7$$!3b$3k(p`ZPHF*$\"3>$*p?.$*G3CF-7$$!3mmO;bjYPHF*$\"37U(*p.jsaBF-7 $$!3'pm9'=(eu$HF*$\"3j,pHL/F8BF-7$$!3\"*\\F\\N)\\u$HF*$\"3SSS%G:q&pAF- 7$$!3pmYUs>WPHF*$\"3H$)3l'HY\\B#F-7$$!3(*\\FRXLVPHF*$\"3.Z3:UHM,AF-7$$ !3?]#=/8Du$HF*$\"3'3$z)\\y3O<#F-7$$!3qmT&*elTPHF*$\"3Q8;$)[c5\\@F-7$$! 3o;Wn(o3u$HF*$\"3\"QB(GTIfI@F-7$$!3KLeV(>+u$HF*$\"3BwDCGg!\\6#F-7$$!3p $3k%y8RPHF*$\"3U4'p(>jJ.@F-7$$!3!*\\K_,PQPHF*$\"3eHA/g/8(4#F-7$$!3ULo@ 5aPPHF*$\"3w7r0'3IX4#F-7$$!3))**f[WoOPHF*$\"3%=8_X'QH'4#F-7$$!3$**\\*e k%et$HF*$\"3&GpN#HfR-@F-7$$!3U](3mN]t$HF*$\"3kA%*4U,U7@F-7$$!3L+&ySNTt $HF*$\"3TaF?.yHG@F-7$$!34nY!\\ELt$HF*$\"3Mo2vAd#o9#F-7$$!3B+])ziCt$HF* $\"3sgVrz01r@F-7$$!3J$3_;!oJPHF*$\"3M56\"=0#*p>#F-7$$!3M+ISX#3t$HF*$\" 3fzu*p+t$HF*$\"3]eUwbG_kAF-7$$!3W]aSlF%4y#F-7$$! 3F]Fpyn@PHF*$\"3sDb.5X(3'GF-7$$!3T]#f6p3s$HF*$\"3LV\"\\dZ,:%HF-7$$!3F+ ++++?PHF*$\"3'zab-cSE.$F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F`[lF_[l-%+ AXESLABELSG6$Q'd[1,6]6\"Q!Fe[l-%%VIEWG6$;$!&w$H!\"#$!&s$HF]\\l%(DEFAUL TG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Digits := 15:\nminimize(evalf(sm),location);\nDigits := 10:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"0b7_r-X4#!#9<#7$<#/&%\"dG6$\"\"\" \"\"'$!01qyYut$H!#7F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "-293.737446787006;\nconvert(%,ratio nal,8);\nevalf[20](eval(sm,d[1,6]=%));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!01qyYut$H!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!&zD&\"$z\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"51Q3\\WoN]%4#!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "d[1,6] = -52579/179;" "6#/&%\"dG6$\"\"\"\"\"',$*&\"&zD&F'\"$z\"! \"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "dd := \{d[1,1]=1,d[1,2]=0,d[1,3]=0 ,d[1,4]=0,d[1,5]=0,d[1,6]=-52579/179\}:\nd_1 := `union`(dd,subs(dd,d1) ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "subs(d_1,matrix([seq([seq(d[j,i],j=1..13)],i=1..6)])) :\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7/$\" \"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+F+F+7/F+F+F+F+F+$\"'5YW!\"%$!'@95F/$\" 'Ie?F*$!'nJ#*!\"\"$\"'iK8F*$!'EoCF*$\"'-\"R\"F/$\"'4XC!\"&7/F+F+F+F+F+ $!'@,'*F/$\"'[/ F/7/F+F+F+F+F+$\"'tN5FO$!'Ix:F/$\"'***H'F*$!'ubGF*$\"'#G7%F*$!';rvF*$ \"'m)o#FO$\"'jn^F/7/F+F+F+F+F+$!'twbF/$\"'?`wF?$!'&\\v$F*$\"'U@-\"\"\"#$\"'7wXF)$!'z/mF)$\"'*\\A\"Fhp$\"'E(z*F/$\"'Pu?F/Q(pprint 76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 " We can check which of the groups of order conditions are satisfied." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "nm := NULL:\nfor ct to nop s(SO6_13) do\n eqn_group := convert(SO6_13[ct],'polynom_order_condit ions',6):\n tt := expand(subs(\{op(e1),op(d1)\},eqn_group));\n tt \+ := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op(i,tt),i=1..nop s(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%]);\nop(\{seq( i,i=1..32)\} minus \{nm\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6@\"\"\" \"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\" #=\"#>\"#?\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#<\"#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Evalua te the \"weight\" polynomials at the node " }{XPPEDIT 18 0 "c[14]=1/2- sqrt(7)/14" "6#/&%\"cG6#\"#9,&*&\"\"\"F*\"\"#!\"\"F**&-%%sqrtG6#\"\"(F *F'F,F," }{TEXT -1 83 " to obtain the linking coefficients in the nex t stage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "eqs14 := \{seq(a[14,j]=add(expand(subs(\{op(d_1),c[1 4]=1/2-7^(1/2)/14\},d[j,i]*c[14]^i)),i=1..6),j=1..13)\}:\ne2 := `union `(eqs14,\{c[14]=1/2-7^(1/2)/14\},e1):" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6801 "e2 := \{a[9,3] = 0, a[13,7] = 384774949 0868980348119500000/15517045062138271618141237517, a[13,8] = -13734512 432397741476562500000/875132892924995907746928783, a[13,11] = 28203554 3183190840068750/12295407629873040425991, a[13,12] = -3068142729369769 36753/1299331183183744997286, c[3] = 341/3200, a[13,6] = 7916386751916 15279648100000/2235604725089973126411512319, a[14,2] = 0, c[2] = 1/20, a[5,3] = -3899844/2907025, c[4] = 1023/6400, a[8,1] = -12211018218693 29/690812928000000, c[10] = 909/1000, a[13,10] = -97983636845777394453 12500000/308722986341456031822630699, a[13,9] = 1227476547031319687842 8812037740635050319234276006986398294443554969616342274215316330684448 207141/489345147493715517650385834143510934888829280686609654482896526 796523353052166757299452852166040, a[9,2] = 0, a[8,2] = 0, a[11,2] = 0 , a[11,3] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12,2] = \+ 0, a[7,2] = 0, a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,3] = 0, a[10 ,3] = 0, a[14,3] = 0, a[14,5] = 0, a[13,5] = 0, a[8,3] = 0, c[12] = 1, c[13] = 1, a[6,3] = 0, a[5,2] = 0, a[13,4] = 0, a[7,3] = 0, a[13,2] = 0, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100, a[8,4] = -125/2, a [6,2] = 0, a[13,3] = 0, a[10,2] = 0, a[7,6] = 5611/283500, a[12,10] = \+ 40279545832706233433100438588458933210937500/8896460842799482846916972 126377338947215101, a[7,5] = -26782109/689364000, c[7] = 31/200, c[8] \+ = 943/1000, c[9] = 7067558016280/7837150160667, a[3,2] = 116281/102400 0, a[7,4] = 8602624/76559175, a[3,1] = -7161/1024000, c[5] = 39/100, a [14,12] = 11352128098668146659861/254668911904014019468056-52158426399 28607924801/127334455952007009734028*7^(1/2), a[2,1] = 1/20, a[4,1] = \+ 1023/25600, a[6,5] = 923521/5106400, a[7,1] = 21173/343200, c[11] = 47 /50, a[6,4] = 31744/135025, a[14,1] = -942013/61397+2856761/491176*7^( 1/2), a[14,6] = -1090956177631910962162375596685/860587625357215461564 8617314+40421151553845087519011727532985/84337587285007115233356449677 2*7^(1/2), a[14,7] = 1274787529143761017157317260602825/26341935917422 215554074218244182-958929531471161732045165837664925/52683871834844431 108148436488364*7^(1/2), a[14,10] = -207966836143121645567339523373437 5/5957776266668670748330679562+224582419744760612736276500828125/17022 21790476763070951622732*7^(1/2), a[14,8] = -87897979289600674896294005 58015625/16279513475811535751534066502+6644470662554184045392548540578 125/32559026951623071503068133004*7^(1/2), a[14,11] = 2970221666030903 358562614446315/4589064694536869684100726-2245277482452996546801919407 985/9178129389073739368201452*7^(1/2), a[10,5] = -18026925980317228116 3724663224981097/38100922558256871086579832832000000, a[10,7] = 318607 235173649312405151265849660869927653414425413/671471671555896530313293 8072935465423910912000000, a[10,8] = 212083202434519082281842245535894 /20022426044775672563822865371173879, a[9,6] = 78012515584389364132309 0552530431036567795592568497182701460674803126770111481625/18311042541 2731972197889874507158786859226102980861859505241443073629143100805376 , a[9,8] = 10332848184452015604056836767286656859124007796970668046446 015775000000/131270355003603364807383424874072791453797202863895016524 9582733679393783, a[9,7] = 6641131229599116421347821358391064699281403 28160577035357155340392950009492511875/1517846559858624813633302310729 5349175279765150089078301139943253016877823170816, a[9,4] = -517229431 1085668458375175655246981230039025336933699114138315270772319372469280 000/124619381004809145897278630571215298365257079410236252921850936749 076487132995191, c[6] = 93/200, a[9,5] = -1207067925846925480797893644 1733187949484571516120469966534514296406891652614970375/27220311547616 57221710478184531100699497284085048389015085076961673446140398628096, \+ a[8,5] = -1024030607959889/168929280000000, a[8,7] = 6070139212132283/ 92502016000000, a[9,1] = -14725142644862158038813847088772642463460444 33307094207829051978044531801133057155/1246894801620032001157059621643 986024803301558393487900440453636168046069686436608, a[8,6] = 15014083 53528689/265697280000000, a[10,6] = 2112767021417280287012828699200394 0810655221489/4679473877997892906145822697976708633673728000, a[5,4] = 3982992/2907025, a[12,1] = -28665569918256639717782953291010338875349 12787724034363/868226711619262703011213925016143612030669233795338240, a[11,5] = -26053085959256534152588089363841/4377552804565683061011299 942400, a[11,4] = -996286030132538159613930889652/16353068885996164905 464325675, a[11,6] = 2098082234509676029222408679497810531264453392563 4933539/3775889992007550803878727839115494641972212962174156800, a[10, 9] = -2698404929400842518721166485087129798562269848229517793703413951 226714583/469545674913934315077000442080871141884676035902717550325616 728175875000000, a[10,1] = -290555733603374150885386184422310364413140 60511/22674759891089577691327962602370597632000000000, a[11,1] = -2342 659845814086836951207140065609179073838476242943917/135848096135105677 7022231400139158760857532162795520000, a[10,4] = -20462749524591049105 403365239069/454251913499893469596231268750, a[11,10] = -3115523743711 1730665923206875/392862141594230515010338956291, a[13,1] = 44901867737 754616851973/1014046409980231013380680, a[12,7] = 16571215593198468021 71283690913610698586256573484808662625/1343148041125514647725915510495 6093505361644432088109056, a[12,8] = 345685379554677052215495825476969 226377187500/74771167436930077221667203179551347546362089, a[12,9] = - 3205890962717072542791434312152727534008102774023210240571361570757249 056167015230160352087048674542196011/947569549683965814783015124451273 6049846577471272576153724492059731926573060172391034910747383240332591 20, a[6,1] = 5611/114400, a[12,4] = -169570887141714686763870543589547 54000/143690415119654683326368228101570221, a[12,5] = -458349397448457 2912949314673356033540575/451957703655250747157313034270335135744, a[1 1,7] = 890722993756379186418929622095833835264322635782294899/13921242 001395112657501941955594013822830119803764736, a[12,6] = 2346305388553 404258656258473446184419154740172519949575/256726716407895402892744978 301151486254183185289662464, a[11,9] = 3007606697681025178342324975654 52434946672266195876496371874262392684852243925359864884962513/4655443 3375013464555850653366045056037608247796155212857518928103156804923641 06674524398280000, a[11,8] = 16102142614312417838907512192924671083312 5/10997207722131034650667041364346422894371443, a[12,11] = -6122933601 070769591613093993993358877250/105051700151023551319824672130202767595 3, a[14,9] = 998747057824857338729725466765942289696336789942851277258 2082944283962013964900384895127596133/41295613981107641868485821081000 930261227361072745605305202526150545875206586329586453004-377489797344 7442754802925721321508929613447293787235365983123527137197728406982036 112477728851/412956139811076418684858210810009302612273610727456053052 02526150545875206586329586453004*7^(1/2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[14,i]=subs(e2,a[14,i]),i=1..13):\nevalf[40](%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6//&%\"aG6$\"#9\"\"\"$\"FF7AS'R>x!zu2*[w #z2Z^%!#Q/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\" \"&F0/&F%6$F'\"\"'$\"E-IJ**fo(3?V[cXYK5Coi;'QaN]f7Fdo" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simp le) order condition gives rise to a group \{list) of equations to be \+ satisfied by the \"d\" coefficients of the weight polynomials for a gi ven stage (corresponding to an \"approximate\" interpolation scheme). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_14 := SimpleOrderConditions(7,14,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "whc h := [1,2,3,6,7,8,12,15,16,21,27,31,32,64]:\nordeqns2 := []:\nfor ct i n whch do\n eqn_group := convert(SO7_14[ct],'polynom_order_condition s',7):\n ordeqns2 := [op(ordeqns2),op(eqn_group)];\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitut e for all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns2 := []:\nfor ct to nops(ordeqns2) do\n eqns2 := [op(eqns 2),expand(subs(e2,ordeqns2[ct]))];\nend do:\nnops(eqns2);\nnops(indets (eqns2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#)*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#)*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d2 := solve(\{op(eqns2)\}):\ninfole vel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 66 "subs(d2,matrix([seq([seq(d[j,i],j=1..14)],i=1. .7)])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 )70$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+F+F+F+70$!'8js!\"&F+F+F+F+$\"',# Q%F/$\"'e0:!\"%$!'*p!\\!\"#$\"'$G9#F7$!'e3JF7$\"'oseF7F+$\"'3dA!\"'$!' oM7F470$\"'UqCF4F+F+F+F+$!'**\\CF4$!'tswF4$\"'1\")e!\"\"$!'ejDFL$\"'>9 PFL$!'NIqFL$!'z;9F4$!'1Q[F/$\"'AJ#)F470$!'LRVF4F+F+F+F+$\"'rDVF4$\"'mq :!\"$$!'74BF*$\"'255F*$!'@i9F*$\"'wgFF*$\"'Q@gF4$\"'cQ=F4$!'-_=Fjn70$ \"',sRF4F+F+F+F+$!%7fFjn$!'@7:Fjn$\"'O1SF*$!'+k$F *$\"'$yT\"F*$!'f[?F*$\"'QCQF*$\"'&eI$F4$\"'mQ5F4$!''=w&F470$\"'uFFF/F+ F+F+F+$\"'g]AF4$!'kN&)F/$\"'2r&*FL$!'A(G%FL$\"'x(='FL$!'LZ6F*F+F+F+Q(p print16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions are satisf ied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "nm := NULL:\nfor ct to nops(SO7_14) do\n eqn_group := convert(SO7_14[ct],'polynom_order _conditions',7):\n tt := expand(subs(\{op(e2),op(d2)\},eqn_group)); \n tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op(i,tt), i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%]);\no p(\{seq(i,i=1..64)\} minus \{nm\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 J\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"# :\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K \"#Z\"#^\"#`\"#d\"#f\"#g\"#i\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6:\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S \"#T\"#U\"#V\"#W\"#X\"#Y\"#[\"#\\\"#]\"#_\"#a\"#b\"#c\"#e\"#h\"#j" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate \+ the \"weight\" polynomials at the node " }{XPPEDIT 18 0 "c[15] = 57/1 25;" "6#/&%\"cG6#\"#:*&\"#d\"\"\"\"$D\"!\"\"" }{TEXT -1 83 " to obtai n the linking coefficients in the next stage in the interpolation sche me." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "eqs15 := \{seq(a[15, j]=add(expand(subs(\{op(d2),c[15]=57/125\},d[j,i]*c[15]^i)),i=1..7),j= 1..14)\}:\ne3 := `union`(eqs15,\{c[15]=57/125\},e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8419 "e3 := \{a[9,6] = 7801 2515584389364132309055253043103656779559256849718270146067480312677011 1481625/18311042541273197219788987450715878685922610298086185950524144 3073629143100805376, a[11,5] = -26053085959256534152588089363841/43775 52804565683061011299942400, a[11,8] = 16102142614312417838907512192924 6710833125/10997207722131034650667041364346422894371443, a[10,7] = 318 607235173649312405151265849660869927653414425413/671471671555896530313 2938072935465423910912000000, a[8,5] = -1024030607959889/1689292800000 00, a[4,1] = 1023/25600, a[2,1] = 1/20, a[11,9] = 30076066976810251783 4232497565452434946672266195876496371874262392684852243925359864884962 513/465544333750134645558506533660450560376082477961552128575189281031 5680492364106674524398280000, a[12,6] = 234630538855340425865625847344 6184419154740172519949575/25672671640789540289274497830115148625418318 5289662464, a[12,5] = -4583493974484572912949314673356033540575/451957 703655250747157313034270335135744, a[10,4] = -204627495245910491054033 65239069/454251913499893469596231268750, a[13,5] = 0, a[7,1] = 21173/3 43200, a[7,2] = 0, a[7,3] = 0, a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), \+ a[8,2] = 0, a[8,3] = 0, a[13,3] = 0, c[13] = 1, a[11,3] = 0, a[14,4] = 0, a[13,4] = 0, a[9,2] = 0, a[14,2] = 0, c[12] = 1, a[10,3] = 0, a[14 ,13] = 3/392-3/392*7^(1/2), a[12,2] = 0, a[11,2] = 0, a[10,2] = 0, a[1 2,3] = 0, a[4,2] = 0, a[14,5] = 0, a[6,2] = 0, a[8,7] = 60701392121322 83/92502016000000, a[8,6] = 1501408353528689/265697280000000, a[6,3] = 0, a[9,3] = 0, c[6] = 93/200, a[9,5] = -12070679258469254807978936441 733187949484571516120469966534514296406891652614970375/272203115476165 7221710478184531100699497284085048389015085076961673446140398628096, a [5,2] = 0, a[13,2] = 0, a[12,8] = 345685379554677052215495825476969226 377187500/74771167436930077221667203179551347546362089, a[12,7] = 1657 121559319846802171283690913610698586256573484808662625/134314804112551 46477259155104956093505361644432088109056, c[5] = 39/100, a[14,12] = 1 1352128098668146659861/254668911904014019468056-5215842639928607924801 /127334455952007009734028*7^(1/2), a[3,1] = -7161/1024000, a[6,4] = 31 744/135025, a[7,4] = 8602624/76559175, c[7] = 31/200, c[8] = 943/1000, c[9] = 7067558016280/7837150160667, c[11] = 47/50, a[6,5] = 923521/51 06400, a[10,6] = 21127670214172802870128286992003940810655221489/46794 73877997892906145822697976708633673728000, a[7,6] = 5611/283500, a[8,1 ] = -1221101821869329/690812928000000, a[13,9] = 122747654703131968784 2881203774063505031923427600698639829444355496961634227421531633068444 8207141/48934514749371551765038583414351093488882928068660965448289652 6796523353052166757299452852166040, a[4,3] = 3069/25600, a[5,1] = 4202 367/11628100, a[8,4] = -125/2, a[3,2] = 116281/1024000, a[12,4] = -169 57088714171468676387054358954754000/1436904151196546833263682281015702 21, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = -3899844/2907025, a[7 ,5] = -26782109/689364000, a[13,8] = -13734512432397741476562500000/87 5132892924995907746928783, a[13,7] = 3847749490868980348119500000/1551 7045062138271618141237517, a[13,12] = -306814272936976936753/129933118 3183744997286, a[10,5] = -180269259803172281163724663224981097/3810092 2558256871086579832832000000, a[11,1] = -23426598458140868369512071400 65609179073838476242943917/1358480961351056777022231400139158760857532 162795520000, a[12,11] = -6122933601070769591613093993993358877250/105 0517001510235513198246721302027675953, a[11,7] = 890722993756379186418 929622095833835264322635782294899/139212420013951126575019419555940138 22830119803764736, c[2] = 1/20, a[13,11] = 282035543183190840068750/12 295407629873040425991, a[13,10] = -9798363684577739445312500000/308722 986341456031822630699, a[11,10] = -31155237437111730665923206875/39286 2141594230515010338956291, a[12,9] = -32058909627170725427914343121527 2753400810277402321024057136157075724905616701523016035208704867454219 6011/94756954968396581478301512445127360498465774712725761537244920597 3192657306017239103491074738324033259120, a[11,4] = -99628603013253815 9613930889652/16353068885996164905464325675, a[12,1] = -28665569918256 63971778295329101033887534912787724034363/8682267116192627030112139250 16143612030669233795338240, a[13,1] = 44901867737754616851973/10140464 09980231013380680, a[10,8] = 212083202434519082281842245535894/2002242 6044775672563822865371173879, a[12,10] = 40279545832706233433100438588 458933210937500/8896460842799482846916972126377338947215101, c[3] = 34 1/3200, a[13,6] = 791638675191615279648100000/223560472508997312641151 2319, a[6,1] = 5611/114400, a[5,4] = 3982992/2907025, a[11,6] = 209808 22345096760292224086794978105312644533925634933539/3775889992007550803 878727839115494641972212962174156800, a[10,9] = -269840492940084251872 1166485087129798562269848229517793703413951226714583/46954567491393431 5077000442080871141884676035902717550325616728175875000000, a[9,8] = 1 0332848184452015604056836767286656859124007796970668046446015775000000 /131270355003603364807383424874072791453797202863895016524958273367939 3783, a[9,4] = -517229431108566845837517565524698123003902533693369911 4138315270772319372469280000/12461938100480914589727863057121529836525 7079410236252921850936749076487132995191, a[10,1] = -29055573360337415 088538618442231036441314060511/226747598910895776913279626023705976320 00000000, a[9,1] = -14725142644862158038813847088772642463460444333070 94207829051978044531801133057155/1246894801620032001157059621643986024 803301558393487900440453636168046069686436608, a[9,7] = 66411312295991 1642134782135839106469928140328160577035357155340392950009492511875/15 1784655985862481363330231072953491752797651500890783011399432530168778 23170816, a[14,11] = 2970221666030903358562614446315/45890646945368696 84100726-2245277482452996546801919407985/9178129389073739368201452*7^( 1/2), a[14,10] = -2079668361431216455673395233734375/59577762666686707 48330679562+224582419744760612736276500828125/170222179047676307095162 2732*7^(1/2), a[14,8] = -8789797928960067489629400558015625/1627951347 5811535751534066502+6644470662554184045392548540578125/325590269516230 71503068133004*7^(1/2), a[14,7] = 1274787529143761017157317260602825/2 6341935917422215554074218244182-958929531471161732045165837664925/5268 3871834844431108148436488364*7^(1/2), a[14,1] = -942013/61397+2856761/ 491176*7^(1/2), a[14,6] = -1090956177631910962162375596685/86058762535 72154615648617314+40421151553845087519011727532985/8433758728500711523 33564496772*7^(1/2), a[15,8] = 598933003512786415989929347488/28972691 439853083012428515625-14751096540440194135375671464064/137194215347539 5989706173828125*7^(1/2), a[15,2] = 0, a[15,3] = 0, a[15,4] = 0, a[15, 5] = 0, a[14,9] = 9987470578248573387297254667659422896963367899428512 772582082944283962013964900384895127596133/412956139811076418684858210 81000930261227361072745605305202526150545875206586329586453004-3774897 9734474427548029257213215089296134472937872353659831235271371977284069 82036112477728851/4129561398110764186848582108100093026122736107274560 5305202526150545875206586329586453004*7^(1/2), a[15,10] = 543652640255 07653851977480738208/4466478390356713423359819140625-14962945283111314 16682934929792/235786943683989396406626953125*7^(1/2), a[15,12] = -192 7555453883917676542797699/276054308640062825012207031250, a[15,11] = - 3598066799092816058284144620546/146286964102678984417724609375+6557086 9112872951387443794088672/5120043743593764454620361328125*7^(1/2), a[1 5,6] = 4116869726480787181292612488487584/2659853483608280800756874389 6484375-245534888302358312351817429197184/8700455320214002619298187255 859375*7^(1/2), a[15,9] = -3067066459661270994000493048546286135407493 899133250841529092716875641340142781006771180215525814345889446383/373 3407192182277814105116532466971854315408940785290943015262808201014351 28911405410349160893280029296875000+1816647611984076940241842059764388 009599469326146254865545587094445401874686830164148895260773137/424475 6546679057970967517479159936835440315081843828746859964350339160106858 78289836883544921875*7^(1/2), a[15,13] = -15630718068/3814697265625+61 67095848/3814697265625*7^(1/2), a[15,7] = 6314766179596965478242603858 43267488/2705957914016858192337687904052734375-33227466731347290455944 055132270208/3922838396059942349838659979248046875*7^(1/2), c[15] = 57 /125, a[15,14] = 172678683744/3814697265625*7^(1/2), a[15,1] = 5719665 1428918572875631366996149/1245822888041363915313720703125000+108917023 47425878099023621466959/8357395207277482931896209716796875*7^(1/2)\}: \+ " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[15,i]=subs(e3,a[15,i]),i=1..1 4):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "60/&%\"aG6$\"#:\" \"\"$\"I)o3s%3O&=G=&Q7?Oe)Gze$\\!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F 0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"I'f*yk4*QhPa&>lOo%=t^7, )F+/&F%6$F'\"\"($\"Ib0;!*[0UZn*)*4$)R_mr]&4@!#S/&F%6$F'\"\")$!H4Gqh=n' >?r@y1Y\"z8SZx(!#Q/&F%6$F'\"\"*$\"I0$f#GJ8kmI)*zIl#4C?Zz5$!#R/&F%6$F' \"#5$!H_zbpUN6_/NO8&y*HL\"*zh%FQ/&F%6$F'\"#6$\"Hnc9XZW&)H`k\\Px]'GiR(G *FQ/&F%6$F'\"#7$!I$=x/?)*Rh7c!QQh,.=E_#)p!#U/&F%6$F'\"#8$\"HnA+k\\-Cml ;nR%))*)yL+)z\"Feo/&F%6$F'\"#9$\"IN1#**)psaQm!3\"R\\,j-Qk(>\"FJ" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 1 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Eac h standard (simple) order condition gives rise to a group \{list) of \+ equations to be satisfied by the \"d\" coefficients of the weight poly nomials for a given stage (corresponding to an \"approximate\" interpo lation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "SO7_15 := SimpleOrderConditions(7,15,'expanded '):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "whch := [1,2,3,6,7,8,12,15,16,21,27,31,32,36,64]:\no rdeqns3 := []:\nfor ct in whch do\n eqn_group := convert(SO7_15[ct], 'polynom_order_conditions',7):\n ordeqns3 := [op(ordeqns3),op(eqn_gr oup)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns3 := []:\nfor ct to nops(ordeqns3) do \n eqns3 := [op(eqns3),expand(subs(e3,ordeqns3[ct]))];\nend do:\nnop s(eqns3);\nnops(indets(eqns3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$ 0\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$0\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[so lve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d3 := solve(\{o p(eqns3)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs(d3,matrix([seq([seq(d[ j,i],j=1..15)],i=1..7)])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)71$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+F+F+F+F+71$!' Hru!\"&F+F+F+F+$!&V2\"!\"%$\"'NR;F2$!'8mD!\"$$\"'U:AF7$!'2-KF7$\"'yCOF 7$!'I**oF/$!'!=H\"F/$!'3'f\"F2$\"&M5*F271$\"'o@FF2F+F+F+F+$\"&c8%F7$!' O(G*F2$\"'l!o#!\"#$!'(eW#FO$\"'C'[$FO$!'P&z$FO$\"'j5pF2$\"'#yM\"F2$\"' Lf7F7$!'y)4\"F771$!'`D`F2F+F+F+F+$!'L_@F7$\"'T/AF7$!'\"*f5!\"\"$\"'U&) **FO$!'179Fao$\"'w-:Fao$!'VmEF7$!'o]`F2$!'7kNF7$\"'x7VF771$\"'LxcF2F+F +F+F+$\"'u5WF7$!'43EF7$\"'?n>Fao$!'`+>Fao$\"'+rEFao$!'I#z#Fao$\"'ge[F7 $\"'c,5F7$\"'8*o%F7$!'mduF771$!'8)4$F2F+F+F+F+$!' " 0 " " {MPLTEXT 1 0 318 "nm := NULL:\nfor ct to nops(SO7_15) do\n eqn_gro up := convert(SO7_15[ct],'polynom_order_conditions',7):\n tt := expa nd(subs(\{op(e3),op(d3)\},eqn_group));\n tt := map(_Z->`if`(lhs(_Z)= rhs(_Z),0,1),tt);\n if add(op(i,tt),i=1..nops(tt))=0 then nm := nm,c t end if;\nend do:\nnm;\nnops([%]);\nop(\{seq(i,i=1..64)\} minus \{nm \});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6R\"\"\"\"\"#\"\"$\"\"%\"\"&\"\" '\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\" #B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#O\"#R\"#V\"#Y\"#Z\"#\\\"#^\"# `\"#a\"#c\"#d\"#f\"#g\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"#L\"#M\"#N\"#P\"#Q\"#S\"#T\" #U\"#W\"#X\"#[\"#]\"#_\"#b\"#e\"#h" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate the \"weight\" polynomials at t he node " }{XPPEDIT 18 0 "c[16] = 14/125;" "6#/&%\"cG6#\"#;*&\"#9\"\" \"\"$D\"!\"\"" }{TEXT -1 83 " to obtain the linking coefficients in t he next stage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 135 "eqs16 := \{seq(a[16,j]=add(expand(subs(\{op(d3),c[ 16]=14/125\},d[j,i]*c[16]^i)),i=1..7),j=1..15)\}:\ne4 := `union`(eqs16 ,\{c[16]=14/125\},e3):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13000 "e4 := \{a[9,6] = 780125155843893641323090552530431 036567795592568497182701460674803126770111481625/183110425412731972197 889874507158786859226102980861859505241443073629143100805376, a[11,5] \+ = -26053085959256534152588089363841/4377552804565683061011299942400, a [11,8] = 161021426143124178389075121929246710833125/109972077221310346 50667041364346422894371443, a[10,7] = 31860723517364931240515126584966 0869927653414425413/6714716715558965303132938072935465423910912000000, a[8,5] = -1024030607959889/168929280000000, a[16,7] = 134784801417938 7864407984855535195535143443353399970936257813063151443875709744174470 25422991363808/1464237214848959077379888528329155065846449561947146728 294932984499575946659768049806174092837890625+373311550922081319948726 1578662228745283054587178425467773365237314452298352825323499384014102 1024/73211860742447953868994426416457753292322478097357336414746649224 97879733298840249030870464189453125*7^(1/2), a[4,1] = 1023/25600, a[2, 1] = 1/20, a[11,9] = 3007606697681025178342324975654524349466722661958 76496371874262392684852243925359864884962513/4655443337501346455585065 336604505603760824779615521285751892810315680492364106674524398280000, a[12,6] = 2346305388553404258656258473446184419154740172519949575/256 726716407895402892744978301151486254183185289662464, a[12,5] = -458349 3974484572912949314673356033540575/45195770365525074715731303427033513 5744, a[10,4] = -20462749524591049105403365239069/45425191349989346959 6231268750, a[13,5] = 0, a[7,1] = 21173/343200, a[7,2] = 0, a[7,3] = 0 , a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), a[8,2] = 0, a[8,3] = 0, a[13, 3] = 0, c[13] = 1, a[11,3] = 0, a[14,4] = 0, a[13,4] = 0, a[9,2] = 0, \+ a[14,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a [12,2] = 0, a[11,2] = 0, a[10,2] = 0, a[12,3] = 0, a[4,2] = 0, a[14,5] = 0, a[6,2] = 0, a[8,7] = 6070139212132283/92502016000000, a[8,6] = 1 501408353528689/265697280000000, a[6,3] = 0, a[9,3] = 0, c[6] = 93/200 , a[9,5] = -1207067925846925480797893644173318794948457151612046996653 4514296406891652614970375/27220311547616572217104781845311006994972840 85048389015085076961673446140398628096, a[5,2] = 0, a[13,2] = 0, a[12, 8] = 345685379554677052215495825476969226377187500/7477116743693007722 1667203179551347546362089, a[12,7] = 165712155931984680217128369091361 0698586256573484808662625/13431480411255146477259155104956093505361644 432088109056, c[5] = 39/100, a[14,12] = 11352128098668146659861/254668 911904014019468056-5215842639928607924801/127334455952007009734028*7^( 1/2), a[3,1] = -7161/1024000, a[6,4] = 31744/135025, a[7,4] = 8602624/ 76559175, c[7] = 31/200, c[8] = 943/1000, c[9] = 7067558016280/7837150 160667, c[11] = 47/50, a[6,5] = 923521/5106400, a[10,6] = 211276702141 72802870128286992003940810655221489/4679473877997892906145822697976708 633673728000, a[7,6] = 5611/283500, a[8,1] = -1221101821869329/6908129 28000000, a[13,9] = 12274765470313196878428812037740635050319234276006 986398294443554969616342274215316330684448207141/489345147493715517650 3858341435109348888292806866096544828965267965233530521667572994528521 66040, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100, a[8,4] = -125/2 , a[3,2] = 116281/1024000, a[12,4] = -16957088714171468676387054358954 754000/143690415119654683326368228101570221, c[10] = 909/1000, c[4] = \+ 1023/6400, a[5,3] = -3899844/2907025, a[7,5] = -26782109/689364000, a[ 13,8] = -13734512432397741476562500000/875132892924995907746928783, a[ 13,7] = 3847749490868980348119500000/15517045062138271618141237517, a[ 13,12] = -306814272936976936753/1299331183183744997286, a[10,5] = -180 269259803172281163724663224981097/38100922558256871086579832832000000, a[11,1] = -2342659845814086836951207140065609179073838476242943917/13 58480961351056777022231400139158760857532162795520000, a[12,11] = -612 2933601070769591613093993993358877250/10505170015102355131982467213020 27675953, a[11,7] = 89072299375637918641892962209583383526432263578229 4899/13921242001395112657501941955594013822830119803764736, c[2] = 1/2 0, a[13,11] = 282035543183190840068750/12295407629873040425991, a[13,1 0] = -9798363684577739445312500000/308722986341456031822630699, a[11,1 0] = -31155237437111730665923206875/392862141594230515010338956291, a[ 12,9] = -3205890962717072542791434312152727534008102774023210240571361 570757249056167015230160352087048674542196011/947569549683965814783015 1244512736049846577471272576153724492059731926573060172391034910747383 24033259120, a[11,4] = -996286030132538159613930889652/163530688859961 64905464325675, a[12,1] = -2866556991825663971778295329101033887534912 787724034363/868226711619262703011213925016143612030669233795338240, a [13,1] = 44901867737754616851973/1014046409980231013380680, a[10,8] = \+ 212083202434519082281842245535894/20022426044775672563822865371173879, a[12,10] = 40279545832706233433100438588458933210937500/8896460842799 482846916972126377338947215101, c[3] = 341/3200, a[13,6] = 79163867519 1615279648100000/2235604725089973126411512319, a[6,1] = 5611/114400, a [5,4] = 3982992/2907025, a[11,6] = 20980822345096760292224086794978105 312644533925634933539/377588999200755080387872783911549464197221296217 4156800, a[10,9] = -26984049294008425187211664850871297985622698482295 17793703413951226714583/4695456749139343150770004420808711418846760359 02717550325616728175875000000, a[9,8] = 103328481844520156040568367672 86656859124007796970668046446015775000000/1312703550036033648073834248 740727914537972028638950165249582733679393783, a[9,4] = -5172294311085 668458375175655246981230039025336933699114138315270772319372469280000/ 1246193810048091458972786305712152983652570794102362529218509367490764 87132995191, a[10,1] = -2905557336033741508853861844223103644131406051 1/22674759891089577691327962602370597632000000000, a[9,1] = -147251426 4486215803881384708877264246346044433307094207829051978044531801133057 155/124689480162003200115705962164398602480330155839348790044045363616 8046069686436608, a[9,7] = 6641131229599116421347821358391064699281403 28160577035357155340392950009492511875/1517846559858624813633302310729 5349175279765150089078301139943253016877823170816, a[16,1] = 308721907 7495121428802855737836440713071381964765974989609100353850622638124319 702720101198911189933/603660574218230035385964271207009410919947344407 67147196732195067996608532430377691588769378662109375-8916810789209489 7373289127357971512982845118526230285336871363105222357390366506181090 6114902/11436214345329734496276674646339100330016999988778468731028169 94752232803493992188909515380859375*7^(1/2), a[14,11] = 29702216660309 03358562614446315/4589064694536869684100726-22452774824529965468019194 07985/9178129389073739368201452*7^(1/2), a[14,10] = -20796683614312164 55673395233734375/5957776266668670748330679562+22458241974476061273627 6500828125/1702221790476763070951622732*7^(1/2), a[14,8] = -8789797928 960067489629400558015625/16279513475811535751534066502+664447066255418 4045392548540578125/32559026951623071503068133004*7^(1/2), a[14,7] = 1 274787529143761017157317260602825/26341935917422215554074218244182-958 929531471161732045165837664925/52683871834844431108148436488364*7^(1/2 ), a[14,1] = -942013/61397+2856761/491176*7^(1/2), a[14,6] = -10909561 77631910962162375596685/8605876253572154615648617314+40421151553845087 519011727532985/843375872850071152333564496772*7^(1/2), a[16,3] = 0, a [16,2] = 0, a[16,4] = 0, a[16,5] = 0, a[15,8] = 5989330035127864159899 29347488/28972691439853083012428515625-1475109654044019413537567146406 4/1371942153475395989706173828125*7^(1/2), a[15,2] = 0, a[15,3] = 0, a [15,4] = 0, a[15,5] = 0, a[14,9] = 99874705782485733872972546676594228 96963367899428512772582082944283962013964900384895127596133/4129561398 1107641868485821081000930261227361072745605305202526150545875206586329 586453004-377489797344744275480292572132150892961344729378723536598312 3527137197728406982036112477728851/41295613981107641868485821081000930 261227361072745605305202526150545875206586329586453004*7^(1/2), a[16,8 ] = -74855688242755874140837522994335294782960180652180921502875834466 293853598523210706487111328/422811107430209503460506370159185920828859 43200894919376755357202712416979579385828158093167+4080137020543686228 2074132593821266756139203379194180287147892059546794287535023023025984 32/1119827706347491057135420295573740149665383274029973921962543361797 4091284107600704550728125*7^(1/2), a[16,12] = -58820694887374956732123 7831706200551548379876180611807174719083297181931575539596530599369641 84/9578826588421893999864045794979276347406434759944469605071338585717 26061567913677743743896484375+5330785904517985582962774414128678356690 592330305476471554875994410016840663607810578383524/373443531712354541 9050310251453908907370929730972502770008319136731875483695569893737792 96875*7^(1/2), a[16,15] = 22029834681848048237734697285465827660777218 38920890760304290760481052156/3363158748474745533741147124297955168071 2577077922629054937124338769013375-10135203558823928958167900165592077 146024657433459350553499494488454213/538105399755959285398583539887672 826891401233246762064878993989420304214*7^(1/2), a[16,10] = -232080254 0679497658014154123603306276080911746562833352085780734430224778846220 269711385730784/186445265626902612878560634061788356019291963762267889 0647626052665702123772234282536764081375+58181585452998409536703173893 35817623809315384767986611466857849527752329675942295310238176/4665797 4381106759979619778293740829834657648589156128394585236553195748843149 006069488590625*7^(1/2), a[15,10] = 54365264025507653851977480738208/4 466478390356713423359819140625-1496294528311131416682934929792/2357869 43683989396406626953125*7^(1/2), a[16,11] = 14152639064598011149478440 582540271096903459559094245113706185285107457868609157190831336140072/ 5801166830186754554514051371059480088582444695362045034875735924602110 577185555412784755859375-230593473763469337865940944144309756620698757 23080237434871217181706005525496230157407214632/4670826755383860349850 2829074552979779246736677633212841189500198084626225326533114208984375 *7^(1/2), a[15,12] = -1927555453883917676542797699/2760543086400628250 12207031250, a[15,11] = -3598066799092816058284144620546/1462869641026 78984417724609375+65570869112872951387443794088672/5120043743593764454 620361328125*7^(1/2), a[16,6] = -8589122431024331358835216437287775024 8457698813390199974517211560164597559644901403704311739138208/10547934 9420598643167761368944195979795281065465375722665494015708354122239689 5919910489297177734375+19796836655578878224716426252377771094127511932 6218727417531422312697579876941533185813932673312/52792467177476798382 2629474195175074050455783109988601929399478019790401600048007963207856 4453125*7^(1/2), a[15,6] = 4116869726480787181292612488487584/26598534 836082808007568743896484375-245534888302358312351817429197184/87004553 20214002619298187255859375*7^(1/2), a[16,9] = 935791277672947918080274 9583759443090703763926683837055780644925837271306263170528758159831759 4137103657009660001416671641483325964108908321303708052246347959789983 8335379/15331924280868264731499337107758835984311473142649072676671181 6080640856626020245065130618903279240264626600690606026028579456343318 0425273922088634890450322720794677734375+23890713666546419820066957912 5422678954629879112926936374114216010517886935041784313250766764292054 989577049086765091486664548987756568897403447748777523160068219132/784 2416511953076589002218469441859838522492656086482187555591615378048932 2772503869632030129554598600832020813609220474976704011927387482042050 5695596138272491455078125*7^(1/2), a[16,14] = 502890395059180530649303 173309978152773646321070353812284039761435107016209192/254205473321003 76821161679478113172165786281774503728665992287528699005126953125-1405 8401811014285262962125828643550336789285498002679613358690084097917655 44/4067287573136060291385868716498107546525805083920596586558766004591 8408203125*7^(1/2), a[16,13] = -14781724992654556932356691694957549259 70922405664549846419638317907395087946/1754564777286668314899506906731 31923190567109284028983890324355658905029296875+5748782022340701550646 91566753530747289127904338643775445570927408156648849/3509129554573336 62979901381346263846381134218568057967780648711317810058593750*7^(1/2) , a[15,9] = -306706645966127099400049304854628613540749389913325084152 9092716875641340142781006771180215525814345889446383/37334071921822778 1410511653246697185431540894078529094301526280820101435128911405410349 160893280029296875000+181664761198407694024184205976438800959946932614 6254865545587094445401874686830164148895260773137/42447565466790579709 6751747915993683544031508184382874685996435033916010685878289836883544 921875*7^(1/2), a[15,13] = -15630718068/3814697265625+6167095848/38146 97265625*7^(1/2), c[16] = 14/125, a[15,7] = 63147661795969654782426038 5843267488/2705957914016858192337687904052734375-332274667313472904559 44055132270208/3922838396059942349838659979248046875*7^(1/2), c[15] = \+ 57/125, a[15,14] = 172678683744/3814697265625*7^(1/2), a[15,1] = 57196 651428918572875631366996149/1245822888041363915313720703125000+1089170 2347425878099023621466959/8357395207277482931896209716796875*7^(1/2)\} : " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[16,i]=subs(e4,a[16,i]) ,i=1..15):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "61/&%\"aG6$ \"#;\"\"\"$\"I')\\tfde/lY6ANQ**pLZ(y!\\!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F '\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"IkUh'pIFa9,%))>B _YzbXy@fJ'4u_4^tg6hFJ/&F%6 $F'\"#5$!If\">z*\\q]k_%pq+!oAWUV[\"*FJ/&F%6$F'\"#6$\"I*z^!ySgs\\\\.Y(* \\2&QmTM8\"!#R/&F%6$F'\"#7$!IN4')GNNPX\"[C'=CdP.[(RO#F+/&F%6$F'\"#8$!I @$H@(R(4gr34(=%)Rl&yf.4%!#U/&F%6$F'\"#9$!Ie,br(=/XNj6RI=r-rSm;(F+/&F%6 $F'\"#:$\"INem=X&3\"*ox18E9n$\\N2n:F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 24 "calculation for stage 17" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order c ondition gives rise to a group \{list) of equations to be satisfied b y the \"d\" coefficients of the weight polynomials for a given stage ( corresponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_ 16 := SimpleOrderConditions(7,16,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "whch := [1, 2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nordeqns4 := []:\nfor ct in \+ whch do\n eqn_group := convert(SO7_16[ct],'polynom_order_conditions' ,7):\n ordeqns4 := [op(ordeqns4),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for \+ all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 " eqns4 := []:\nfor ct to nops(ordeqns4) do\n eqns4 := [op(eqns4),expa nd(subs(e4,ordeqns4[ct]))];\nend do:\nnops(eqns4);\nnops(indets(eqns4) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d4 := solve(\{op(eqns4)\}):\ninfole vel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 66 "subs(d4,matrix([seq([seq(d[j,i],j=1..16)],i=1. .7)])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 )72$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+F+F+F+F+F+72$!'@-**!\"&F+F+F+F+$ !'Q*=\"!\"%$!'1r:F2$!']K>!\"$$\"'M'4#F7$!'$=(GF7$\"'TeFF7$!'6+UF/$!'&G _*!\"'$!'5oGF/$\"'fI9F2$\"''zh#F272$\"' " 0 "" {MPLTEXT 1 0 285 "nm := NULL :\nfor ct to nops(SO7_16) do\n eqn_group := convert(SO7_16[ct],'poly nom_order_conditions',7):\n tt := expand(subs(\{op(e4),op(d4)\},eqn_ group));\n tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(o p(i,tt),i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops( [%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"& \"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\" #A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R \"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c \"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Ev aluate the \"weight\" polynomials at the node " }{XPPEDIT 18 0 "c[17] = 373/1000;" "6#/&%\"cG6#\"#<*&\"$t$\"\"\"\"%+5!\"\"" }{TEXT -1 83 " \+ to obtain the linking coefficients in the next stage in the interpola tion scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "eqs17 := \{ seq(a[17,j]=add(expand(subs(\{op(d4),c[17]=373/1000\},d[j,i]*c[17]^i)) ,i=1..7),j=1..16)\}:\ne5 := `union`(eqs17,\{c[17]=373/1000\},e4):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15719 "e5 := \{a [9,6] = 78012515584389364132309055253043103656779559256849718270146067 4803126770111481625/18311042541273197219788987450715878685922610298086 1859505241443073629143100805376, a[11,5] = -26053085959256534152588089 363841/4377552804565683061011299942400, a[11,8] = 16102142614312417838 9075121929246710833125/10997207722131034650667041364346422894371443, a [10,7] = 318607235173649312405151265849660869927653414425413/671471671 5558965303132938072935465423910912000000, a[8,5] = -1024030607959889/1 68929280000000, a[16,7] = 13478480141793878644079848555351955351434433 5339997093625781306315144387570974417447025422991363808/14642372148489 5907737988852832915506584644956194714672829493298449957594665976804980 6174092837890625+37331155092208131994872615786622287452830545871784254 677733652373144522983528253234993840141021024/732118607424479538689944 2641645775329232247809735733641474664922497879733298840249030870464189 453125*7^(1/2), a[4,1] = 1023/25600, a[2,1] = 1/20, a[11,9] = 30076066 9768102517834232497565452434946672266195876496371874262392684852243925 359864884962513/465544333750134645558506533660450560376082477961552128 5751892810315680492364106674524398280000, a[12,6] = 234630538855340425 8656258473446184419154740172519949575/25672671640789540289274497830115 1486254183185289662464, a[12,5] = -45834939744845729129493146733560335 40575/451957703655250747157313034270335135744, a[10,4] = -204627495245 91049105403365239069/454251913499893469596231268750, a[13,5] = 0, a[7, 1] = 21173/343200, a[7,2] = 0, a[7,3] = 0, a[14,3] = 0, c[14] = 1/2-1/ 14*7^(1/2), a[8,2] = 0, a[8,3] = 0, a[13,3] = 0, c[13] = 1, a[11,3] = \+ 0, a[14,4] = 0, a[13,4] = 0, a[9,2] = 0, a[14,2] = 0, c[12] = 1, a[10, 3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12,2] = 0, a[11,2] = 0, a[10 ,2] = 0, a[12,3] = 0, a[4,2] = 0, a[14,5] = 0, a[6,2] = 0, a[8,7] = 60 70139212132283/92502016000000, a[8,6] = 1501408353528689/2656972800000 00, a[6,3] = 0, a[9,3] = 0, c[6] = 93/200, a[9,5] = -12070679258469254 807978936441733187949484571516120469966534514296406891652614970375/272 2031154761657221710478184531100699497284085048389015085076961673446140 398628096, a[5,2] = 0, a[13,2] = 0, a[12,8] = 345685379554677052215495 825476969226377187500/74771167436930077221667203179551347546362089, a[ 12,7] = 1657121559319846802171283690913610698586256573484808662625/134 31480411255146477259155104956093505361644432088109056, c[5] = 39/100, \+ a[14,12] = 11352128098668146659861/254668911904014019468056-5215842639 928607924801/127334455952007009734028*7^(1/2), a[3,1] = -7161/1024000, a[6,4] = 31744/135025, a[7,4] = 8602624/76559175, c[7] = 31/200, c[8] = 943/1000, c[9] = 7067558016280/7837150160667, c[11] = 47/50, a[6,5] = 923521/5106400, a[10,6] = 21127670214172802870128286992003940810655 221489/4679473877997892906145822697976708633673728000, a[7,6] = 5611/2 83500, a[8,1] = -1221101821869329/690812928000000, a[13,9] = 122747654 7031319687842881203774063505031923427600698639829444355496961634227421 5316330684448207141/48934514749371551765038583414351093488882928068660 9654482896526796523353052166757299452852166040, a[4,3] = 3069/25600, a [5,1] = 4202367/11628100, a[8,4] = -125/2, a[3,2] = 116281/1024000, a[ 12,4] = -16957088714171468676387054358954754000/1436904151196546833263 68228101570221, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = -3899844/ 2907025, a[7,5] = -26782109/689364000, a[13,8] = -13734512432397741476 562500000/875132892924995907746928783, a[13,7] = 384774949086898034811 9500000/15517045062138271618141237517, a[13,12] = -3068142729369769367 53/1299331183183744997286, a[10,5] = -18026925980317228116372466322498 1097/38100922558256871086579832832000000, a[11,1] = -23426598458140868 36951207140065609179073838476242943917/1358480961351056777022231400139 158760857532162795520000, a[12,11] = -61229336010707695916130939939933 58877250/1050517001510235513198246721302027675953, a[11,7] = 890722993 756379186418929622095833835264322635782294899/139212420013951126575019 41955594013822830119803764736, c[2] = 1/20, a[13,11] = 282035543183190 840068750/12295407629873040425991, a[13,10] = -97983636845777394453125 00000/308722986341456031822630699, a[11,10] = -31155237437111730665923 206875/392862141594230515010338956291, a[12,9] = -32058909627170725427 9143431215272753400810277402321024057136157075724905616701523016035208 7048674542196011/94756954968396581478301512445127360498465774712725761 5372449205973192657306017239103491074738324033259120, a[11,4] = -99628 6030132538159613930889652/16353068885996164905464325675, a[12,1] = -28 66556991825663971778295329101033887534912787724034363/8682267116192627 03011213925016143612030669233795338240, a[13,1] = 44901867737754616851 973/1014046409980231013380680, a[10,8] = 21208320243451908228184224553 5894/20022426044775672563822865371173879, a[12,10] = 40279545832706233 433100438588458933210937500/889646084279948284691697212637733894721510 1, c[3] = 341/3200, a[13,6] = 791638675191615279648100000/223560472508 9973126411512319, a[6,1] = 5611/114400, a[5,4] = 3982992/2907025, a[11 ,6] = 20980822345096760292224086794978105312644533925634933539/3775889 992007550803878727839115494641972212962174156800, a[10,9] = -269840492 9400842518721166485087129798562269848229517793703413951226714583/46954 5674913934315077000442080871141884676035902717550325616728175875000000 , a[9,8] = 10332848184452015604056836767286656859124007796970668046446 015775000000/131270355003603364807383424874072791453797202863895016524 9582733679393783, a[9,4] = -517229431108566845837517565524698123003902 5336933699114138315270772319372469280000/12461938100480914589727863057 1215298365257079410236252921850936749076487132995191, a[10,1] = -29055 573360337415088538618442231036441314060511/226747598910895776913279626 02370597632000000000, a[9,1] = -14725142644862158038813847088772642463 46044433307094207829051978044531801133057155/1246894801620032001157059 621643986024803301558393487900440453636168046069686436608, a[9,7] = 66 4113122959911642134782135839106469928140328160577035357155340392950009 492511875/151784655985862481363330231072953491752797651500890783011399 43253016877823170816, a[16,1] = 30872190774951214288028557378364407130 71381964765974989609100353850622638124319702720101198911189933/6036605 7421823003538596427120700941091994734440767147196732195067996608532430 377691588769378662109375-891681078920948973732891273579715129828451185 262302853368713631052223573903665061810906114902/114362143453297344962 7667464633910033001699998877846873102816994752232803493992188909515380 859375*7^(1/2), a[14,11] = 2970221666030903358562614446315/45890646945 36869684100726-2245277482452996546801919407985/91781293890737393682014 52*7^(1/2), a[14,10] = -2079668361431216455673395233734375/59577762666 68670748330679562+224582419744760612736276500828125/170222179047676307 0951622732*7^(1/2), a[14,8] = -8789797928960067489629400558015625/1627 9513475811535751534066502+6644470662554184045392548540578125/325590269 51623071503068133004*7^(1/2), a[14,7] = 127478752914376101715731726060 2825/26341935917422215554074218244182-95892953147116173204516583766492 5/52683871834844431108148436488364*7^(1/2), a[14,1] = -942013/61397+28 56761/491176*7^(1/2), a[14,6] = -1090956177631910962162375596685/86058 76253572154615648617314+40421151553845087519011727532985/8433758728500 71152333564496772*7^(1/2), a[16,3] = 0, a[16,2] = 0, a[16,4] = 0, a[16 ,5] = 0, a[15,8] = 598933003512786415989929347488/28972691439853083012 428515625-14751096540440194135375671464064/137194215347539598970617382 8125*7^(1/2), a[15,2] = 0, a[15,3] = 0, a[15,4] = 0, a[15,5] = 0, a[17 ,15] = -1927193513548850260413881071/23077873387050840510431232000+835 4781001061139440464823/369245974192813448166899712*7^(1/2), a[14,9] = \+ 9987470578248573387297254667659422896963367899428512772582082944283962 013964900384895127596133/412956139811076418684858210810009302612273610 72745605305202526150545875206586329586453004-3774897973447442754802925 721321508929613447293787235365983123527137197728406982036112477728851/ 4129561398110764186848582108100093026122736107274560530520252615054587 5206586329586453004*7^(1/2), a[17,16] = 61561095821428811034532925699/ 1195262539043064700790636544000+197966374530935184348002389/4781050156 172258803162546176*7^(1/2), a[17,1] = 83288558830021174184836229037264 20956080895292363251/1983315923723223823165391130206468227200000000000 00000-4772929606928922621289088245796233257786356432779/14691229064616 47276418808244597383872000000000000000*7^(1/2), a[16,8] = -74855688242 7558741408375229943352947829601806521809215028758344662938535985232107 06487111328/4228111074302095034605063701591859208288594320089491937675 5357202712416979579385828158093167+40801370205436862282074132593821266 75613920337919418028714789205954679428753502302302598432/1119827706347 4910571354202955737401496653832740299739219625433617974091284107600704 550728125*7^(1/2), a[16,12] = -588206948873749567321237831706200551548 37987618061180717471908329718193157553959653059936964184/9578826588421 8939998640457949792763474064347599444696050713385857172606156791367774 3743896484375+53307859045179855829627744141286783566905923303054764715 54875994410016840663607810578383524/3734435317123545419050310251453908 90737092973097250277000831913673187548369556989373779296875*7^(1/2), a [17,10] = 56160702543048579126936480152903088846798498167/123661065966 112908213890615348396782899806208000-704890678912714132819876715191341 334116124219/16357283857951442885435266580475764933836800000*7^(1/2), \+ a[17,14] = 106657925843659476662976498724178969/2688984574690119078864 000000000000000+67590823874770618986810696149161/143412510650139684206 0800000000000*7^(1/2), a[16,15] = 220298346818480482377346972854658276 6077721838920890760304290760481052156/33631587484747455337411471242979 551680712577077922629054937124338769013375-101352035588239289581679001 65592077146024657433459350553499494488454213/5381053997559592853985835 39887672826891401233246762064878993989420304214*7^(1/2), a[16,10] = -2 3208025406794976580141541236033062760809117465628333520857807344302247 78846220269711385730784/1864452656269026128785606340617883560192919637 622678890647626052665702123772234282536764081375+581815854529984095367 0317389335817623809315384767986611466857849527752329675942295310238176 /466579743811067599796197782937408298346576485891561283945852365531957 48843149006069488590625*7^(1/2), a[15,10] = 54365264025507653851977480 738208/4466478390356713423359819140625-1496294528311131416682934929792 /235786943683989396406626953125*7^(1/2), a[17,6] = 9608501774836052499 507564485849861825840507225298757/699598820217037388471203324684932177 00284160000000000-31215118821227947550703658120770299949941405900809/6 81870195143311294806241057197789646201600000000000*7^(1/2), c[17] = 37 3/1000, a[16,11] = 141526390645980111494784405825402710969034595590942 45113706185285107457868609157190831336140072/5801166830186754554514051 371059480088582444695362045034875735924602110577185555412784755859375- 2305934737634693378659409441443097566206987572308023743487121718170600 5525496230157407214632/46708267553838603498502829074552979779246736677 633212841189500198084626225326533114208984375*7^(1/2), a[15,12] = -192 7555453883917676542797699/276054308640062825012207031250, a[17,13] = 5 929009119114323566893394932293/1968360743640880256000000000000000-2816 129290207834445979834640567/3936721487281760512000000000000000*7^(1/2) , a[15,11] = -3598066799092816058284144620546/146286964102678984417724 609375+65570869112872951387443794088672/512004374359376445462036132812 5*7^(1/2), a[17,7] = 1025633351096621944615574068216622797246156111054 69/618576496058236960959302704321559561229568000000000-265492203346789 44450486694876789442771916528311224941/4855825494057160143530526228924 24255565210880000000000*7^(1/2), a[17,9] = -37400732889192332684832452 8689796601085558570798816077884060194802159144857724872873387178283663 57415478453578758816356450881/1914164901466188608010876856268576598392 4233982965078906908937587726438607819575170347827948246797820425712832 0000000000000000-33177519731408172000388101825291614226145544535634954 162616522934065875624456931916785200056606590002524998996767787117019/ 1740149910423807825464433505698705998538566725724098082446267053429676 237074506833667984358931527074584155712000000000000000*7^(1/2), a[16,6 ] = -85891224310243313588352164372877750248457698813390199974517211560 164597559644901403704311739138208/105479349420598643167761368944195979 7952810654653757226654940157083541222396895919910489297177734375+19796 8366555788782247164262523777710941275119326218727417531422312697579876 941533185813932673312/527924671774767983822629474195175074050455783109 9886019293994780197904016000480079632078564453125*7^(1/2), a[17,12] = \+ 2902460704378612529880216802791069594834177328387/12706441245863181188 8314056657262072000000000000000-17095105338496523291024743569399504075 413989/2573391305711385775313314734457600000000000000*7^(1/2), a[17,11 ] = -9531620235114508657502072531656361570671453261/980296201865448172 2288970934359649280000000000+36056630455459788103338323194055323527901 56659/15000633888194479828453883398581529600000000000*7^(1/2), a[15,6] = 4116869726480787181292612488487584/26598534836082808007568743896484 375-245534888302358312351817429197184/87004553202140026192981872558593 75*7^(1/2), a[16,9] = 935791277672947918080274958375944309070376392668 3837055780644925837271306263170528758159831759413710365700966000141667 16414833259641089083213037080522463479597899838335379/1533192428086826 4731499337107758835984311473142649072676671181608064085662602024506513 0618903279240264626600690606026028579456343318042527392208863489045032 2720794677734375+23890713666546419820066957912542267895462987911292693 6374114216010517886935041784313250766764292054989577049086765091486664 548987756568897403447748777523160068219132/784241651195307658900221846 9441859838522492656086482187555591615378048932277250386963203012955459 8600832020813609220474976704011927387482042050569559613827249145507812 5*7^(1/2), a[17,8] = 248147914310376513887773097942267349975604062017/ 350540358797316839315026245887139178767630336000-304605634346256759097 7832367296624503866287449/17082863489147994118665996388262143214796800 000*7^(1/2), a[16,14] = 5028903950591805306493031733099781527736463210 70353812284039761435107016209192/2542054733210037682116167947811317216 5786281774503728665992287528699005126953125-14058401811014285262962125 82864355033678928549800267961335869008409791765544/4067287573136060291 3858687164981075465258050839205965865587660045918408203125*7^(1/2), a[ 16,13] = -147817249926545569323566916949575492597092240566454984641963 8317907395087946/17545647772866683148995069067313192319056710928402898 3890324355658905029296875+57487820223407015506469156675353074728912790 4338643775445570927408156648849/35091295545733366297990138134626384638 1134218568057967780648711317810058593750*7^(1/2), a[15,9] = -306706645 9661270994000493048546286135407493899133250841529092716875641340142781 006771180215525814345889446383/373340719218227781410511653246697185431 540894078529094301526280820101435128911405410349160893280029296875000+ 1816647611984076940241842059764388009599469326146254865545587094445401 874686830164148895260773137/424475654667905797096751747915993683544031 508184382874685996435033916010685878289836883544921875*7^(1/2), a[17,2 ] = 0, a[17,3] = 0, a[17,4] = 0, a[17,5] = 0, a[15,13] = -15630718068/ 3814697265625+6167095848/3814697265625*7^(1/2), c[16] = 14/125, a[15,7 ] = 631476617959696547824260385843267488/27059579140168581923376879040 52734375-33227466731347290455944055132270208/3922838396059942349838659 979248046875*7^(1/2), c[15] = 57/125, a[15,14] = 172678683744/38146972 65625*7^(1/2), a[15,1] = 57196651428918572875631366996149/124582288804 1363915313720703125000+10891702347425878099023621466959/83573952072774 82931896209716796875*7^(1/2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "se q(a[17,i]=subs(e5,a[17,i]),i=1..16):\nevalf[40](%);" }}{PARA 12 "" 1 " " {XPPMATH 20 "62/&%\"aG6$\"#<\"\"\"$\"I2i!H\\BVNcfm%3a)*fZ0!*RL!#T/&F %6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F '\"\"'$\"H'[o?L:%oa'fnd?6YQ;SA;!#S/&F%6$F'\"\"($\"H!)*)pb.xYZO[zx$**\\ K'**[6#FD/&F%6$F'\"\")$\"I#[ED7PR#\\X&ewJl4(>^NhBFD/&F%6$F'\"\"*$!I$R# [6)z%*Rez0XH`SE#GH3!)zRmH@uQcme8S$FD/&F%6$F' \"#6$!I&GlXw-qDl1Q55&=4=7ojLFD/&F%6$F'\"#7$\"H-D;gF$))>xw#e\\xS\\ATmE& F+/&F%6$F'\"#8$\"I4(4;2wS`y2(GMf!fu^?&>6!#U/&F%6$F'\"#9$\"IsQgya[u.H9@ [+7@]&*fV;FD/&F%6$F'\"#:$!Iuw9&pke-ni)HXof\"HJ#RkBF+/&F%6$F'\"#;$\"IBg W)3vR!R7r_U&H%=([a0h\"FD" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 18" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order conditi on gives rise to a group \{list) of equations to be satisfied by the \+ \"d\" coefficients of the weight polynomials for a given stage (corres ponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_17 := \+ SimpleOrderConditions(7,17,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "whch := [1,2,3,6,7,8 ,12,15,16,21,27,31,32,33,36,64]:\nordeqns5 := []:\nfor ct in whch do\n eqn_group := convert(SO7_17[ct],'polynom_order_conditions',7):\n \+ ordeqns5 := [op(ordeqns5),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all know n coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns5 := []:\nfor ct to nops(ordeqns5) do\n eqns5 := [op(eqns5),expand(subs( e5,ordeqns5[ct]))];\nend do:\nnops(eqns5);\nnops(indets(eqns5));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$>\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Solve the system of equations." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "d5 := solve(\{op(eqns5)\},indets(eqns5) minus \{seq(d [1,i],i=1..7)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "W e would like to ensure that " }{XPPEDIT 18 0 "a[18,17] = 0;" "6#/&%\" aG6$\"#=\"#<\"\"!" }{TEXT -1 29 " as in the published scheme. " }} {PARA 0 "" 0 "" {TEXT -1 22 "We use the fact that " }{XPPEDIT 18 0 "a [18,17] = Sum(d[17,i]*c[18]^i,i = 1 .. 7);" "6#/&%\"aG6$\"#=\"#<-%$Sum G6$*&&%\"dG6$F(%\"iG\"\"\")&%\"cG6#F'F0F1/F0;F1\"\"(" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[18] = 9/20 ;" "6#/&%\"cG6#\"#=*&\"\"*\"\"\"\"#?!\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "c_18 \+ := 9/20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c_18G#\"\"*\"#?" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "eq := add(subs(\{op(d5),c[1 8]=c_18\},d[17,i]*c[18]^i),i=1..7)=0:\ndd := \{d[1,1]=1,seq(d[1,i]=0,i =2..6)\}:\nsol := \{d[1,7]=expand(rationalize(solve(subs(dd,eq))))\}: \nsol;\ndd_5 := `union`(subs(sol,dd),sol):\nd_5 := `union`(subs(dd_5,d 5),dd_5):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/&%\"dG6$\"\"\"\"\"(,&# \"Ll#Gx\"Q)zNpTu=BN\"Q'\\r')p2%)\"JRXNQvPJ!H?u`Xn;t;J>Ex!\"\"*(\"JvidV '*G1ri&p\"e#)[!emX>7#)F(F-F.F)#F(\"\"#F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "subs(d_5,matrix([seq( [seq(d[j,i],j=1..16)],i=1..7)])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)72$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+F+F+ F+F+F+72F+F+F+F+F+$\"':QS!\"$$\"'3W_F/$\"'9$o&!\"#$!'+NeF4$\"',,\")F4$ !'-!4)F4$\"':y7F/$\"'-%o#!\"%$!'u!G\"F4$!'^u(*F/$!'48'F4$\"'W#f%F4$\"'1c7F472F+F+F+F+F+$\"'y?YF4$\"'S*)fF4$\"'g#R'F M$!'f:lFM$\"'Si!*FM$!'(o4*FM$\"'(RW\"F4$\"')4+$F/$!'pz:FM$!'Mp6FM$!'%R 1$F472F+F+F+F+F+$!'(oJ'F4$!'(e<)F4$!'!Gi)FM$\"'BP()FM$!'-<7F*$\"'kE7F* $!'Ra>F4$!'$f-%F/$\"'%4A#FM$\"'ZO;FM$\"'3MTF472F+F+F+F+F+$\"'zDWF4$\"' nq6FM$!'(*oGF472$!'L;6F/F+F+F+F+$!'c@fF4$!'E\"o(F4$!'R\\#)FM$\" 'EL%)FM$!'4s6F*$\"'3u6F*$!'1g=F4$!'KwQF/$\"'a!*=FM$\"'.]9FM$\"'rlUF4Q( pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions are satisf ied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "nm := NULL:\nfor ct to nops(SO7_17) do\n eqn_group := convert(SO7_17[ct],'polynom_order _conditions',7):\n tt := expand(subs(\{op(e5),op(d_5)\},eqn_group)); \n tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op(i,tt), i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%]);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\" (\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"# C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T \"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e \"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate \+ the \"weight\" polynomials at the node " }{XPPEDIT 18 0 "c[18] = 9/20 ;" "6#/&%\"cG6#\"#=*&\"\"*\"\"\"\"#?!\"\"" }{TEXT -1 83 " to obtain t he linking coefficients in the next stage in the interpolation scheme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "eqs18 := \{seq(a[18,j]= add(expand(subs(\{op(d_5),c[18]=c_18\},d[j,i]*c[18]^i)),i=1..7),j=1..1 7)\}:\ne6 := `union`(eqs18,\{c[18]=c_18\},e5):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e6 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18072 "e6 := \{a[9,6] = 78012515 5843893641323090552530431036567795592568497182701460674803126770111481 625/183110425412731972197889874507158786859226102980861859505241443073 629143100805376, a[11,5] = -26053085959256534152588089363841/437755280 4565683061011299942400, a[11,8] = 161021426143124178389075121929246710 833125/10997207722131034650667041364346422894371443, a[18,17] = 0, a[1 0,7] = 318607235173649312405151265849660869927653414425413/67147167155 58965303132938072935465423910912000000, a[8,5] = -1024030607959889/168 929280000000, a[18,5] = 0, a[16,7] = 134784801417938786440798485553519 553514344335339997093625781306315144387570974417447025422991363808/146 4237214848959077379888528329155065846449561947146728294932984499575946 659768049806174092837890625+373311550922081319948726157866222874528305 45871784254677733652373144522983528253234993840141021024/7321186074244 7953868994426416457753292322478097357336414746649224978797332988402490 30870464189453125*7^(1/2), a[4,1] = 1023/25600, a[2,1] = 1/20, a[11,9] = 3007606697681025178342324975654524349466722661958764963718742623926 84852243925359864884962513/4655443337501346455585065336604505603760824 779615521285751892810315680492364106674524398280000, a[12,6] = 2346305 388553404258656258473446184419154740172519949575/256726716407895402892 744978301151486254183185289662464, a[12,5] = -458349397448457291294931 4673356033540575/451957703655250747157313034270335135744, a[10,4] = -2 0462749524591049105403365239069/454251913499893469596231268750, a[13,5 ] = 0, a[7,1] = 21173/343200, a[7,2] = 0, a[7,3] = 0, a[14,3] = 0, c[1 4] = 1/2-1/14*7^(1/2), a[8,2] = 0, a[8,3] = 0, a[13,3] = 0, c[13] = 1, a[11,3] = 0, a[14,4] = 0, a[13,4] = 0, a[9,2] = 0, a[14,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12,2] = 0, a[11,2 ] = 0, a[10,2] = 0, a[12,3] = 0, a[4,2] = 0, a[14,5] = 0, a[6,2] = 0, \+ a[8,7] = 6070139212132283/92502016000000, a[8,6] = 1501408353528689/26 5697280000000, a[6,3] = 0, a[9,3] = 0, c[6] = 93/200, a[9,5] = -120706 7925846925480797893644173318794948457151612046996653451429640689165261 4970375/27220311547616572217104781845311006994972840850483890150850769 61673446140398628096, a[5,2] = 0, a[13,2] = 0, a[12,8] = 3456853795546 77052215495825476969226377187500/7477116743693007722166720317955134754 6362089, a[12,7] = 165712155931984680217128369091361069858625657348480 8662625/13431480411255146477259155104956093505361644432088109056, c[5] = 39/100, a[14,12] = 11352128098668146659861/254668911904014019468056 -5215842639928607924801/127334455952007009734028*7^(1/2), a[3,1] = -71 61/1024000, a[6,4] = 31744/135025, a[7,4] = 8602624/76559175, c[7] = 3 1/200, c[8] = 943/1000, c[9] = 7067558016280/7837150160667, c[11] = 47 /50, a[6,5] = 923521/5106400, a[10,6] = 211276702141728028701282869920 03940810655221489/4679473877997892906145822697976708633673728000, a[7, 6] = 5611/283500, a[8,1] = -1221101821869329/690812928000000, a[13,9] \+ = 12274765470313196878428812037740635050319234276006986398294443554969 616342274215316330684448207141/489345147493715517650385834143510934888 829280686609654482896526796523353052166757299452852166040, a[4,3] = 30 69/25600, a[5,1] = 4202367/11628100, a[8,4] = -125/2, a[3,2] = 116281/ 1024000, a[12,4] = -16957088714171468676387054358954754000/14369041511 9654683326368228101570221, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] \+ = -3899844/2907025, a[7,5] = -26782109/689364000, a[13,8] = -137345124 32397741476562500000/875132892924995907746928783, a[13,7] = 3847749490 868980348119500000/15517045062138271618141237517, a[13,12] = -30681427 2936976936753/1299331183183744997286, a[10,5] = -180269259803172281163 724663224981097/38100922558256871086579832832000000, a[11,1] = -234265 9845814086836951207140065609179073838476242943917/13584809613510567770 22231400139158760857532162795520000, a[12,11] = -612293360107076959161 3093993993358877250/1050517001510235513198246721302027675953, a[11,7] \+ = 890722993756379186418929622095833835264322635782294899/1392124200139 5112657501941955594013822830119803764736, c[2] = 1/20, a[13,11] = 2820 35543183190840068750/12295407629873040425991, a[13,10] = -979836368457 7739445312500000/308722986341456031822630699, a[11,10] = -311552374371 11730665923206875/392862141594230515010338956291, a[12,9] = -320589096 2717072542791434312152727534008102774023210240571361570757249056167015 230160352087048674542196011/947569549683965814783015124451273604984657 747127257615372449205973192657306017239103491074738324033259120, a[11, 4] = -996286030132538159613930889652/16353068885996164905464325675, a[ 12,1] = -2866556991825663971778295329101033887534912787724034363/86822 6711619262703011213925016143612030669233795338240, a[13,1] = 449018677 37754616851973/1014046409980231013380680, a[10,8] = 212083202434519082 281842245535894/20022426044775672563822865371173879, a[12,10] = 402795 45832706233433100438588458933210937500/8896460842799482846916972126377 338947215101, c[3] = 341/3200, a[13,6] = 791638675191615279648100000/2 235604725089973126411512319, a[6,1] = 5611/114400, a[5,4] = 3982992/29 07025, a[11,6] = 20980822345096760292224086794978105312644533925634933 539/3775889992007550803878727839115494641972212962174156800, a[10,9] = -26984049294008425187211664850871297985622698482295177937034139512267 14583/4695456749139343150770004420808711418846760359027175503256167281 75875000000, a[9,8] = 103328481844520156040568367672866568591240077969 70668046446015775000000/1312703550036033648073834248740727914537972028 638950165249582733679393783, a[9,4] = -5172294311085668458375175655246 981230039025336933699114138315270772319372469280000/124619381004809145 897278630571215298365257079410236252921850936749076487132995191, a[10, 1] = -29055573360337415088538618442231036441314060511/2267475989108957 7691327962602370597632000000000, a[9,1] = -147251426448621580388138470 8877264246346044433307094207829051978044531801133057155/12468948016200 32001157059621643986024803301558393487900440453636168046069686436608, \+ a[9,7] = 6641131229599116421347821358391064699281403281605770353571553 40392950009492511875/1517846559858624813633302310729534917527976515008 9078301139943253016877823170816, a[16,1] = 308721907749512142880285573 7836440713071381964765974989609100353850622638124319702720101198911189 933/603660574218230035385964271207009410919947344407671471967321950679 96608532430377691588769378662109375-8916810789209489737328912735797151 29828451185262302853368713631052223573903665061810906114902/1143621434 5329734496276674646339100330016999988778468731028169947522328034939921 88909515380859375*7^(1/2), a[14,11] = 2970221666030903358562614446315/ 4589064694536869684100726-2245277482452996546801919407985/917812938907 3739368201452*7^(1/2), a[14,10] = -2079668361431216455673395233734375/ 5957776266668670748330679562+224582419744760612736276500828125/1702221 790476763070951622732*7^(1/2), a[14,8] = -8789797928960067489629400558 015625/16279513475811535751534066502+664447066255418404539254854057812 5/32559026951623071503068133004*7^(1/2), a[14,7] = 1274787529143761017 157317260602825/26341935917422215554074218244182-958929531471161732045 165837664925/52683871834844431108148436488364*7^(1/2), a[14,1] = -9420 13/61397+2856761/491176*7^(1/2), a[14,6] = -10909561776319109621623755 96685/8605876253572154615648617314+40421151553845087519011727532985/84 3375872850071152333564496772*7^(1/2), a[16,3] = 0, a[16,2] = 0, a[16,4 ] = 0, a[16,5] = 0, a[15,8] = 598933003512786415989929347488/289726914 39853083012428515625-14751096540440194135375671464064/1371942153475395 989706173828125*7^(1/2), a[15,2] = 0, a[15,3] = 0, a[15,4] = 0, a[15,5 ] = 0, a[17,15] = -1927193513548850260413881071/2307787338705084051043 1232000+8354781001061139440464823/369245974192813448166899712*7^(1/2), a[14,9] = 99874705782485733872972546676594228969633678994285127725820 82944283962013964900384895127596133/4129561398110764186848582108100093 0261227361072745605305202526150545875206586329586453004-37748979734474 4275480292572132150892961344729378723536598312352713719772840698203611 2477728851/41295613981107641868485821081000930261227361072745605305202 526150545875206586329586453004*7^(1/2), a[17,16] = 6156109582142881103 4532925699/1195262539043064700790636544000+197966374530935184348002389 /4781050156172258803162546176*7^(1/2), a[17,1] = 832885588300211741848 3622903726420956080895292363251/19833159237232238231653911302064682272 0000000000000000-4772929606928922621289088245796233257786356432779/146 9122906461647276418808244597383872000000000000000*7^(1/2), a[16,8] = - 7485568824275587414083752299433529478296018065218092150287583446629385 3598523210706487111328/42281110743020950346050637015918592082885943200 894919376755357202712416979579385828158093167+408013702054368622820741 3259382126675613920337919418028714789205954679428753502302302598432/11 1982770634749105713542029557374014966538327402997392196254336179740912 84107600704550728125*7^(1/2), a[16,12] = -5882069488737495673212378317 0620055154837987618061180717471908329718193157553959653059936964184/95 7882658842189399986404579497927634740643475994446960507133858571726061 567913677743743896484375+533078590451798558296277441412867835669059233 0305476471554875994410016840663607810578383524/37344353171235454190503 1025145390890737092973097250277000831913673187548369556989373779296875 *7^(1/2), a[17,10] = 56160702543048579126936480152903088846798498167/1 23661065966112908213890615348396782899806208000-7048906789127141328198 76715191341334116124219/1635728385795144288543526658047576493383680000 0*7^(1/2), a[17,14] = 106657925843659476662976498724178969/26889845746 90119078864000000000000000+67590823874770618986810696149161/1434125106 501396842060800000000000*7^(1/2), a[16,15] = 2202983468184804823773469 728546582766077721838920890760304290760481052156/336315874847474553374 11471242979551680712577077922629054937124338769013375-1013520355882392 8958167900165592077146024657433459350553499494488454213/53810539975595 9285398583539887672826891401233246762064878993989420304214*7^(1/2), a[ 16,10] = -232080254067949765801415412360330627608091174656283335208578 0734430224778846220269711385730784/18644526562690261287856063406178835 60192919637622678890647626052665702123772234282536764081375+5818158545 2998409536703173893358176238093153847679866114668578495277523296759422 95310238176/4665797438110675997961977829374082983465764858915612839458 5236553195748843149006069488590625*7^(1/2), a[15,10] = 543652640255076 53851977480738208/4466478390356713423359819140625-14962945283111314166 82934929792/235786943683989396406626953125*7^(1/2), a[17,6] = 96085017 74836052499507564485849861825840507225298757/6995988202170373884712033 2468493217700284160000000000-31215118821227947550703658120770299949941 405900809/681870195143311294806241057197789646201600000000000*7^(1/2), c[17] = 373/1000, a[16,11] = 1415263906459801114947844058254027109690 3459559094245113706185285107457868609157190831336140072/58011668301867 5455451405137105948008858244469536204503487573592460211057718555541278 4755859375-23059347376346933786594094414430975662069875723080237434871 217181706005525496230157407214632/467082675538386034985028290745529797 79246736677633212841189500198084626225326533114208984375*7^(1/2), a[18 ,11] = -35975392605857817344162998122286881735/45601927020111218678499 805531687849984+67001279346027425501202673610603822817/228009635100556 093392499027658439249920*7^(1/2), a[15,12] = -192755545388391767654279 7699/276054308640062825012207031250, a[17,13] = 5929009119114323566893 394932293/1968360743640880256000000000000000-2816129290207834445979834 640567/3936721487281760512000000000000000*7^(1/2), a[15,11] = -3598066 799092816058284144620546/146286964102678984417724609375+65570869112872 951387443794088672/5120043743593764454620361328125*7^(1/2), a[18,2] = \+ 0, a[18,3] = 0, a[17,7] = 10256333510966219446155740682166227972461561 1105469/618576496058236960959302704321559561229568000000000-2654922033 4678944450486694876789442771916528311224941/48558254940571601435305262 2892424255565210880000000000*7^(1/2), a[17,9] = -374007328891923326848 3245286897966010855585707988160778840601948021591448577248728733871782 8366357415478453578758816356450881/19141649014661886080108768562685765 9839242339829650789069089375877264386078195751703478279482467978204257 128320000000000000000-331775197314081720003881018252916142261455445356 3495416261652293406587562445693191678520005660659000252499899676778711 7019/17401499104238078254644335056987059985385667257240980824462670534 29676237074506833667984358931527074584155712000000000000000*7^(1/2), a [16,6] = -858912243102433135883521643728777502484576988133901999745172 11560164597559644901403704311739138208/1054793494205986431677613689441 959797952810654653757226654940157083541222396895919910489297177734375+ 1979683665557887822471642625237777109412751193262187274175314223126975 79876941533185813932673312/5279246717747679838226294741951750740504557 831099886019293994780197904016000480079632078564453125*7^(1/2), a[17,1 2] = 2902460704378612529880216802791069594834177328387/127064412458631 811888314056657262072000000000000000-170951053384965232910247435693995 04075413989/2573391305711385775313314734457600000000000000*7^(1/2), a[ 17,11] = -9531620235114508657502072531656361570671453261/9802962018654 481722288970934359649280000000000+360566304554597881033383231940553235 2790156659/15000633888194479828453883398581529600000000000*7^(1/2), a[ 15,6] = 4116869726480787181292612488487584/265985348360828080075687438 96484375-245534888302358312351817429197184/870045532021400261929818725 5859375*7^(1/2), a[16,9] = 9357912776729479180802749583759443090703763 9266838370557806449258372713062631705287581598317594137103657009660001 4166716414833259641089083213037080522463479597899838335379/15331924280 8682647314993371077588359843114731426490726766711816080640856626020245 0651306189032792402646266006906060260285794563433180425273922088634890 450322720794677734375+238907136665464198200669579125422678954629879112 9269363741142160105178869350417843132507667642920549895770490867650914 86664548987756568897403447748777523160068219132/7842416511953076589002 2184694418598385224926560864821875555916153780489322772503869632030129 5545986008320208136092204749767040119273874820420505695596138272491455 078125*7^(1/2), a[17,8] = 24814791431037651388777309794226734997560406 2017/350540358797316839315026245887139178767630336000-3046056343462567 590977832367296624503866287449/170828634891479941186659963882621432147 96800000*7^(1/2), a[16,14] = 50289039505918053064930317330997815277364 6321070353812284039761435107016209192/25420547332100376821161679478113 172165786281774503728665992287528699005126953125-140584018110142852629 6212582864355033678928549800267961335869008409791765544/40672875731360 602913858687164981075465258050839205965865587660045918408203125*7^(1/2 ), a[16,13] = -1478172499265455693235669169495754925970922405664549846 419638317907395087946/175456477728666831489950690673131923190567109284 028983890324355658905029296875+574878202234070155064691566753530747289 127904338643775445570927408156648849/350912955457333662979901381346263 846381134218568057967780648711317810058593750*7^(1/2), c[18] = 9/20, a [18,15] = -137805077367430365234375/4567495774520478851022848+50537274 3991484208984375/18269983098081915404091392*7^(1/2), a[18,1] = 1614143 524969292517000674483565805268889123/372177802970283976692764755298003 91424000000-29563900439689757729410425758264271687459/7443556059405679 533855295105960078284800000*7^(1/2), a[15,9] = -3067066459661270994000 4930485462861354074938991332508415290927168756413401427810067711802155 25814345889446383/3733407192182277814105116532466971854315408940785290 94301526280820101435128911405410349160893280029296875000+1816647611984 0769402418420597643880095994693261462548655455870944454018746868301641 48895260773137/4244756546679057970967517479159936835440315081843828746 85996435033916010685878289836883544921875*7^(1/2), a[17,2] = 0, a[17,3 ] = 0, a[17,4] = 0, a[17,5] = 0, a[15,13] = -15630718068/3814697265625 +6167095848/3814697265625*7^(1/2), a[18,12] = 125045555379421210983190 86045547445973/579210996962424213735904531770995200000-167192247856247 91190523597280063453/2058713044569108620250651787566080000*7^(1/2), c[ 16] = 14/125, a[18,6] = -580046684768304199644873721295749756809267/10 364426966178331681054864069406402622264320*7^(1/2)+3061845663376807113 84412537461373905511815/2072885393235666336210972813881280524452864, a [18,7] = 5274418898058305991498806821071039010281465/29882003040351754 729418622947226108034782208-259654663641643518512998087650564525921651 57/388466039524572811482442098313939404452168704*7^(1/2), a[18,10] = 6 7477811248813361131627146239991609375/43175334466689328883124760959023 5122688-718116474757235797245927435911971003125/1363106988162620240452 9388817063137444864*7^(1/2), a[18,9] = 7367415963180026585066952709494 5838295548932773003657322843981886980490481027080901750474469091930372 60623375725413/1531331921172950886408701485014861278713938718637206312 55271500701811508862556601362782623585974382563405702656000000-3244802 5281820224488625199754712079672076342032563550205311186269577863651133 841452427139849535918572768985697763/139211992833904626037154680455896 4798830853380579278465957013642743740989659605466934387487145221659667 324569600000*7^(1/2), a[18,4] = 0, a[18,13] = 25110392089861989207/106 39787803464217600000-2754209330453161502559/3149377189825408409600000* 7^(1/2), a[15,7] = 631476617959696547824260385843267488/27059579140168 58192337687904052734375-33227466731347290455944055132270208/3922838396 059942349838659979248046875*7^(1/2), a[18,14] = 3477092423121768089251 0971/717062553250698421030400000+66104663026735794404097/1147300085201 117473648640*7^(1/2), c[15] = 57/125, a[15,14] = 172678683744/38146972 65625*7^(1/2), a[18,16] = 27007176860389880859375/67300818639812201621 0944+630252495402507333984375/12450651448365257299902464*7^(1/2), a[18 ,8] = 27899837334492834306211688588370551859375/4773153033732527768450 7931084850106041344-9309627614823891860310196472043443353125/427071587 22869985296664990970655358036992*7^(1/2), a[15,1] = 571966514289185728 75631366996149/1245822888041363915313720703125000+10891702347425878099 023621466959/8357395207277482931896209716796875*7^(1/2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[18,i]=subs(e6,a[18,i]),i=1..17):\neva lf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "63/&%\"aG6$\"#=\"\"\"$\"I- fdn&eWB3:@pRrWe\"y>'G$!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F' \"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$!FCW,4^j[*py9G#Q_mm]g$!#S/&F%6$F' \"\"($!F$=L`5xVzI>a!oe_2c]O$FD/&F%6$F'\"\")$\"G&RA\"GAW<#\\y(=QjdY16vx FD/&F%6$F'\"\"*$!I;Vk)z@DT%yBdl+ZFUZpb8F+/&F%6$F'\"#5$\"H1Ldq^%*G_x.Ec \"3+&p`.p\"FD/&F%6$F'\"#6$!HlS`CqoP!p!efW%QjFn\"R9\"FD/&F%6$F'\"#7$\"G *4NtS/N0mI.)oobn#eE-\"F+/&F%6$F'\"#8$\"G/#==f#RnQ(*zp*4.^PXqi%!#U/&F%6 $F'\"#9$\"I^V)pRZUu.u?,qs>,!fK4?FD/&F%6$F'\"#:$\"I#\\pTB8NqRZ&*\\:$eYn 'G9I%F+/&F%6$F'\"#;$\"IA)Ri2&\\$*))[TQud'4D)3dS " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 1 9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Eac h standard (simple) order condition gives rise to a group \{list) of \+ equations to be satisfied by the \"d\" coefficients of the weight poly nomials for a given stage (corresponding to an \"approximate\" interpo lation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "SO7_18 := SimpleOrderConditions(7,18,'expanded '):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "whch := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]: \nordeqns6 := []:\nfor ct in whch do\n eqn_group := convert(SO7_18[c t],'polynom_order_conditions',7):\n ordeqns6 := [op(ordeqns6),op(eqn _group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 38 "Substitute for all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns6 := []:\nfor ct to nops(ordeqns6) d o\n eqns6 := [op(eqns6),expand(subs(e6,ordeqns6[ct]))];\nend do:\nno ps(eqns6);\nnops(indets(eqns6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" $7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$E\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Solve the system of equat ions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "d6 := solve(\{op(eqns 6)\},indets(eqns6) minus \{seq(d[1,i],i=1..7),seq(d[9,i],i=1..7)\}):\n infolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We would like to ensure that " }{XPPEDIT 18 0 "a[19, \+ 18] = 0;" "6#/&%\"aG6$\"#>\"#=\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[19,17] = 0;" "6#/&%\"aG6$\"#>\"#<\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[19] = 7/10;" "6#/&%\"cG6#\"#>*&\"\"(\"\"\"\"#5!\"\" " }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "c_19 := \+ 7/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c_19G#\"\"(\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "e q1 := add(subs(\{op(d6),c[19]=c_19\},d[17,i]*c[19]^i),i=1..7)=0:\neq2 \+ := add(subs(\{op(d6),c[19]=c_19\},d[18,i]*c[19]^i),i=1..7)=0:\ndd := \+ \{d[1,1]=1,seq(d[1,i]=0,i=2..6),seq(d[9,i]=0,i=1..6)\}:\nsol := solve( subs(dd,\{eq1,eq2\}));\ndd_6 := `union`(subs(sol,dd),sol):\nd_6 := `un ion`(subs(dd_6,d6),dd_6):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<$ /&%\"dG6$\"\"*\"\"(,&#\"\\r?[oVa'Q`BQSl`M36)=+\\)3'3T6!fCEx*)R vh#*['otr*Ha_N8,%f#R+#\"jqjHQ3>[ui%4(GB!3](\\7['Rt'\\&*o6c+n$)Gur^!fd1 \\*=3&41_!3u*>[n:j#46=(!\"\"*(\"jqv?McY'>,9i(z'z7h()o6KK73@3!QY7%*)o-G 5ZSKN.a^Hw:)GY!pnH1nu\"o1H\"\"\"\"F/F0F+#F3\"\"#F3/&F(6$F3F+,&*(\"GvYb d)fSFXef^Fqt[Ir?$F3\"IPr&3Dtzq(e$)f[znr;i=Z5F0F+F4F3#\"L0OA[n4li\"[Qi* 4?+*)H.ptb\"\"K$*))*>TWp!o2)*3&GKr%f\\>=z>F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "subs(d_6,mat rix([seq([seq(d[j,i],j=1..18)],i=1..7)])):\nevalf[6](%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7)74$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F +F+F+F+F+F+F+F+F+74F+F+F+F+F+$\"'j]X!\"%$\"'G[aF/$\"'E->!\"$F+$\"'B@nF /$!'$yd#F4$\"',Ao!\"&$\"'SnS!\"'$\"'1)>&F4$!'cW%)!\"#$!'_(z&F/$!'M=:FC $\"'E%R*FC74F+F+F+F+F+$!'&3E#F4$!'z1FF4$!'o]%*F4F+$!'CRLF4$\"'r!G\"FC$ !'G*Q$F/$!'YH>F;$!'#*o?FC$\"'CgQ!\"\"$\"'YqBF4$\"'SZlFC$!'tyUFgn74F+F+ F+F+F+$\"'^(>'F4$\"'+?uF4$\"'q!f#FCF+$\"'n`\"*F4$!'x5NFC$\"'$4H*F/$\"' PL\\F;$\"'x3VFC$!'X2&*Fgn$!'`ccF4$!'L-:Fgn$\"'+\\5F*74F+F+F+F+F+$!'l:& *F4$!'ER6FC$!'txRFCF+$!'Y09FC$\"'U!R&FC$!'_E9F4$!';)z'F;$!'v^ZFC$\"'x \"G\"F*$\"'ZRyF4$\"'@p=Fgn$!'o19F*74F+F+F+F+F+$\"'?&f(F4$\"'R$4*F4$\"' '\\<$FCF+$\"'!=7\"FC$!'_-VFC$\"'iQ6F4$\"'*4^%F;$\"'*&\"\"(\"\"\" \"#5!\"\"" }{TEXT -1 83 " to obtain the linking coefficients in the n ext stage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "eqs19 := \{seq(a[19,j]=add(expand(subs(\{op(d_6),c[1 9]=c_19\},d[j,i]*c[19]^i)),i=1..7),j=1..18)\}:\ne7 := `union`(eqs19,\{ c[19]=c_19\},e6):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e7 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20353 "e7 := \{a[9,6] = 780125155843893641323090552530431036567795 592568497182701460674803126770111481625/183110425412731972197889874507 158786859226102980861859505241443073629143100805376, a[11,5] = -260530 85959256534152588089363841/4377552804565683061011299942400, a[11,8] = \+ 161021426143124178389075121929246710833125/109972077221310346506670413 64346422894371443, a[18,17] = 0, a[10,7] = 318607235173649312405151265 849660869927653414425413/671471671555896530313293807293546542391091200 0000, a[8,5] = -1024030607959889/168929280000000, a[18,5] = 0, a[16,7] = 1347848014179387864407984855535195535143443353399970936257813063151 44387570974417447025422991363808/1464237214848959077379888528329155065 846449561947146728294932984499575946659768049806174092837890625+373311 5509220813199487261578662228745283054587178425467773365237314452298352 8253234993840141021024/73211860742447953868994426416457753292322478097 35733641474664922497879733298840249030870464189453125*7^(1/2), a[4,1] \+ = 1023/25600, a[2,1] = 1/20, a[11,9] = 3007606697681025178342324975654 52434946672266195876496371874262392684852243925359864884962513/4655443 3375013464555850653366045056037608247796155212857518928103156804923641 06674524398280000, a[12,6] = 23463053885534042586562584734461844191547 40172519949575/256726716407895402892744978301151486254183185289662464, a[12,5] = -4583493974484572912949314673356033540575/45195770365525074 7157313034270335135744, a[10,4] = -20462749524591049105403365239069/45 4251913499893469596231268750, a[13,5] = 0, a[7,1] = 21173/343200, a[7, 2] = 0, a[7,3] = 0, a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), a[8,2] = 0, a[8,3] = 0, a[13,3] = 0, c[13] = 1, a[11,3] = 0, a[14,4] = 0, a[13,4] = 0, a[9,2] = 0, a[14,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/39 2-3/392*7^(1/2), a[12,2] = 0, a[11,2] = 0, a[10,2] = 0, a[12,3] = 0, a [4,2] = 0, a[14,5] = 0, a[6,2] = 0, a[8,7] = 6070139212132283/92502016 000000, a[8,6] = 1501408353528689/265697280000000, a[6,3] = 0, a[9,3] \+ = 0, c[6] = 93/200, a[9,5] = -1207067925846925480797893644173318794948 4571516120469966534514296406891652614970375/27220311547616572217104781 84531100699497284085048389015085076961673446140398628096, a[5,2] = 0, \+ a[13,2] = 0, a[12,8] = 345685379554677052215495825476969226377187500/7 4771167436930077221667203179551347546362089, a[12,7] = 165712155931984 6802171283690913610698586256573484808662625/13431480411255146477259155 104956093505361644432088109056, c[5] = 39/100, a[14,12] = 113521280986 68146659861/254668911904014019468056-5215842639928607924801/1273344559 52007009734028*7^(1/2), a[3,1] = -7161/1024000, a[19,17] = 0, a[19,18] = 0, a[6,4] = 31744/135025, a[7,4] = 8602624/76559175, c[7] = 31/200, c[8] = 943/1000, c[9] = 7067558016280/7837150160667, c[11] = 47/50, a [6,5] = 923521/5106400, a[10,6] = 211276702141728028701282869920039408 10655221489/4679473877997892906145822697976708633673728000, a[7,6] = 5 611/283500, a[8,1] = -1221101821869329/690812928000000, a[13,9] = 1227 4765470313196878428812037740635050319234276006986398294443554969616342 274215316330684448207141/489345147493715517650385834143510934888829280 686609654482896526796523353052166757299452852166040, a[4,3] = 3069/256 00, a[5,1] = 4202367/11628100, a[8,4] = -125/2, a[3,2] = 116281/102400 0, a[12,4] = -16957088714171468676387054358954754000/14369041511965468 3326368228101570221, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = -389 9844/2907025, a[7,5] = -26782109/689364000, a[13,8] = -137345124323977 41476562500000/875132892924995907746928783, a[13,7] = 3847749490868980 348119500000/15517045062138271618141237517, a[13,12] = -30681427293697 6936753/1299331183183744997286, a[10,5] = -180269259803172281163724663 224981097/38100922558256871086579832832000000, a[11,1] = -234265984581 4086836951207140065609179073838476242943917/13584809613510567770222314 00139158760857532162795520000, a[12,11] = -612293360107076959161309399 3993358877250/1050517001510235513198246721302027675953, a[11,7] = 8907 22993756379186418929622095833835264322635782294899/1392124200139511265 7501941955594013822830119803764736, c[2] = 1/20, a[13,11] = 2820355431 83190840068750/12295407629873040425991, a[13,10] = -979836368457773944 5312500000/308722986341456031822630699, a[11,10] = -311552374371117306 65923206875/392862141594230515010338956291, a[12,9] = -320589096271707 2542791434312152727534008102774023210240571361570757249056167015230160 352087048674542196011/947569549683965814783015124451273604984657747127 257615372449205973192657306017239103491074738324033259120, a[11,4] = - 996286030132538159613930889652/16353068885996164905464325675, a[12,1] \+ = -2866556991825663971778295329101033887534912787724034363/86822671161 9262703011213925016143612030669233795338240, a[13,1] = 449018677377546 16851973/1014046409980231013380680, a[10,8] = 212083202434519082281842 245535894/20022426044775672563822865371173879, a[12,10] = 402795458327 06233433100438588458933210937500/8896460842799482846916972126377338947 215101, c[3] = 341/3200, a[13,6] = 791638675191615279648100000/2235604 725089973126411512319, a[6,1] = 5611/114400, a[5,4] = 3982992/2907025, a[11,6] = 20980822345096760292224086794978105312644533925634933539/37 75889992007550803878727839115494641972212962174156800, a[10,9] = -2698 404929400842518721166485087129798562269848229517793703413951226714583/ 4695456749139343150770004420808711418846760359027175503256167281758750 00000, a[9,8] = 103328481844520156040568367672866568591240077969706680 46446015775000000/1312703550036033648073834248740727914537972028638950 165249582733679393783, a[9,4] = -5172294311085668458375175655246981230 039025336933699114138315270772319372469280000/124619381004809145897278 630571215298365257079410236252921850936749076487132995191, a[10,1] = - 29055573360337415088538618442231036441314060511/2267475989108957769132 7962602370597632000000000, a[9,1] = -147251426448621580388138470887726 4246346044433307094207829051978044531801133057155/12468948016200320011 57059621643986024803301558393487900440453636168046069686436608, a[9,7] = 6641131229599116421347821358391064699281403281605770353571553403929 50009492511875/1517846559858624813633302310729534917527976515008907830 1139943253016877823170816, a[16,1] = 308721907749512142880285573783644 0713071381964765974989609100353850622638124319702720101198911189933/60 3660574218230035385964271207009410919947344407671471967321950679966085 32430377691588769378662109375-8916810789209489737328912735797151298284 51185262302853368713631052223573903665061810906114902/1143621434532973 4496276674646339100330016999988778468731028169947522328034939921889095 15380859375*7^(1/2), a[14,11] = 2970221666030903358562614446315/458906 4694536869684100726-2245277482452996546801919407985/917812938907373936 8201452*7^(1/2), a[14,10] = -2079668361431216455673395233734375/595777 6266668670748330679562+224582419744760612736276500828125/1702221790476 763070951622732*7^(1/2), a[14,8] = -8789797928960067489629400558015625 /16279513475811535751534066502+6644470662554184045392548540578125/3255 9026951623071503068133004*7^(1/2), a[14,7] = 1274787529143761017157317 260602825/26341935917422215554074218244182-958929531471161732045165837 664925/52683871834844431108148436488364*7^(1/2), a[14,1] = -942013/613 97+2856761/491176*7^(1/2), a[14,6] = -1090956177631910962162375596685/ 8605876253572154615648617314+40421151553845087519011727532985/84337587 2850071152333564496772*7^(1/2), a[16,3] = 0, a[16,2] = 0, a[16,4] = 0, a[16,5] = 0, a[15,8] = 598933003512786415989929347488/289726914398530 83012428515625-14751096540440194135375671464064/1371942153475395989706 173828125*7^(1/2), a[15,2] = 0, c[19] = 7/10, a[19,2] = 0, a[19,3] = 0 , a[19,4] = 0, a[19,5] = 0, a[15,3] = 0, a[15,4] = 0, a[15,5] = 0, a[1 7,15] = -1927193513548850260413881071/23077873387050840510431232000+83 54781001061139440464823/369245974192813448166899712*7^(1/2), a[14,9] = 998747057824857338729725466765942289696336789942851277258208294428396 2013964900384895127596133/41295613981107641868485821081000930261227361 072745605305202526150545875206586329586453004-377489797344744275480292 5721321508929613447293787235365983123527137197728406982036112477728851 /412956139811076418684858210810009302612273610727456053052025261505458 75206586329586453004*7^(1/2), a[17,16] = 61561095821428811034532925699 /1195262539043064700790636544000+197966374530935184348002389/478105015 6172258803162546176*7^(1/2), a[17,1] = 8328855883002117418483622903726 420956080895292363251/198331592372322382316539113020646822720000000000 000000-4772929606928922621289088245796233257786356432779/1469122906461 647276418808244597383872000000000000000*7^(1/2), a[16,8] = -7485568824 2755874140837522994335294782960180652180921502875834466293853598523210 706487111328/422811107430209503460506370159185920828859432008949193767 55357202712416979579385828158093167+4080137020543686228207413259382126 675613920337919418028714789205954679428753502302302598432/111982770634 7491057135420295573740149665383274029973921962543361797409128410760070 4550728125*7^(1/2), a[16,12] = -58820694887374956732123783170620055154 837987618061180717471908329718193157553959653059936964184/957882658842 1893999864045794979276347406434759944469605071338585717260615679136777 43743896484375+5330785904517985582962774414128678356690592330305476471 554875994410016840663607810578383524/373443531712354541905031025145390 890737092973097250277000831913673187548369556989373779296875*7^(1/2), \+ a[19,6] = 18737553720631362810484177932773607619285/624641803765211953 99214582561154658660968+411104910959722573223373581396660538073/115674 40810466888036891589363176788640920*7^(1/2), a[19,8] = 363871561817901 1339460790182555028109375/1286934470443626789073610218981802083704+659 8147583912342477482708328167459375/47664239646060251447170748851177854 952*7^(1/2), a[17,10] = 5616070254304857912693648015290308884679849816 7/123661065966112908213890615348396782899806208000-7048906789127141328 19876715191341334116124219/1635728385795144288543526658047576493383680 0000*7^(1/2), a[17,14] = 106657925843659476662976498724178969/26889845 74690119078864000000000000000+67590823874770618986810696149161/1434125 106501396842060800000000000*7^(1/2), a[16,15] = 2202983468184804823773 469728546582766077721838920890760304290760481052156/336315874847474553 37411471242979551680712577077922629054937124338769013375-1013520355882 3928958167900165592077146024657433459350553499494488454213/53810539975 5959285398583539887672826891401233246762064878993989420304214*7^(1/2), a[16,10] = -232080254067949765801415412360330627608091174656283335208 5780734430224778846220269711385730784/18644526562690261287856063406178 83560192919637622678890647626052665702123772234282536764081375+5818158 5452998409536703173893358176238093153847679866114668578495277523296759 42295310238176/4665797438110675997961977829374082983465764858915612839 4585236553195748843149006069488590625*7^(1/2), a[15,10] = 543652640255 07653851977480738208/4466478390356713423359819140625-14962945283111314 16682934929792/235786943683989396406626953125*7^(1/2), a[17,6] = 96085 01774836052499507564485849861825840507225298757/6995988202170373884712 0332468493217700284160000000000-31215118821227947550703658120770299949 941405900809/681870195143311294806241057197789646201600000000000*7^(1/ 2), a[19,15] = -7521783452765791015625/428202728861294892283392*7^(1/2 )+31644645563653568359375/428202728861294892283392, c[17] = 373/1000, \+ a[19,9] = -49096185532777203730343075531898691297274942930245779566091 680054608715446145234727210629741675867657970337027809/213634475610065 6928583567919942607810705829685598781128003229641487325737479863300262 0343692239729710296554500000+25297096223544341667330724788953850190572 3513232149327032484290496093192153494909971810817233379463450726430467 /170907580488052554286685433595408624856466374847902490240258371318986 05899838906402096274953791783768237243600000*7^(1/2), a[16,11] = 14152 6390645980111494784405825402710969034595590942451137061852851074578686 09157190831336140072/5801166830186754554514051371059480088582444695362 045034875735924602110577185555412784755859375-230593473763469337865940 94414430975662069875723080237434871217181706005525496230157407214632/4 6708267553838603498502829074552979779246736677633212841189500198084626 225326533114208984375*7^(1/2), a[18,11] = -359753926058578173441629981 22286881735/45601927020111218678499805531687849984+6700127934602742550 1202673610603822817/228009635100556093392499027658439249920*7^(1/2), a [15,12] = -1927555453883917676542797699/276054308640062825012207031250 , a[17,13] = 5929009119114323566893394932293/1968360743640880256000000 000000000-2816129290207834445979834640567/3936721487281760512000000000 000000*7^(1/2), a[15,11] = -3598066799092816058284144620546/1462869641 02678984417724609375+65570869112872951387443794088672/5120043743593764 454620361328125*7^(1/2), a[18,2] = 0, a[18,3] = 0, a[17,7] = 102563335 109662194461557406821662279724615611105469/618576496058236960959302704 321559561229568000000000-265492203346789444504866948767894427719165283 11224941/485582549405716014353052622892424255565210880000000000*7^(1/2 ), a[19,1] = 174871013236725476061563224349045572286953/33645537991733 26128584145667314544314000000+62859757552457393127856737157320807163/2 4922620734617230582104782720848476400000*7^(1/2), a[17,9] = -374007328 8919233268483245286897966010855585707988160778840601948021591448577248 7287338717828366357415478453578758816356450881/19141649014661886080108 7685626857659839242339829650789069089375877264386078195751703478279482 467978204257128320000000000000000-331775197314081720003881018252916142 2614554453563495416261652293406587562445693191678520005660659000252499 8996767787117019/17401499104238078254644335056987059985385667257240980 8244626705342967623707450683366798435893152707458415571200000000000000 0*7^(1/2), a[16,6] = -858912243102433135883521643728777502484576988133 90199974517211560164597559644901403704311739138208/1054793494205986431 6776136894419597979528106546537572266549401570835412223968959199104892 97177734375+1979683665557887822471642625237777109412751193262187274175 31422312697579876941533185813932673312/5279246717747679838226294741951 750740504557831099886019293994780197904016000480079632078564453125*7^( 1/2), a[17,12] = 2902460704378612529880216802791069594834177328387/127 064412458631811888314056657262072000000000000000-170951053384965232910 24743569399504075413989/2573391305711385775313314734457600000000000000 *7^(1/2), a[19,13] = 307244454352750466699/24604509295511003200000+136 64215448604158347/24604509295511003200000*7^(1/2), a[17,11] = -9531620 235114508657502072531656361570671453261/980296201865448172228897093435 9649280000000000+3605663045545978810333832319405532352790156659/150006 33888194479828453883398581529600000000000*7^(1/2), a[15,6] = 411686972 6480787181292612488487584/26598534836082808007568743896484375-24553488 8302358312351817429197184/8700455320214002619298187255859375*7^(1/2), \+ a[16,9] = 935791277672947918080274958375944309070376392668383705578064 4925837271306263170528758159831759413710365700966000141667164148332596 41089083213037080522463479597899838335379/1533192428086826473149933710 7758835984311473142649072676671181608064085662602024506513061890327924 0264626600690606026028579456343318042527392208863489045032272079467773 4375+23890713666546419820066957912542267895462987911292693637411421601 0517886935041784313250766764292054989577049086765091486664548987756568 897403447748777523160068219132/784241651195307658900221846944185983852 2492656086482187555591615378048932277250386963203012955459860083202081 36092204749767040119273874820420505695596138272491455078125*7^(1/2), a [19,12] = 1706949206714180811702245481891303829/2388428805613379922712 6702379184600000+70186441751105240857319275322903/13609280943665982465 599260614920000*7^(1/2), a[17,8] = 24814791431037651388777309794226734 9975604062017/350540358797316839315026245887139178767630336000-3046056 343462567590977832367296624503866287449/170828634891479941186659963882 62143214796800000*7^(1/2), a[16,14] = 50289039505918053064930317330997 8152773646321070353812284039761435107016209192/25420547332100376821161 679478113172165786281774503728665992287528699005126953125-140584018110 1428526296212582864355033678928549800267961335869008409791765544/40672 8757313606029138586871649810754652580508392059658655876600459184082031 25*7^(1/2), a[16,13] = -1478172499265455693235669169495754925970922405 664549846419638317907395087946/175456477728666831489950690673131923190 567109284028983890324355658905029296875+574878202234070155064691566753 530747289127904338643775445570927408156648849/350912955457333662979901 381346263846381134218568057967780648711317810058593750*7^(1/2), c[18] \+ = 9/20, a[18,15] = -137805077367430365234375/4567495774520478851022848 +505372743991484208984375/18269983098081915404091392*7^(1/2), a[18,1] \+ = 1614143524969292517000674483565805268889123/372177802970283976692764 75529800391424000000-29563900439689757729410425758264271687459/7443556 059405679533855295105960078284800000*7^(1/2), a[15,9] = -3067066459661 2709940004930485462861354074938991332508415290927168756413401427810067 71180215525814345889446383/3733407192182277814105116532466971854315408 94078529094301526280820101435128911405410349160893280029296875000+1816 6476119840769402418420597643880095994693261462548655455870944454018746 86830164148895260773137/4244756546679057970967517479159936835440315081 84382874685996435033916010685878289836883544921875*7^(1/2), a[17,2] = \+ 0, a[17,3] = 0, a[17,4] = 0, a[17,5] = 0, a[15,13] = -15630718068/3814 697265625+6167095848/3814697265625*7^(1/2), a[18,12] = 125045555379421 21098319086045547445973/579210996962424213735904531770995200000-167192 24785624791190523597280063453/2058713044569108620250651787566080000*7^ (1/2), c[16] = 14/125, a[18,6] = -580046684768304199644873721295749756 809267/10364426966178331681054864069406402622264320*7^(1/2)+3061845663 37680711384412537461373905511815/2072885393235666336210972813881280524 452864, a[18,7] = 5274418898058305991498806821071039010281465/29882003 040351754729418622947226108034782208-259654663641643518512998087650564 52592165157/388466039524572811482442098313939404452168704*7^(1/2), a[1 9,7] = 162699268454889496761023351141284149876845/43355584768367501281 5225556153950228183224+1415606313445844287537299717843980987691/333504 49821821154831940427396457709860248*7^(1/2), a[19,10] = 29317433704285 468516504401709896572546875/8625911409466581209116253860797766664328+8 22167960519864311728648427688384375/2457524618081647068124288849230132 9528*7^(1/2), a[18,10] = 67477811248813361131627146239991609375/431753 344666893288831247609590235122688-718116474757235797245927435911971003 125/13631069881626202404529388817063137444864*7^(1/2), a[18,9] = 73674 1596318002658506695270949458382955489327730036573228439818869804904810 2708090175047446909193037260623375725413/15313319211729508864087014850 1486127871393871863720631255271500701811508862556601362782623585974382 563405702656000000-324480252818202244886251997547120796720763420325635 50205311186269577863651133841452427139849535918572768985697763/1392119 9283390462603715468045589647988308533805792784659570136427437409896596 05466934387487145221659667324569600000*7^(1/2), a[18,4] = 0, a[18,13] \+ = 25110392089861989207/10639787803464217600000-2754209330453161502559/ 3149377189825408409600000*7^(1/2), a[15,7] = 6314766179596965478242603 85843267488/2705957914016858192337687904052734375-33227466731347290455 944055132270208/3922838396059942349838659979248046875*7^(1/2), a[18,14 ] = 34770924231217680892510971/717062553250698421030400000+66104663026 735794404097/1147300085201117473648640*7^(1/2), c[15] = 57/125, a[15,1 4] = 172678683744/3814697265625*7^(1/2), a[18,16] = 270071768603898808 59375/673008186398122016210944+630252495402507333984375/12450651448365 257299902464*7^(1/2), a[19,11] = -547551544801059530995430107901463107 5/1374165211543530027142293247048629408-474867897757895015033631646490 95523/254475039174727782804128379083079520*7^(1/2), a[18,8] = 27899837 334492834306211688588370551859375/477315303373252776845079310848501060 41344-9309627614823891860310196472043443353125/42707158722869985296664 990970655358036992*7^(1/2), a[19,14] = -517517744336749012103429/16806 153591813244242900000-327959225091317538901/8963281915633730262880*7^( 1/2), a[15,1] = 57196651428918572875631366996149/124582288804136391531 3720703125000+10891702347425878099023621466959/83573952072774829318962 09716796875*7^(1/2), a[19,16] = -127299854452380859375/130273278268330 6776636-1340064012820380859375/41687449045865816852352*7^(1/2)\}: " } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[19,i]=subs(e7,a[19,i]),i=1..1 8):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "64/&%\"aG6$\"#>\" \"\"$\"Iw7Qxuk&ewH;F$>OqM4wke!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/& F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"IcvPfyIJ#HzKixWU.PB+%R!#S/ &F%6$F'\"\"($\"Ii7*z_%=Cz7L3zJz5Ovpv[FD/&F%6$F'\"\")$\"I..V6M!)o!Hut[Z !*)p^%zO>$!#R/&F%6$F'\"\"*$!IfGwq-6;_b()=s=td=#y*eAFQ/&F%6$F'\"#5$\"Ie Yy%e'=N)o$*=z5`EDaxs[$FQ/&F%6$F'\"#6$!Iv0Pyh*eY*>p]f^jC\\vKyWFQ/&F%6$F '\"#7$\"IYwgYTf;;$pb`W+\"*[%[A6&)F+/&F%6$F'\"#8$\"I'z6lov>C&z$[j'z$fa= lcR\"F+/&F%6$F'\"#9$!I:MeH,/\"y\"pPk)\\e/#fC*fF\"FD/&F%6$F'\"#:$\"ITYO O!GjA];9Wu?*=_ufUFF+/&F%6$F'\"#;$!I+sGjAMhHLe_wqx)Hblw#=FD/&F%6$F'\"#< F0/&F%6$F'\"#=F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "ca lculation for stage 20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order condition gives rise t o a group \{list) of equations to be satisfied by the \"d\" coefficie nts of the weight polynomials for a given stage (corresponding to an \+ \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_19 := SimpleOrderCon ditions(7,19,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "whch := [1,2,3,6,7,8,12,15,16,21,2 7,31,32,33,36,64]:\nordeqns7 := []:\nfor ct in whch do\n eqn_group : = convert(SO7_19[ct],'polynom_order_conditions',7):\n ordeqns7 := [o p(ordeqns7),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known coefficients ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns7 := []:\nfor ct t o nops(ordeqns7) do\n eqns7 := [op(eqns7),expand(subs(e7,ordeqns7[ct ]))];\nend do:\nnops(eqns7);\nnops(indets(eqns7));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$L\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Solve the system of equations." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "inf olevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "d7 := solve(\{op(eqns7)\},indets(eqns7) minus \{seq(seq(d[j,i],i=1..7),j=[1 ,9,13])\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "We would like to ensure that " } {XPPEDIT 18 0 "a[20,19]=0" "6#/&%\"aG6$\"#?\"#>\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "a[20,18] = 0;" "6#/&%\"aG6$\"#?\"#=\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[20,17] = 0;" "6#/&%\"aG6$\"#?\"#<\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[20] = 9/10;" "6#/&%\"cG6#\"#?* &\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "c_20 := 9/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c _20G#\"\"*\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 382 "eq1 := add(subs(\{op(d7),c[20]=c_20\},d[17,i] *c[20]^i),i=1..7)=0:\neq2 := add(subs(\{op(d7),c[20]=c_20\},d[18,i]*c[ 20]^i),i=1..7)=0:\neq3 := add(subs(\{op(d7),c[20]=c_20\},d[19,i]*c[20] ^i),i=1..7)=0:\ndd := \{d[1,1]=1,seq(d[1,i]=0,i=2..6),seq(d[9,i]=0,i=1 ..6),seq(d[13,i]=0,i=1..6)\}:\nsol := solve(subs(dd,\{eq1,eq2,eq3\})); \ndd_7 := `union`(subs(sol,dd),sol):\nd_7 := `union`(subs(dd_7,d7),dd_ 7):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<%/&%\"dG6$\"\"\"\"\"(,& #\"KNv$36&\\I;@\"f2oKKokH$og8\"JRXNQvPJ!H?u`Xn;t;J>Ex!\"\"*(\"HD&*Q>t? q%f!4QiL:)fK))y]F*F/F0F+#F*\"\"#F*/&F(6$\"\"*F+,&#\"hq&oa5\"e8EauTu#[R&[@S)QQ#\"fq*HK6iJ`\"\\#ow'=G y]nGhtg^H\"o(oYh(4<$o_J&\\\"oVoiz0KX'==*f6^q_p:)F**(\"gqvleXM/on9'R=Gc VW8T*3s:4]'*Gya]D)QnU?u\"*)[L^(>_!*>n&yemTfWUQ5F*\"hqneO(\\Vf?A_Ij,Lex iC#H/.u#[$pSG@UYaQSX$*[TeoF\"zbH:+tFoo#R%z\"p#F0F+F3F*/&F(6$\"#8F+,&# \"6vAc9<43QL$[\"8w'4#>(H/_Pl*3\"F0*(\"5v1ZtysL=x:F*FGF0F+F3F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "subs(d_7,matrix([seq([seq(d[j,i],j=1..19)],i=1..7)])):\nevalf[5](% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)75$\"\"\"\"\"!$F*F *F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+75F+F+F+F+F+$\"&jj#!\"$$\"&j:$F/$\" &@5\"!\"#F+$\"&O*QF/$!&M\\\"F4$\"&=&R!\"%F+$\"&z]'F4$!&u(**!\"\"$!&Vx% F/$!&B#=F@$\"&F6\"F*$\"&dF)F;75F+F+F+F+F+$!&FN\"F4$!&'>;F4$!&Wl&F4F+$! &!)*>F4$\"&=m(F4$!&r-#F/F+$!&-p#F@$\"&ve%F*$\"&_)=F4$\"&**)zF@$!&15&F* $!&O$RF/75F+F+F+F+F+$\"&b(QF4$\"&+k%F4$\"&,i\"F@F+$\"&Us&F4$!&a>#F@$\" &)4eF/F+$\"&u*eF@$!&n8\"F)$!&cT%F4$!&5(=F*$\"&#f7F)$\"&P+\"F475F+F+F+F +F+$!&fJ'F4$!&>c(F4$!&.k#F@F+$!&#H$*F4$\"&!yNF@$!&*o%*F/F+$!&3%pF@$\"& !Q:F)$\"&'HhF4$\"&uP#F*$!&jp\"F)$!&JQ\"F475F+F+F+F+F+$\"&@Z&F4$\"&;b'F 4$\"&uG#F@F+$\"&C3)F4$!&**4$F@$\"&L?)F/F+$\"&**)RF@$!&60\"F)$!&Bm%F4$! &F^\"F*$\"&N:\"F)$\"&y<*F/75$!&Pu\"F;F+F+F+F+$!&<(>F4$!&FO#F4$!&HW)F4$ \"&F$HF/$!&%*G$F4$\"&h9\"F@$!&'*)HF/$!&G0%!\"'$!&)f&*F4$\"&9)HF*$\"&.g \"F4$\"&L.%F@$!&lD$F*$!&(3BF/Q(pprint76\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate the \"weight\" polynom ials at the node " }{XPPEDIT 18 0 "c[20] = 9/10;" "6#/&%\"cG6#\"#?*& \"\"*\"\"\"\"#5!\"\"" }{TEXT -1 82 " to obtain the linking coefficient s in the next stage in the interpolation scheme." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 132 "eqs20 := \{seq(a[20,j]=add(expand(subs(\{op(d _7),c[20]=c_20\},d[j,i]*c[20]^i)),i=1..7),j=1..19)\}:\ne8 := `union`(e qs20,\{c[20]=c_20\},e7):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e8 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22667 "e8 := \{a[9,6] = 780125155843893641323090552530431 036567795592568497182701460674803126770111481625/183110425412731972197 889874507158786859226102980861859505241443073629143100805376, a[11,5] \+ = -26053085959256534152588089363841/4377552804565683061011299942400, a [11,8] = 161021426143124178389075121929246710833125/109972077221310346 50667041364346422894371443, a[18,17] = 0, a[10,7] = 318607235173649312 405151265849660869927653414425413/671471671555896530313293807293546542 3910912000000, a[8,5] = -1024030607959889/168929280000000, a[18,5] = 0 , a[16,7] = 1347848014179387864407984855535195535143443353399970936257 81306315144387570974417447025422991363808/1464237214848959077379888528 3291550658464495619471467282949329844995759466597680498061740928378906 25+3733115509220813199487261578662228745283054587178425467773365237314 4522983528253234993840141021024/73211860742447953868994426416457753292 32247809735733641474664922497879733298840249030870464189453125*7^(1/2) , a[4,1] = 1023/25600, a[2,1] = 1/20, a[11,9] = 3007606697681025178342 3249756545243494667226619587649637187426239268485224392535986488496251 3/46554433375013464555850653366045056037608247796155212857518928103156 80492364106674524398280000, a[12,6] = 23463053885534042586562584734461 84419154740172519949575/2567267164078954028927449783011514862541831852 89662464, a[12,5] = -4583493974484572912949314673356033540575/45195770 3655250747157313034270335135744, a[20,8] = -48838465320982269123224176 957197891703125/6339343872926013442473709597206654708616+5757597614705 7867393261247514690821875/333649677522421760130195241958244984664*7^(1 /2), a[10,4] = -20462749524591049105403365239069/454251913499893469596 231268750, a[13,5] = 0, a[7,1] = 21173/343200, a[7,2] = 0, a[7,3] = 0, a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), a[8,2] = 0, a[8,3] = 0, a[13,3 ] = 0, c[13] = 1, a[11,3] = 0, a[14,4] = 0, a[13,4] = 0, a[9,2] = 0, a [14,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[ 12,2] = 0, a[11,2] = 0, a[10,2] = 0, a[12,3] = 0, a[4,2] = 0, a[14,5] \+ = 0, a[6,2] = 0, a[8,7] = 6070139212132283/92502016000000, a[8,6] = 15 01408353528689/265697280000000, a[6,3] = 0, a[9,3] = 0, c[6] = 93/200, a[9,5] = -12070679258469254807978936441733187949484571516120469966534 514296406891652614970375/272203115476165722171047818453110069949728408 5048389015085076961673446140398628096, a[5,2] = 0, a[13,2] = 0, a[12,8 ] = 345685379554677052215495825476969226377187500/74771167436930077221 667203179551347546362089, a[12,7] = 1657121559319846802171283690913610 698586256573484808662625/134314804112551464772591551049560935053616444 32088109056, c[5] = 39/100, a[14,12] = 11352128098668146659861/2546689 11904014019468056-5215842639928607924801/127334455952007009734028*7^(1 /2), a[3,1] = -7161/1024000, a[19,17] = 0, a[19,18] = 0, a[6,4] = 3174 4/135025, a[7,4] = 8602624/76559175, c[7] = 31/200, c[8] = 943/1000, c [9] = 7067558016280/7837150160667, c[11] = 47/50, a[6,5] = 923521/5106 400, a[10,6] = 21127670214172802870128286992003940810655221489/4679473 877997892906145822697976708633673728000, a[7,6] = 5611/283500, a[8,1] \+ = -1221101821869329/690812928000000, a[13,9] = 12274765470313196878428 8120377406350503192342760069863982944435549696163422742153163306844482 07141/4893451474937155176503858341435109348888292806866096544828965267 96523353052166757299452852166040, a[4,3] = 3069/25600, a[5,1] = 420236 7/11628100, a[8,4] = -125/2, a[3,2] = 116281/1024000, a[12,4] = -16957 088714171468676387054358954754000/143690415119654683326368228101570221 , c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = -3899844/2907025, a[7,5 ] = -26782109/689364000, a[13,8] = -13734512432397741476562500000/8751 32892924995907746928783, a[13,7] = 3847749490868980348119500000/155170 45062138271618141237517, a[13,12] = -306814272936976936753/12993311831 83744997286, a[10,5] = -180269259803172281163724663224981097/381009225 58256871086579832832000000, a[11,1] = -2342659845814086836951207140065 609179073838476242943917/135848096135105677702223140013915876085753216 2795520000, a[12,11] = -6122933601070769591613093993993358877250/10505 17001510235513198246721302027675953, a[11,7] = 89072299375637918641892 9622095833835264322635782294899/13921242001395112657501941955594013822 830119803764736, c[2] = 1/20, a[13,11] = 282035543183190840068750/1229 5407629873040425991, a[13,10] = -9798363684577739445312500000/30872298 6341456031822630699, a[11,10] = -31155237437111730665923206875/3928621 41594230515010338956291, a[12,9] = -3205890962717072542791434312152727 5340081027740232102405713615707572490561670152301603520870486745421960 11/9475695496839658147830151244512736049846577471272576153724492059731 92657306017239103491074738324033259120, a[11,4] = -9962860301325381596 13930889652/16353068885996164905464325675, a[12,1] = -2866556991825663 971778295329101033887534912787724034363/868226711619262703011213925016 143612030669233795338240, a[13,1] = 44901867737754616851973/1014046409 980231013380680, a[10,8] = 212083202434519082281842245535894/200224260 44775672563822865371173879, a[12,10] = 4027954583270623343310043858845 8933210937500/8896460842799482846916972126377338947215101, c[3] = 341/ 3200, a[13,6] = 791638675191615279648100000/22356047250899731264115123 19, a[6,1] = 5611/114400, a[5,4] = 3982992/2907025, a[11,6] = 20980822 345096760292224086794978105312644533925634933539/377588999200755080387 8727839115494641972212962174156800, a[10,9] = -26984049294008425187211 66485087129798562269848229517793703413951226714583/4695456749139343150 77000442080871141884676035902717550325616728175875000000, a[9,8] = 103 32848184452015604056836767286656859124007796970668046446015775000000/1 3127035500360336480738342487407279145379720286389501652495827336793937 83, a[9,4] = -51722943110856684583751756552469812300390253369336991141 38315270772319372469280000/1246193810048091458972786305712152983652570 79410236252921850936749076487132995191, a[10,1] = -2905557336033741508 8538618442231036441314060511/22674759891089577691327962602370597632000 000000, a[9,1] = -1472514264486215803881384708877264246346044433307094 207829051978044531801133057155/124689480162003200115705962164398602480 3301558393487900440453636168046069686436608, a[9,7] = 6641131229599116 42134782135839106469928140328160577035357155340392950009492511875/1517 8465598586248136333023107295349175279765150089078301139943253016877823 170816, a[20,4] = 0, a[20,5] = 0, a[16,1] = 30872190774951214288028557 3783644071307138196476597498960910035385062263812431970272010119891118 9933/60366057421823003538596427120700941091994734440767147196732195067 996608532430377691588769378662109375-891681078920948973732891273579715 129828451185262302853368713631052223573903665061810906114902/114362143 4532973449627667464633910033001699998877846873102816994752232803493992 188909515380859375*7^(1/2), a[20,17] = 0, a[20,18] = 0, a[20,19] = 0, \+ a[20,11] = 4051052479952572002583490413428916431/356265054844618895925 779730716311328-414373616330969707208269617880554327/17813252742230944 79628898653581556640*7^(1/2), a[14,11] = 29702216660309033585626144463 15/4589064694536869684100726-2245277482452996546801919407985/917812938 9073739368201452*7^(1/2), a[14,10] = -20796683614312164556733952337343 75/5957776266668670748330679562+224582419744760612736276500828125/1702 221790476763070951622732*7^(1/2), a[14,8] = -8789797928960067489629400 558015625/16279513475811535751534066502+664447066255418404539254854057 8125/32559026951623071503068133004*7^(1/2), a[14,7] = 1274787529143761 017157317260602825/26341935917422215554074218244182-958929531471161732 045165837664925/52683871834844431108148436488364*7^(1/2), a[14,1] = -9 42013/61397+2856761/491176*7^(1/2), a[14,6] = -10909561776319109621623 75596685/8605876253572154615648617314+40421151553845087519011727532985 /843375872850071152333564496772*7^(1/2), a[16,3] = 0, a[16,2] = 0, a[1 6,4] = 0, a[20,1] = 16764524349182732858436037634017802692437/29076390 8570534356791222465076565558000000+18283979735335210457126140927463498 0229/58152781714106871358244493015313111600000*7^(1/2), a[16,5] = 0, a [15,8] = 598933003512786415989929347488/28972691439853083012428515625- 14751096540440194135375671464064/1371942153475395989706173828125*7^(1/ 2), a[15,2] = 0, c[19] = 7/10, a[19,2] = 0, a[19,3] = 0, a[19,4] = 0, \+ a[19,5] = 0, a[15,3] = 0, a[15,4] = 0, a[15,5] = 0, a[20,12] = -100783 5712877603197752652668370693383/10695657614362947128645964365089400000 +1137412042683412511904456412390473/1769206522676577720527903879939600 00*7^(1/2), a[17,15] = -1927193513548850260413881071/23077873387050840 510431232000+8354781001061139440464823/369245974192813448166899712*7^( 1/2), c[20] = 9/10, a[14,9] = 9987470578248573387297254667659422896963 367899428512772582082944283962013964900384895127596133/412956139811076 4186848582108100093026122736107274560530520252615054587520658632958645 3004-37748979734474427548029257213215089296134472937872353659831235271 37197728406982036112477728851/4129561398110764186848582108100093026122 7361072745605305202526150545875206586329586453004*7^(1/2), a[20,10] = \+ -12909998295917152763260545013409460140625/745449134151432943997700950 933140329016+4441236398714324867916688020197971875/1064927334502047062 85385850133305761288*7^(1/2), a[17,16] = 61561095821428811034532925699 /1195262539043064700790636544000+197966374530935184348002389/478105015 6172258803162546176*7^(1/2), a[17,1] = 8328855883002117418483622903726 420956080895292363251/198331592372322382316539113020646822720000000000 000000-4772929606928922621289088245796233257786356432779/1469122906461 647276418808244597383872000000000000000*7^(1/2), a[16,8] = -7485568824 2755874140837522994335294782960180652180921502875834466293853598523210 706487111328/422811107430209503460506370159185920828859432008949193767 55357202712416979579385828158093167+4080137020543686228207413259382126 675613920337919418028714789205954679428753502302302598432/111982770634 7491057135420295573740149665383274029973921962543361797409128410760070 4550728125*7^(1/2), a[16,12] = -58820694887374956732123783170620055154 837987618061180717471908329718193157553959653059936964184/957882658842 1893999864045794979276347406434759944469605071338585717260615679136777 43743896484375+5330785904517985582962774414128678356690592330305476471 554875994410016840663607810578383524/373443531712354541905031025145390 890737092973097250277000831913673187548369556989373779296875*7^(1/2), \+ a[19,6] = 18737553720631362810484177932773607619285/624641803765211953 99214582561154658660968+411104910959722573223373581396660538073/115674 40810466888036891589363176788640920*7^(1/2), a[19,8] = 363871561817901 1339460790182555028109375/1286934470443626789073610218981802083704+659 8147583912342477482708328167459375/47664239646060251447170748851177854 952*7^(1/2), a[17,10] = 5616070254304857912693648015290308884679849816 7/123661065966112908213890615348396782899806208000-7048906789127141328 19876715191341334116124219/1635728385795144288543526658047576493383680 0000*7^(1/2), a[17,14] = 106657925843659476662976498724178969/26889845 74690119078864000000000000000+67590823874770618986810696149161/1434125 106501396842060800000000000*7^(1/2), a[16,15] = 2202983468184804823773 469728546582766077721838920890760304290760481052156/336315874847474553 37411471242979551680712577077922629054937124338769013375-1013520355882 3928958167900165592077146024657433459350553499494488454213/53810539975 5959285398583539887672826891401233246762064878993989420304214*7^(1/2), a[16,10] = -232080254067949765801415412360330627608091174656283335208 5780734430224778846220269711385730784/18644526562690261287856063406178 83560192919637622678890647626052665702123772234282536764081375+5818158 5452998409536703173893358176238093153847679866114668578495277523296759 42295310238176/4665797438110675997961977829374082983465764858915612839 4585236553195748843149006069488590625*7^(1/2), a[15,10] = 543652640255 07653851977480738208/4466478390356713423359819140625-14962945283111314 16682934929792/235786943683989396406626953125*7^(1/2), a[17,6] = 96085 01774836052499507564485849861825840507225298757/6995988202170373884712 0332468493217700284160000000000-31215118821227947550703658120770299949 941405900809/681870195143311294806241057197789646201600000000000*7^(1/ 2), a[19,15] = -7521783452765791015625/428202728861294892283392*7^(1/2 )+31644645563653568359375/428202728861294892283392, c[17] = 373/1000, \+ a[19,9] = -49096185532777203730343075531898691297274942930245779566091 680054608715446145234727210629741675867657970337027809/213634475610065 6928583567919942607810705829685598781128003229641487325737479863300262 0343692239729710296554500000+25297096223544341667330724788953850190572 3513232149327032484290496093192153494909971810817233379463450726430467 /170907580488052554286685433595408624856466374847902490240258371318986 05899838906402096274953791783768237243600000*7^(1/2), a[16,11] = 14152 6390645980111494784405825402710969034595590942451137061852851074578686 09157190831336140072/5801166830186754554514051371059480088582444695362 045034875735924602110577185555412784755859375-230593473763469337865940 94414430975662069875723080237434871217181706005525496230157407214632/4 6708267553838603498502829074552979779246736677633212841189500198084626 225326533114208984375*7^(1/2), a[18,11] = -359753926058578173441629981 22286881735/45601927020111218678499805531687849984+6700127934602742550 1202673610603822817/228009635100556093392499027658439249920*7^(1/2), a [15,12] = -1927555453883917676542797699/276054308640062825012207031250 , a[17,13] = 5929009119114323566893394932293/1968360743640880256000000 000000000-2816129290207834445979834640567/3936721487281760512000000000 000000*7^(1/2), a[15,11] = -3598066799092816058284144620546/1462869641 02678984417724609375+65570869112872951387443794088672/5120043743593764 454620361328125*7^(1/2), a[18,2] = 0, a[18,3] = 0, a[17,7] = 102563335 109662194461557406821662279724615611105469/618576496058236960959302704 321559561229568000000000-265492203346789444504866948767894427719165283 11224941/485582549405716014353052622892424255565210880000000000*7^(1/2 ), a[19,1] = 174871013236725476061563224349045572286953/33645537991733 26128584145667314544314000000+62859757552457393127856737157320807163/2 4922620734617230582104782720848476400000*7^(1/2), a[17,9] = -374007328 8919233268483245286897966010855585707988160778840601948021591448577248 7287338717828366357415478453578758816356450881/19141649014661886080108 7685626857659839242339829650789069089375877264386078195751703478279482 467978204257128320000000000000000-331775197314081720003881018252916142 2614554453563495416261652293406587562445693191678520005660659000252499 8996767787117019/17401499104238078254644335056987059985385667257240980 8244626705342967623707450683366798435893152707458415571200000000000000 0*7^(1/2), a[16,6] = -858912243102433135883521643728777502484576988133 90199974517211560164597559644901403704311739138208/1054793494205986431 6776136894419597979528106546537572266549401570835412223968959199104892 97177734375+1979683665557887822471642625237777109412751193262187274175 31422312697579876941533185813932673312/5279246717747679838226294741951 750740504557831099886019293994780197904016000480079632078564453125*7^( 1/2), a[17,12] = 2902460704378612529880216802791069594834177328387/127 064412458631811888314056657262072000000000000000-170951053384965232910 24743569399504075413989/2573391305711385775313314734457600000000000000 *7^(1/2), a[19,13] = 307244454352750466699/24604509295511003200000+136 64215448604158347/24604509295511003200000*7^(1/2), a[20,15] = -3125509 446488994140625/142734242953764964094464*7^(1/2)-892630495172128710937 5/142734242953764964094464, a[17,11] = -953162023511450865750207253165 6361570671453261/9802962018654481722288970934359649280000000000+360566 3045545978810333832319405532352790156659/15000633888194479828453883398 581529600000000000*7^(1/2), a[15,6] = 41168697264807871812926124884875 84/26598534836082808007568743896484375-2455348883023583123518174291971 84/8700455320214002619298187255859375*7^(1/2), a[16,9] = 9357912776729 4791808027495837594430907037639266838370557806449258372713062631705287 5815983175941371036570096600014166716414833259641089083213037080522463 479597899838335379/153319242808682647314993371077588359843114731426490 7267667118160806408566260202450651306189032792402646266006906060260285 794563433180425273922088634890450322720794677734375+238907136665464198 2006695791254226789546298791129269363741142160105178869350417843132507 6676429205498957704908676509148666454898775656889740344774877752316006 8219132/78424165119530765890022184694418598385224926560864821875555916 1537804893227725038696320301295545986008320208136092204749767040119273 874820420505695596138272491455078125*7^(1/2), a[20,7] = 13248444953061 53570227909695456980315126025/3034890933785725089706578893077651597282 568+160585056017693402348695205888904986106867/30348909337857250897065 78893077651597282568*7^(1/2), a[19,12] = 17069492067141808117022454818 91303829/23884288056133799227126702379184600000+7018644175110524085731 9275322903/13609280943665982465599260614920000*7^(1/2), a[17,8] = 2481 47914310376513887773097942267349975604062017/3505403587973168393150262 45887139178767630336000-3046056343462567590977832367296624503866287449 /17082863489147994118665996388262143214796800000*7^(1/2), a[16,14] = 5 0289039505918053064930317330997815277364632107035381228403976143510701 6209192/25420547332100376821161679478113172165786281774503728665992287 528699005126953125-140584018110142852629621258286435503367892854980026 7961335869008409791765544/40672875731360602913858687164981075465258050 839205965865587660045918408203125*7^(1/2), a[16,13] = -147817249926545 5693235669169495754925970922405664549846419638317907395087946/17545647 7728666831489950690673131923190567109284028983890324355658905029296875 +574878202234070155064691566753530747289127904338643775445570927408156 648849/350912955457333662979901381346263846381134218568057967780648711 317810058593750*7^(1/2), c[18] = 9/20, a[18,15] = -1378050773674303652 34375/4567495774520478851022848+505372743991484208984375/1826998309808 1915404091392*7^(1/2), a[18,1] = 1614143524969292517000674483565805268 889123/37217780297028397669276475529800391424000000-295639004396897577 29410425758264271687459/7443556059405679533855295105960078284800000*7^ (1/2), a[15,9] = -3067066459661270994000493048546286135407493899133250 841529092716875641340142781006771180215525814345889446383/373340719218 2277814105116532466971854315408940785290943015262808201014351289114054 10349160893280029296875000+1816647611984076940241842059764388009599469 326146254865545587094445401874686830164148895260773137/424475654667905 7970967517479159936835440315081843828746859964350339160106858782898368 83544921875*7^(1/2), a[17,2] = 0, a[17,3] = 0, a[17,4] = 0, a[17,5] = \+ 0, a[15,13] = -15630718068/3814697265625+6167095848/3814697265625*7^(1 /2), a[18,12] = 12504555537942121098319086045547445973/579210996962424 213735904531770995200000-16719224785624791190523597280063453/205871304 4569108620250651787566080000*7^(1/2), c[16] = 14/125, a[18,6] = -58004 6684768304199644873721295749756809267/10364426966178331681054864069406 402622264320*7^(1/2)+306184566337680711384412537461373905511815/207288 5393235666336210972813881280524452864, a[18,7] = 527441889805830599149 8806821071039010281465/29882003040351754729418622947226108034782208-25 965466364164351851299808765056452592165157/388466039524572811482442098 313939404452168704*7^(1/2), a[20,14] = -215042962723876515301101/56020 51197271081414300000-408828436443173054007/8963281915633730262880*7^(1 /2), a[19,7] = 162699268454889496761023351141284149876845/433555847683 675012815225556153950228183224+141560631344584428753729971784398098769 1/33350449821821154831940427396457709860248*7^(1/2), a[20,9] = 7601362 2495733182749428117639777017485540799906967181834750692711033454516581 7153657233758784359977604937627710651/54379684700743994545763547053084 5624543302101788780650764458454196773824085783385521245112166102210807 54866000000+2207445332547394246865233259955947665935051316026032661827 550919737260651075417192858710850665382231070639000183/119635306341636 7880006798035167860373995264623935317431681808599232902412988723448146 73924676542486377660705200000*7^(1/2), a[19,10] = 29317433704285468516 504401709896572546875/8625911409466581209116253860797766664328+8221679 60519864311728648427688384375/24575246180816470681242888492301329528*7 ^(1/2), a[20,16] = -14699461699626275390625/97270714440353572655488-38 97836105080869140625/97270714440353572655488*7^(1/2), a[18,10] = 67477 811248813361131627146239991609375/431753344666893288831247609590235122 688-718116474757235797245927435911971003125/13631069881626202404529388 817063137444864*7^(1/2), a[20,2] = 0, a[20,3] = 0, a[18,9] = 736741596 3180026585066952709494583829554893277300365732284398188698049048102708 090175047446909193037260623375725413/153133192117295088640870148501486 1278713938718637206312552715007018115088625566013627826235859743825634 05702656000000-3244802528182022448862519975471207967207634203256355020 5311186269577863651133841452427139849535918572768985697763/13921199283 3904626037154680455896479883085338057927846595701364274374098965960546 6934387487145221659667324569600000*7^(1/2), a[18,4] = 0, a[18,13] = 25 110392089861989207/10639787803464217600000-2754209330453161502559/3149 377189825408409600000*7^(1/2), a[15,7] = 63147661795969654782426038584 3267488/2705957914016858192337687904052734375-332274667313472904559440 55132270208/3922838396059942349838659979248046875*7^(1/2), a[18,14] = \+ 34770924231217680892510971/717062553250698421030400000+661046630267357 94404097/1147300085201117473648640*7^(1/2), c[15] = 57/125, a[20,13] = -522000512739051727257/24604509295511003200000+17033580426103348329/2 4604509295511003200000*7^(1/2), a[15,14] = 172678683744/3814697265625* 7^(1/2), a[18,16] = 27007176860389880859375/673008186398122016210944+6 30252495402507333984375/12450651448365257299902464*7^(1/2), a[19,11] = -5475515448010595309954301079014631075/137416521154353002714229324704 8629408-47486789775789501503363164649095523/25447503917472778280412837 9083079520*7^(1/2), a[18,8] = 2789983733449283430621168858837055185937 5/47731530337325277684507931084850106041344-93096276148238918603101964 72043443353125/42707158722869985296664990970655358036992*7^(1/2), a[20 ,6] = 7231245850603923375549468974817821336739/16194417134653643251648 225108447504097288+3587335118885057093042630057609435084277/8097208567 3268216258241125542237520486440*7^(1/2), a[19,14] = -51751774433674901 2103429/16806153591813244242900000-327959225091317538901/8963281915633 730262880*7^(1/2), a[15,1] = 57196651428918572875631366996149/12458228 88041363915313720703125000+10891702347425878099023621466959/8357395207 277482931896209716796875*7^(1/2), a[19,16] = -127299854452380859375/13 02732782683306776636-1340064012820380859375/41687449045865816852352*7^ (1/2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[20,i]=subs(e8,a[20 ,i]),i=1..19):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "65/&%\" aG6$\"#?\"\"\"$\"I\"*)z^-u_[-F`*f^q(f!3a(f'!#T/&F%6$F'\"\"#$\"\"!F1/&F %6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"IfO*>qGA_3P& Qjy*\\,XFuj&!#S/&F%6$F'\"\"($\"IA.#))RVO!G)*)=3Ag/H*GKldFD/&F%6$F'\"\" )$!I\"pEl#H$4/72'[P3Tp2MYZs!#R/&F%6$F'\"\"*$\"IrBW?/p\"3Vk3)4v&oJ/8FS \"!#Q/&F%6$F'\"#5$!IY9::%3dzbEyywLfBh23s\"FX/&F%6$F'\"#6$\"Im<;Zr90z0- $4w*4ihPav5FX/&F%6$F'\"#7$!I,Lc]YAE7>3&RWYn>67>s(F+/&F%6$F'\"#8$!I*Qv= xS)=6w)Q`o/BGN+%Q>F+/&F%6$F'\"#9$!I3xO5\"=t@%>TP\"429-nI1f\"FD/&F%6$F' \"#:$!I?`s5t'4\\!)RH#oASKD-t/7FD/&F%6$F'\"#;$!I-u**>N>N.\\[')QgvrZuRrD FD/&F%6$F'\"#F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 21" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simp le) order condition gives rise to a group \{list) of equations to be \+ satisfied by the \"d\" coefficients of the weight polynomials for a gi ven stage (corresponding to an \"approximate\" interpolation scheme). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_20 := SimpleOrderConditions(7,20,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "whc h := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nordeqns8 := []:\nfo r ct in whch do\n eqn_group := convert(SO7_20[ct],'polynom_order_con ditions',7):\n ordeqns8 := [op(ordeqns8),op(eqn_group)];\nend do:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substit ute for all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns8 := []:\nfor ct to nops(ordeqns8) do\n eqns8 := [op(eq ns8),expand(subs(e8,ordeqns8[ct]))];\nend do:\nnops(eqns8);\nnops(inde ts(eqns8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$S\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Solve the system of equations." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 106 "d8 := solve(\{op(eqns8)\},indets(eqns8) min us \{seq(seq(d[j,i],i=1..7),j=[1,9,12,13])\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We wou ld like to ensure that " }{XPPEDIT 18 0 "a[21,19] = 0;" "6#/&%\"aG6$ \"#@\"#>\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[21,18] = 0;" "6#/&% \"aG6$\"#@\"#=\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[21,19]=0" "6# /&%\"aG6$\"#@\"#>\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[21, 20 ] = 0;" "6#/&%\"aG6$\"#@\"#?\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 " c[21] = 97/1000;" "6#/&%\"cG6#\"#@*&\"#(*\"\"\"\"%+5!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c_21 := 97/1000;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c_21G#\"#(*\"%+5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 480 "eq1 := a dd(subs(\{op(d8),c[21]=c_21\},d[17,i]*c[21]^i),i=1..7)=0:\neq2 := add( subs(\{op(d8),c[21]=c_21\},d[18,i]*c[21]^i),i=1..7)=0:\neq3 := add(sub s(\{op(d8),c[21]=c_21\},d[19,i]*c[21]^i),i=1..7)=0:\neq4 := add(subs( \{op(d8),c[21]=c_21\},d[20,i]*c[21]^i),i=1..7)=0:\ndd := \{d[1,1]=1,se q(d[1,i]=0,i=2..6),seq(d[9,i]=0,i=1..6),seq(d[12,i]=0,i=1..6),seq(d[13 ,i]=0,i=1..6)\}:\nsol := solve(subs(dd,\{eq1,eq2,eq3,eq4\}));\ndd_8 := `union`(subs(sol,dd),sol):\nd_8 := expand(`union`(subs(dd_8,d8),dd_8) ):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<&/&%\"dG6$\"#7\"\"(,&#\" R+](yUXIh#*p(*=63aJ5EVPyFKY=\"MLnB`F:W&Qk%GuL[^KH\"*y$*R7!\"\"*(\"N]7` %)f6Me_PK_-4zS41W!*y1#\"\"\"\"J$>@Xbz+q5k>E\")*)=y)f%>)z$F0F+#F3\"\"#F 3/&F(6$F3F+,&#\"Vv$f1l2_9LU9xs*=\\aY$\\Ady-`*Q](\"QV/FzA9/\")3`P2Kgu)) =p-!*)R7,\"F0*(\"Pvo/(z*\\_hr$>\"f#zc8Vk`u&Q6.$F3\"M@L>99EanB]Du#\\VGh c1*RQ6F0F+F5F3/&F(6$\"#8F+,&#\"@vVBVIT['*))p4'>><%)\"DJqT*GF\"G!Q0vl@=F3\";o_&3qoTLh2mr5$F0F+F5F3/&F(6$\"\"*F+,&#\"g rDJl#*=M^VnWl,2Xi[D%>TWEs,Ln$=zy`B[1VQhPiLU!fTYFzy$*o@=0H6n$H-,M>(4$\" `ri0W](3jVk&)*\\%o/3R$oy)Qo&4fk%z_3>H_Kz2NSy`+$*=zn17ktBQ@-fm$fbo(pYF3 *(\"drD\"y?VtEO2(*\\*3?)He()Gr/Gvwjn`,(yk5O^;fc*f$fh#pT#R,]&4DXM-B%>]o (y(fKt&F3\"`r8!*[K(*4d?())\\#4^x#4%R!p(3m9Kz&>F&y+o7mKhX(o8O([O=QBvNHV R " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "d_8 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7017 "d_8 := \{d[11,4] = \+ 0, d[4,4] = 0, d[14,4] = -49944488129380999000/7625255514473691+455985 33047946550000/7625255514473691*7^(1/2), d[10,3] = 0, d[6,1] = 0, d[3, 2] = 0, d[3,7] = 0, d[14,2] = -19514630672107725/31379652322937+178165 51234151250/31379652322937*7^(1/2), d[20,7] = -10353021078824656250000 000/77485466768510711979*7^(1/2)+38441485506713075000000000/7748546676 8510711979, d[19,3] = 126297650/80703*7^(1/2)-37602144710/8796627, d[9 ,1] = 0, d[3,3] = 0, d[6,5] = 0, d[10,5] = 0, d[14,6] = -4715584525334 3750000/7625255514473691+43052545910937500000/7625255514473691*7^(1/2) , d[12,7] = -1846322778374326103154081118976992613045427875000/1239937 8912932514833742846438544152753236733+20678904406094079090252323752583 4115984531250/37981945987818898126196410700079554521193*7^(1/2), d[1,7 ] = -75038953027857224934654491897277144233145207650659375/10112398900 2691888746032073753088104142279270443+30311385745364431356792591193716 152499797046875/11383990656612843492742550236754261414193321*7^(1/2), \+ d[19,7] = -1875259437950102500000000/25179876387637713+207060421576493 1250000000/75539629162913139*7^(1/2), d[9,4] = 0, d[9,5] = 0, d[9,6] = 0, d[9,2] = 0, d[9,3] = 0, d[3,5] = 0, d[5,4] = 0, d[3,6] = 0, d[5,1] = 0, d[8,1] = 0, d[2,2] = 0, d[4,7] = 0, d[5,6] = 0, d[13,7] = -84171 91960969889648413043234375/264109116471340433957269778+182165750538028 12728941703125/31071660761334168700855268*7^(1/2), d[9,7] = 3097193401 0229367112905182168937879274641590423362376138430648235378791836733017 22644411942548624507016544674351341892653125/4669768555936659022138237 3641206677918930053784035077932522919085279464590956838878683390804684 4998564436308750440562+57332597787685019423023445250955001392416926159 3599565916513610647870153676376752804712887582982008949970736267343207 8125/36691038653788035173943293575233818364873613687456132661268007852 7195793214660876903940927751092498872057099732489013*7^(1/2), d[13,2] \+ = 0, d[13,5] = 0, d[13,4] = 0, d[13,3] = 0, d[4,3] = 0, d[13,6] = 0, d [18,1] = 0, d[4,6] = 0, d[6,2] = 0, d[19,1] = 0, d[11,5] = 0, d[7,4] = 0, d[2,1] = 0, d[10,7] = -5991318309675943986149313133370161825150935 66894531250000/64814431762349485749053274400377793835741885220759+7124 52527580852315761159145787226007404785156250000/2017255890518191277592 6945035909677508789880243*7^(1/2), d[6,3] = 0, d[18,4] = 51860545600/4 667949+2111918656000/32675643*7^(1/2), d[4,1] = 0, d[3,4] = 0, d[11,1] = 0, d[11,2] = 0, d[5,7] = 0, d[2,3] = 0, d[4,5] = 0, d[15,3] = 36334 5751953125/14477866209*7^(1/2)-1569572998046875/535681049733, d[20,4] \+ = 164993645000/9023220747*7^(1/2)-22525692625/291071637, d[8,2] = 0, d [16,2] = -162204527343750/5677737381589-3597509765625/1622210680454*7^ (1/2), d[14,3] = 20835893147585019350/7625255514473691-190228431175917 57500/7625255514473691*7^(1/2), d[3,1] = 0, d[17,1] = 0, d[12,1] = 0, \+ d[2,4] = 0, d[1,4] = 0, d[15,5] = 1333376757812500/59520116637+4804125 976562500/59520116637*7^(1/2), d[1,5] = 0, d[17,6] = -4203170000000000 00000/39409020822118563-249250000000000000000/39409020822118563*7^(1/2 ), d[1,1] = 1, d[8,6] = 0, d[15,1] = 0, d[1,3] = 0, d[4,2] = 0, d[1,2] = 0, d[16,3] = 13443792822265625/1379690183726127*7^(1/2)+13030750073 2421875/1379690183726127, d[7,7] = -2036072455324413244555046839332144 235015692882136718750000/325770513641280980934653015578402702354561849 8842497+20853876405515951597113252829089886732301321794531250000/46538 6448058972829906647165112003860506516928406071*7^(1/2), d[11,3] = 0, d [13,1] = 0, d[16,6] = -181627685546875000/1379690183726127-30426025390 625000/1379690183726127*7^(1/2), d[19,2] = -16577325/46543*7^(1/2)+315 734571/325801, d[15,4] = -2633663037109375/535681049733-32225321289062 500/535681049733*7^(1/2), d[2,5] = 0, d[5,3] = 0, d[8,5] = 0, d[8,7] = -1205404778832880234373343291831201781008811218261718750000/183728597 094935485707637035297742367455639137745803+439819070222159130775052749 598485575559082031250000/300962532302873991691052853207761835070747354 9*7^(1/2), d[6,6] = 0, d[19,4] = -32998729000/8796627*7^(1/2)+12939899 725/1256661, d[16,5] = 3597064648437500/17033212144767+686303710937500 /21899844186129*7^(1/2), d[1,6] = 0, d[2,6] = 0, d[12,2] = 0, d[17,2] \+ = -16931784000000000/23168148631463-14735400000000000/23168148631463*7 ^(1/2), d[5,2] = 0, d[20,1] = 0, d[6,4] = 0, d[2,7] = 0, d[20,2] = 193 402125/111397787*7^(1/2)-6583682665/1002580083, d[6,7] = -226773442871 572359032157632172314794134775249517968750000/469351024421833644007258 749632746869132121193365179+931716126638250064214687470667469466082884 69843750000/2483338753554675365117771162078025762603815837911*7^(1/2), d[15,6] = -13931884765625000/535681049733-30426025390625000/535681049 733*7^(1/2), d[18,5] = -5493440000/172887-44977600000/518661*7^(1/2), \+ d[19,5] = 702775000/139629*7^(1/2)-40750420000/2932209, d[10,2] = 0, d [16,7] = 50531969462480935782011370147609710693359375/3695626322800088 82350699766204137734371*7^(1/2)+28236476030319408576448361361806343078 61328125/739125264560017764701399532408275468742, d[20,5] = -245971250 00/1002580083*7^(1/2)+37492550000/334193361, d[17,7] = 165648337261194 50000000000000000000/338418670994729972076890691*7^(1/2)+1804385877668 1440200000000000000000/338418670994729972076890691, d[14,7] = 26402098 38669252395844543388687657848394187500/5520255185559626606255824616473 0797939-2414240754361416940713337325473694209023437500/552025518555962 66062558246164730797939*7^(1/2), d[15,2] = 11277287109375/6613346293-7 5547705078125/13226692586*7^(1/2), d[12,6] = 0, d[12,3] = 0, d[12,4] = 0, d[12,5] = 0, d[10,1] = 0, d[17,4] = -1757807800000000000/181608390 885339-263989832000000000000/39409020822118563*7^(1/2), d[8,3] = 0, d[ 7,1] = 0, d[17,3] = 143031662000000000000/39409020822118563+1010381200 000000000/361550649744207*7^(1/2), d[7,2] = 0, d[20,3] = -1860330250/2 43870831*7^(1/2)+91158900050/3007740249, d[7,6] = 0, d[7,3] = 0, d[19, 6] = -31156250000/8796627*7^(1/2)+86612500000/8796627, d[18,2] = -6672 51776/518661+353649600/57629*7^(1/2), d[15,7] = 3756959278592109807117 608249581623077392578125/8547175600486592321033140365559941228*7^(1/2) -11746245388677249627782625531031699371337890625/726509926041360347287 81693107259500438, d[17,5] = 8856843200000000000/625540013049501+56222 00000000000000/625540013049501*7^(1/2), d[5,5] = 0, d[18,7] = -1325186 69808955600000000000/280596864557260251*7^(1/2)+3719688788556508000000 0000/280596864557260251, d[20,6] = 155781250000/9023220747*7^(1/2)-781 750000000/9023220747, d[18,6] = 1085800000000/32675643+1994000000000/3 2675643*7^(1/2), d[14,1] = 0, d[11,6] = 0, d[16,4] = -3628915610351562 5/197098597675161-32225321289062500/1379690183726127*7^(1/2), d[16,1] \+ = 0, d[18,3] = -881052406400/32675643*7^(1/2)+2606815360/3630627, d[14 ,5] = 7445685666132175000/847250612719299-6797794043503750000/84725061 2719299*7^(1/2), d[7,5] = 0, d[11,7] = 2409033728143605897115148197055 8402588486047460937500/2581342802687397986555350696054987258248354531- 57246100635282868964005337900506786794179687500/2905935835514350992407 23932911740094365457*7^(1/2), d[10,6] = 0, d[8,4] = 0, d[10,4] = 0\}: \+ " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(d_8,matrix([seq([seq(d[j,i],j =1..10)],i=1..7)])):\nevalf[8](%);\nsubs(d_8,matrix([seq([seq(d[j,i],j =11..20)],i=1..7)])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7)7,$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+7,F+F+F+F+F+F+F+F+F+ F+F,F,F,F,7,$!)K/]t!\"#F+F+F+F+$!)s)*QQF0$!)ZYk]F0$!)o9uh!\"\"$\")kxtm F7$!)0O]\"*F7Q(pprint86\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7)7,$\"\"!F)F(F(F(F(F(F(F(F(F(7,F(F(F($\")&3I!))!\"&$!)nmS8!\"$$!)* )eVM!\"'$!)!\"(7,F(F(F($!)A\"z' QF6$\")4%pM'F0$\")IF-7F-$\")rJ-6F0$!)G4iqF0$!(?4M\"F6$\")1a75F37,F(F(F ($\")Par#*F6$!)qyS;!\"#$!)Q8fCF-$!)ZASFF0$\")E7@=FR$\"('f?PF6$!)x+,HF3 7,F(F(F($!)^(RC\"F0$\"):_fBFR$\")!G4%HF-$\")z!Qz$F0$!)\"3@h#FR$!'F5eF0 $\"))zxs%F37,F(F(F($\")\"oQv)F6$!)A$Gw\"FR$!).!***=F-$!)q!*RFF0$\")4%o %>FR$\"(WFv%F6$!)[+'4%F37,$\")uF6))!\"\"$!)#**\\M\"FR$!)*)*=.$F0$!).A) y'F)$\")WF,5\"\"#$\")8-#=%Fhp$\")t@G=\"\"\"$!)o&p6\"Faq$!(zA&>F)$\")(p gU\"FRQ(pprint96\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 47 "Evaluate the \"weight\" polynomials at the node " } {XPPEDIT 18 0 "c[21] = 97/1000;" "6#/&%\"cG6#\"#@*&\"#(*\"\"\"\"%+5!\" \"" }{TEXT -1 82 " to obtain the linking coefficients in the next stag e in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "eqs21 := \{seq(a[21,j]=add(expand(subs(\{op(d_8),c[21]=c_21\},d [j,i]*c[21]^i)),i=1..7),j=1..20)\}:\ne9 := `union`(eqs21,\{c[21]=c_21 \},e8):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e9 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25409 "e9 := \{a[9,6] = 7801251558438936413230905525304310365677955925 68497182701460674803126770111481625/1831104254127319721978898745071587 86859226102980861859505241443073629143100805376, a[11,5] = -2605308595 9256534152588089363841/4377552804565683061011299942400, a[11,8] = 1610 21426143124178389075121929246710833125/1099720772213103465066704136434 6422894371443, a[21,20] = 0, a[18,17] = 0, a[10,7] = 31860723517364931 2405151265849660869927653414425413/67147167155589653031329380729354654 23910912000000, a[8,5] = -1024030607959889/168929280000000, a[18,5] = \+ 0, a[16,7] = 134784801417938786440798485553519553514344335339997093625 781306315144387570974417447025422991363808/146423721484895907737988852 8329155065846449561947146728294932984499575946659768049806174092837890 625+373311550922081319948726157866222874528305458717842546777336523731 44522983528253234993840141021024/7321186074244795386899442641645775329 232247809735733641474664922497879733298840249030870464189453125*7^(1/2 ), a[4,1] = 1023/25600, a[2,1] = 1/20, a[11,9] = 300760669768102517834 2324975654524349466722661958764963718742623926848522439253598648849625 13/4655443337501346455585065336604505603760824779615521285751892810315 680492364106674524398280000, a[12,6] = 2346305388553404258656258473446 184419154740172519949575/256726716407895402892744978301151486254183185 289662464, a[12,5] = -4583493974484572912949314673356033540575/4519577 03655250747157313034270335135744, a[20,8] = -4883846532098226912322417 6957197891703125/6339343872926013442473709597206654708616+575759761470 57867393261247514690821875/333649677522421760130195241958244984664*7^( 1/2), a[10,4] = -20462749524591049105403365239069/45425191349989346959 6231268750, a[13,5] = 0, a[7,1] = 21173/343200, a[7,2] = 0, a[7,3] = 0 , a[14,3] = 0, c[14] = 1/2-1/14*7^(1/2), a[8,2] = 0, a[8,3] = 0, a[13, 3] = 0, c[13] = 1, a[11,3] = 0, a[14,4] = 0, a[13,4] = 0, a[9,2] = 0, \+ a[14,2] = 0, c[12] = 1, a[10,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a [12,2] = 0, a[11,2] = 0, a[10,2] = 0, a[12,3] = 0, a[4,2] = 0, a[14,5] = 0, a[6,2] = 0, a[8,7] = 6070139212132283/92502016000000, a[8,6] = 1 501408353528689/265697280000000, a[6,3] = 0, a[9,3] = 0, c[6] = 93/200 , a[9,5] = -1207067925846925480797893644173318794948457151612046996653 4514296406891652614970375/27220311547616572217104781845311006994972840 85048389015085076961673446140398628096, a[5,2] = 0, a[13,2] = 0, a[12, 8] = 345685379554677052215495825476969226377187500/7477116743693007722 1667203179551347546362089, a[12,7] = 165712155931984680217128369091361 0698586256573484808662625/13431480411255146477259155104956093505361644 432088109056, c[5] = 39/100, a[14,12] = 11352128098668146659861/254668 911904014019468056-5215842639928607924801/127334455952007009734028*7^( 1/2), a[3,1] = -7161/1024000, a[19,17] = 0, a[19,18] = 0, a[6,4] = 317 44/135025, a[7,4] = 8602624/76559175, c[7] = 31/200, c[8] = 943/1000, \+ c[9] = 7067558016280/7837150160667, c[11] = 47/50, a[6,5] = 923521/510 6400, a[10,6] = 21127670214172802870128286992003940810655221489/467947 3877997892906145822697976708633673728000, a[7,6] = 5611/283500, a[8,1] = -1221101821869329/690812928000000, a[13,9] = 1227476547031319687842 8812037740635050319234276006986398294443554969616342274215316330684448 207141/489345147493715517650385834143510934888829280686609654482896526 796523353052166757299452852166040, a[4,3] = 3069/25600, a[5,1] = 42023 67/11628100, a[8,4] = -125/2, a[3,2] = 116281/1024000, a[12,4] = -1695 7088714171468676387054358954754000/14369041511965468332636822810157022 1, c[10] = 909/1000, c[4] = 1023/6400, a[5,3] = -3899844/2907025, a[7, 5] = -26782109/689364000, a[13,8] = -13734512432397741476562500000/875 132892924995907746928783, a[13,7] = 3847749490868980348119500000/15517 045062138271618141237517, a[13,12] = -306814272936976936753/1299331183 183744997286, a[10,5] = -180269259803172281163724663224981097/38100922 558256871086579832832000000, a[11,1] = -234265984581408683695120714006 5609179073838476242943917/13584809613510567770222314001391587608575321 62795520000, a[12,11] = -6122933601070769591613093993993358877250/1050 517001510235513198246721302027675953, a[11,7] = 8907229937563791864189 29622095833835264322635782294899/1392124200139511265750194195559401382 2830119803764736, c[2] = 1/20, a[13,11] = 282035543183190840068750/122 95407629873040425991, a[13,10] = -9798363684577739445312500000/3087229 86341456031822630699, a[11,10] = -31155237437111730665923206875/392862 141594230515010338956291, a[12,9] = -320589096271707254279143431215272 7534008102774023210240571361570757249056167015230160352087048674542196 011/947569549683965814783015124451273604984657747127257615372449205973 192657306017239103491074738324033259120, a[11,4] = -996286030132538159 613930889652/16353068885996164905464325675, a[12,1] = -286655699182566 3971778295329101033887534912787724034363/86822671161926270301121392501 6143612030669233795338240, a[13,1] = 44901867737754616851973/101404640 9980231013380680, a[10,8] = 212083202434519082281842245535894/20022426 044775672563822865371173879, a[12,10] = 402795458327062334331004385884 58933210937500/8896460842799482846916972126377338947215101, c[3] = 341 /3200, a[13,6] = 791638675191615279648100000/2235604725089973126411512 319, a[6,1] = 5611/114400, a[5,4] = 3982992/2907025, a[11,6] = 2098082 2345096760292224086794978105312644533925634933539/37758899920075508038 78727839115494641972212962174156800, a[10,9] = -2698404929400842518721 166485087129798562269848229517793703413951226714583/469545674913934315 077000442080871141884676035902717550325616728175875000000, a[9,8] = 10 332848184452015604056836767286656859124007796970668046446015775000000/ 1312703550036033648073834248740727914537972028638950165249582733679393 783, a[9,4] = -5172294311085668458375175655246981230039025336933699114 138315270772319372469280000/124619381004809145897278630571215298365257 079410236252921850936749076487132995191, a[10,1] = -290555733603374150 88538618442231036441314060511/2267475989108957769132796260237059763200 0000000, a[9,1] = -147251426448621580388138470887726424634604443330709 4207829051978044531801133057155/12468948016200320011570596216439860248 03301558393487900440453636168046069686436608, a[9,7] = 664113122959911 642134782135839106469928140328160577035357155340392950009492511875/151 7846559858624813633302310729534917527976515008907830113994325301687782 3170816, a[20,4] = 0, a[20,5] = 0, a[16,1] = 3087219077495121428802855 7378364407130713819647659749896091003538506226381243197027201011989111 89933/6036605742182300353859642712070094109199473444076714719673219506 7996608532430377691588769378662109375-89168107892094897373289127357971 5129828451185262302853368713631052223573903665061810906114902/11436214 3453297344962766746463391003300169999887784687310281699475223280349399 2188909515380859375*7^(1/2), a[20,17] = 0, a[20,18] = 0, a[20,19] = 0, a[20,11] = 4051052479952572002583490413428916431/35626505484461889592 5779730716311328-414373616330969707208269617880554327/1781325274223094 479628898653581556640*7^(1/2), a[14,11] = 2970221666030903358562614446 315/4589064694536869684100726-2245277482452996546801919407985/91781293 89073739368201452*7^(1/2), a[14,10] = -2079668361431216455673395233734 375/5957776266668670748330679562+224582419744760612736276500828125/170 2221790476763070951622732*7^(1/2), a[14,8] = -878979792896006748962940 0558015625/16279513475811535751534066502+66444706625541840453925485405 78125/32559026951623071503068133004*7^(1/2), a[14,7] = 127478752914376 1017157317260602825/26341935917422215554074218244182-95892953147116173 2045165837664925/52683871834844431108148436488364*7^(1/2), a[14,1] = - 942013/61397+2856761/491176*7^(1/2), a[14,6] = -1090956177631910962162 375596685/8605876253572154615648617314+4042115155384508751901172753298 5/843375872850071152333564496772*7^(1/2), a[16,3] = 0, a[16,2] = 0, a[ 16,4] = 0, a[20,1] = 16764524349182732858436037634017802692437/2907639 08570534356791222465076565558000000+1828397973533521045712614092746349 80229/58152781714106871358244493015313111600000*7^(1/2), a[16,5] = 0, \+ a[15,8] = 598933003512786415989929347488/28972691439853083012428515625 -14751096540440194135375671464064/1371942153475395989706173828125*7^(1 /2), a[21,4] = 0, a[15,2] = 0, c[19] = 7/10, a[19,2] = 0, a[19,3] = 0, a[19,4] = 0, a[19,5] = 0, a[15,3] = 0, a[15,4] = 0, a[15,5] = 0, a[20 ,12] = -1007835712877603197752652668370693383/106956576143629471286459 64365089400000+1137412042683412511904456412390473/17692065226765777205 2790387993960000*7^(1/2), a[17,15] = -1927193513548850260413881071/230 77873387050840510431232000+8354781001061139440464823/36924597419281344 8166899712*7^(1/2), a[21,19] = 0, c[20] = 9/10, a[14,9] = 998747057824 8573387297254667659422896963367899428512772582082944283962013964900384 895127596133/412956139811076418684858210810009302612273610727456053052 02526150545875206586329586453004-3774897973447442754802925721321508929 613447293787235365983123527137197728406982036112477728851/412956139811 0764186848582108100093026122736107274560530520252615054587520658632958 6453004*7^(1/2), a[20,10] = -12909998295917152763260545013409460140625 /745449134151432943997700950933140329016+44412363987143248679166880201 97971875/106492733450204706285385850133305761288*7^(1/2), a[17,16] = 6 1561095821428811034532925699/1195262539043064700790636544000+197966374 530935184348002389/4781050156172258803162546176*7^(1/2), a[17,1] = 832 8855883002117418483622903726420956080895292363251/19833159237232238231 6539113020646822720000000000000000-47729296069289226212890882457962332 57786356432779/1469122906461647276418808244597383872000000000000000*7^ (1/2), a[16,8] = -7485568824275587414083752299433529478296018065218092 1502875834466293853598523210706487111328/42281110743020950346050637015 918592082885943200894919376755357202712416979579385828158093167+408013 7020543686228207413259382126675613920337919418028714789205954679428753 502302302598432/111982770634749105713542029557374014966538327402997392 19625433617974091284107600704550728125*7^(1/2), a[21,5] = 0, a[16,12] \+ = -5882069488737495673212378317062005515483798761806118071747190832971 8193157553959653059936964184/95788265884218939998640457949792763474064 3475994446960507133858571726061567913677743743896484375+53307859045179 8558296277441412867835669059233030547647155487599441001684066360781057 8383524/37344353171235454190503102514539089073709297309725027700083191 3673187548369556989373779296875*7^(1/2), a[19,6] = 1873755372063136281 0484177932773607619285/62464180376521195399214582561154658660968+41110 4910959722573223373581396660538073/11567440810466888036891589363176788 640920*7^(1/2), a[19,8] = 3638715618179011339460790182555028109375/128 6934470443626789073610218981802083704+65981475839123424774827083281674 59375/47664239646060251447170748851177854952*7^(1/2), a[17,10] = 56160 702543048579126936480152903088846798498167/123661065966112908213890615 348396782899806208000-704890678912714132819876715191341334116124219/16 357283857951442885435266580475764933836800000*7^(1/2), a[17,14] = 1066 57925843659476662976498724178969/2688984574690119078864000000000000000 +67590823874770618986810696149161/1434125106501396842060800000000000*7 ^(1/2), a[16,15] = 220298346818480482377346972854658276607772183892089 0760304290760481052156/33631587484747455337411471242979551680712577077 922629054937124338769013375-101352035588239289581679001655920771460246 57433459350553499494488454213/5381053997559592853985835398876728268914 01233246762064878993989420304214*7^(1/2), a[16,10] = -2320802540679497 6580141541236033062760809117465628333520857807344302247788462202697113 85730784/1864452656269026128785606340617883560192919637622678890647626 052665702123772234282536764081375+581815854529984095367031738933581762 3809315384767986611466857849527752329675942295310238176/46657974381106 7599796197782937408298346576485891561283945852365531957488431490060694 88590625*7^(1/2), a[15,10] = 54365264025507653851977480738208/44664783 90356713423359819140625-1496294528311131416682934929792/23578694368398 9396406626953125*7^(1/2), a[17,6] = 9608501774836052499507564485849861 825840507225298757/699598820217037388471203324684932177002841600000000 00-31215118821227947550703658120770299949941405900809/6818701951433112 94806241057197789646201600000000000*7^(1/2), a[19,15] = -7521783452765 791015625/428202728861294892283392*7^(1/2)+31644645563653568359375/428 202728861294892283392, c[17] = 373/1000, a[19,9] = -490961855327772037 3034307553189869129727494293024577956609168005460871544614523472721062 9741675867657970337027809/21363447561006569285835679199426078107058296 855987811280032296414873257374798633002620343692239729710296554500000+ 2529709622354434166733072478895385019057235132321493270324842904960931 92153494909971810817233379463450726430467/1709075804880525542866854335 9540862485646637484790249024025837131898605899838906402096274953791783 768237243600000*7^(1/2), a[16,11] = 1415263906459801114947844058254027 1096903459559094245113706185285107457868609157190831336140072/58011668 3018675455451405137105948008858244469536204503487573592460211057718555 5412784755859375-23059347376346933786594094414430975662069875723080237 434871217181706005525496230157407214632/467082675538386034985028290745 52979779246736677633212841189500198084626225326533114208984375*7^(1/2) , a[18,11] = -35975392605857817344162998122286881735/45601927020111218 678499805531687849984+67001279346027425501202673610603822817/228009635 100556093392499027658439249920*7^(1/2), a[21,7] = -4904295867429735479 812719542086693147459239507974243/971165098811432028706105245784848511 13042176000000000+251154077567359473378865402712200632658205615058873/ 69368935629388002050436088984632036509315840000000000*7^(1/2), a[15,12 ] = -1927555453883917676542797699/276054308640062825012207031250, a[17 ,13] = 5929009119114323566893394932293/1968360743640880256000000000000 000-2816129290207834445979834640567/3936721487281760512000000000000000 *7^(1/2), a[15,11] = -3598066799092816058284144620546/1462869641026789 84417724609375+65570869112872951387443794088672/5120043743593764454620 361328125*7^(1/2), a[18,2] = 0, a[18,3] = 0, a[17,7] = 102563335109662 194461557406821662279724615611105469/618576496058236960959302704321559 561229568000000000-265492203346789444504866948767894427719165283112249 41/485582549405716014353052622892424255565210880000000000*7^(1/2), a[1 9,1] = 174871013236725476061563224349045572286953/33645537991733261285 84145667314544314000000+62859757552457393127856737157320807163/2492262 0734617230582104782720848476400000*7^(1/2), a[17,9] = -374007328891923 3268483245286897966010855585707988160778840601948021591448577248728733 8717828366357415478453578758816356450881/19141649014661886080108768562 6857659839242339829650789069089375877264386078195751703478279482467978 204257128320000000000000000-331775197314081720003881018252916142261455 4453563495416261652293406587562445693191678520005660659000252499899676 7787117019/17401499104238078254644335056987059985385667257240980824462 67053429676237074506833667984358931527074584155712000000000000000*7^(1 /2), a[16,6] = -858912243102433135883521643728777502484576988133901999 74517211560164597559644901403704311739138208/1054793494205986431677613 6894419597979528106546537572266549401570835412223968959199104892971777 34375+1979683665557887822471642625237777109412751193262187274175314223 12697579876941533185813932673312/5279246717747679838226294741951750740 504557831099886019293994780197904016000480079632078564453125*7^(1/2), \+ a[17,12] = 2902460704378612529880216802791069594834177328387/127064412 458631811888314056657262072000000000000000-170951053384965232910247435 69399504075413989/2573391305711385775313314734457600000000000000*7^(1/ 2), a[19,13] = 307244454352750466699/24604509295511003200000+136642154 48604158347/24604509295511003200000*7^(1/2), a[20,15] = -3125509446488 994140625/142734242953764964094464*7^(1/2)-8926304951721287109375/1427 34242953764964094464, a[17,11] = -953162023511450865750207253165636157 0671453261/9802962018654481722288970934359649280000000000+360566304554 5978810333832319405532352790156659/15000633888194479828453883398581529 600000000000*7^(1/2), a[15,6] = 4116869726480787181292612488487584/265 98534836082808007568743896484375-245534888302358312351817429197184/870 0455320214002619298187255859375*7^(1/2), a[16,9] = 9357912776729479180 8027495837594430907037639266838370557806449258372713062631705287581598 3175941371036570096600014166716414833259641089083213037080522463479597 899838335379/153319242808682647314993371077588359843114731426490726766 7118160806408566260202450651306189032792402646266006906060260285794563 433180425273922088634890450322720794677734375+238907136665464198200669 5791254226789546298791129269363741142160105178869350417843132507667642 9205498957704908676509148666454898775656889740344774877752316006821913 2/78424165119530765890022184694418598385224926560864821875555916153780 4893227725038696320301295545986008320208136092204749767040119273874820 420505695596138272491455078125*7^(1/2), a[20,7] = 13248444953061535702 27909695456980315126025/3034890933785725089706578893077651597282568+16 0585056017693402348695205888904986106867/30348909337857250897065788930 77651597282568*7^(1/2), a[19,12] = 17069492067141808117022454818913038 29/23884288056133799227126702379184600000+7018644175110524085731927532 2903/13609280943665982465599260614920000*7^(1/2), a[21,8] = -185821651 993207932931584606746046915502275047467/350540358797316839315026245887 139178767630336000+1695030325952632939095028253070192959666341/1435534 74698722639652655431834135657267200000*7^(1/2), a[17,8] = 248147914310 376513887773097942267349975604062017/350540358797316839315026245887139 178767630336000-3046056343462567590977832367296624503866287449/1708286 3489147994118665996388262143214796800000*7^(1/2), a[16,14] = 502890395 059180530649303173309978152773646321070353812284039761435107016209192/ 2542054733210037682116167947811317216578628177450372866599228752869900 5126953125-14058401811014285262962125828643550336789285498002679613358 69008409791765544/4067287573136060291385868716498107546525805083920596 5865587660045918408203125*7^(1/2), a[16,13] = -14781724992654556932356 69169495754925970922405664549846419638317907395087946/1754564777286668 31489950690673131923190567109284028983890324355658905029296875+5748782 02234070155064691566753530747289127904338643775445570927408156648849/3 5091295545733366297990138134626384638113421856805796778064871131781005 8593750*7^(1/2), c[18] = 9/20, a[18,15] = -137805077367430365234375/45 67495774520478851022848+505372743991484208984375/182699830980819154040 91392*7^(1/2), a[18,1] = 1614143524969292517000674483565805268889123/3 7217780297028397669276475529800391424000000-29563900439689757729410425 758264271687459/7443556059405679533855295105960078284800000*7^(1/2), a [15,9] = -306706645966127099400049304854628613540749389913325084152909 2716875641340142781006771180215525814345889446383/37334071921822778141 0511653246697185431540894078529094301526280820101435128911405410349160 893280029296875000+181664761198407694024184205976438800959946932614625 4865545587094445401874686830164148895260773137/42447565466790579709675 1747915993683544031508184382874685996435033916010685878289836883544921 875*7^(1/2), a[17,2] = 0, a[17,3] = 0, a[17,4] = 0, a[17,5] = 0, c[21] = 97/1000, a[15,13] = -15630718068/3814697265625+6167095848/381469726 5625*7^(1/2), a[18,12] = 12504555537942121098319086045547445973/579210 996962424213735904531770995200000-16719224785624791190523597280063453/ 2058713044569108620250651787566080000*7^(1/2), c[16] = 14/125, a[18,6] = -580046684768304199644873721295749756809267/10364426966178331681054 864069406402622264320*7^(1/2)+3061845663376807113844125374613739055118 15/2072885393235666336210972813881280524452864, a[21,17] = 0, a[18,7] \+ = 5274418898058305991498806821071039010281465/298820030403517547294186 22947226108034782208-25965466364164351851299808765056452592165157/3884 66039524572811482442098313939404452168704*7^(1/2), a[20,14] = -2150429 62723876515301101/5602051197271081414300000-408828436443173054007/8963 281915633730262880*7^(1/2), a[19,7] = 16269926845488949676102335114128 4149876845/433555847683675012815225556153950228183224+1415606313445844 287537299717843980987691/33350449821821154831940427396457709860248*7^( 1/2), a[20,9] = 760136224957331827494281176397770174855407999069671818 347506927110334545165817153657233758784359977604937627710651/543796847 0074399454576354705308456245433021017887806507644584541967738240857833 8552124511216610221080754866000000+22074453325473942468652332599559476 6593505131602603266182755091973726065107541719285871085066538223107063 9000183/11963530634163678800067980351678603739952646239353174316818085 9923290241298872344814673924676542486377660705200000*7^(1/2), a[19,10] = 29317433704285468516504401709896572546875/8625911409466581209116253 860797766664328+822167960519864311728648427688384375/24575246180816470 681242888492301329528*7^(1/2), a[20,16] = -14699461699626275390625/972 70714440353572655488-3897836105080869140625/97270714440353572655488*7^ (1/2), a[18,10] = 67477811248813361131627146239991609375/4317533446668 93288831247609590235122688-718116474757235797245927435911971003125/136 31069881626202404529388817063137444864*7^(1/2), a[21,12] = -1389764081 73792274436613993987636187969155447007/1155131022351198289893764151429 6552000000000000000+12452339939644108170251783308035668462710109/28307 304362825243528446462079033600000000000000*7^(1/2), a[20,2] = 0, a[20, 3] = 0, a[18,9] = 7367415963180026585066952709494583829554893277300365 732284398188698049048102708090175047446909193037260623375725413/153133 1921172950886408701485014861278713938718637206312552715007018115088625 56601362782623585974382563405702656000000-3244802528182022448862519975 4712079672076342032563550205311186269577863651133841452427139849535918 572768985697763/139211992833904626037154680455896479883085338057927846 5957013642743740989659605466934387487145221659667324569600000*7^(1/2), a[18,4] = 0, a[18,13] = 25110392089861989207/10639787803464217600000- 2754209330453161502559/3149377189825408409600000*7^(1/2), a[21,9] = 93 2527766727193968529039548888116819504328717738933110676780701589372967 6538270690675607029580802555685974029110968277751441/17401499104238078 2546443350569870599853856672572409808244626705342967623707450683366798 43589315270745841557120000000000000000+3452431440539701425607856616743 9078918480054926938741220534250000549185766022584749693078779316977422 14575780825236635077/2734521287808840868586966937526537997703461997566 4398438441339411037769439742250243354039926066854029179589760000000000 00000*7^(1/2), a[15,7] = 631476617959696547824260385843267488/27059579 14016858192337687904052734375-33227466731347290455944055132270208/3922 838396059942349838659979248046875*7^(1/2), a[18,14] = 3477092423121768 0892510971/717062553250698421030400000+66104663026735794404097/1147300 085201117473648640*7^(1/2), a[21,2] = 0, c[15] = 57/125, a[21,18] = 0, a[20,13] = -522000512739051727257/24604509295511003200000+17033580426 103348329/24604509295511003200000*7^(1/2), a[15,14] = 172678683744/381 4697265625*7^(1/2), a[21,3] = 0, a[18,16] = 27007176860389880859375/67 3008186398122016210944+630252495402507333984375/1245065144836525729990 2464*7^(1/2), a[19,11] = -5475515448010595309954301079014631075/137416 5211543530027142293247048629408-47486789775789501503363164649095523/25 4475039174727782804128379083079520*7^(1/2), a[21,16] = 885306012611864 85384283727/794194378101704120126668800-43552334168255874111619/158838 87562034082402533376*7^(1/2), a[21,11] = 11605298354228832197609691845 68238130969933929267/1539065036928753630399368436694464936960000000000 -13788891158460838760507551342230229937828779/866298005701200962737458 311772185600000000000*7^(1/2), a[18,8] = 27899837334492834306211688588 370551859375/47731530337325277684507931084850106041344-930962761482389 1860310196472043443353125/42707158722869985296664990970655358036992*7^ (1/2), a[20,6] = 7231245850603923375549468974817821336739/161944171346 53643251648225108447504097288+3587335118885057093042630057609435084277 /80972085673268216258241125542237520486440*7^(1/2), a[21,6] = -2731150 494692639137450971086219916669458048413074647/699598820217037388471203 32468493217700284160000000000+5610570902745148706685436423366540932079 271761663/1850790529674702085902654298108286182547200000000000*7^(1/2) , a[21,13] = -5068630986289004268922772722703/196836074364088025600000 0000000000+10969584299598762557863199021/23157185219304473600000000000 0000*7^(1/2), a[21,10] = -92360399217853983892568288906247950520885499 837/123661065966112908213890615348396782899806208000+27457396047905748 74213067326550839092703897/9621931681147907579667803870868097019904000 00*7^(1/2), a[19,14] = -517517744336749012103429/168061535918132442429 00000-327959225091317538901/8963281915633730262880*7^(1/2), a[15,1] = \+ 57196651428918572875631366996149/1245822888041363915313720703125000+10 891702347425878099023621466959/8357395207277482931896209716796875*7^(1 /2), a[21,15] = 431350947507910891770735349/10197199868696883016237056 000-14379963913139300630917/9597364582302948721164288*7^(1/2), a[19,16 ] = -127299854452380859375/1302732782683306776636-13400640128203808593 75/41687449045865816852352*7^(1/2), a[21,1] = 139591841849675389198129 068733378286244267757188782701/376830025507412526401424314739228963168 0000000000000000+18252789022600571816707935394667217847717786499/84842 964202952274322058834794377792000000000000000*7^(1/2), a[21,14] = -706 2844350975267650463930634528499/2688984574690119078864000000000000000- 4475836791364596609381966077731/1434125106501396842060800000000000*7^( 1/2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(a[21,i]=subs(e9,a[21 ,i]),i=1..20):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "66/&%\" aG6$\"#@\"\"\"$\"I&*y8[/\\%*R<\\*R*\\8827HhP!#T/&F%6$F'\"\"#$\"\"!F1/& F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$!Iv8qro5a\"\\ T;'eO*>>!e$=5$F+/&F%6$F'\"\"($!IxE0Z3<;Ql7*y/R[xd+?4%F+/&F%6$F'\"\")$! I^^?_f)*oUq*pT0nuHm/'))\\!#S/&F%6$F'\"\"*$\"I/7l\\)e103Dh[x6W]%oH#R&FP /&F%6$F'\"#5$!I5J`G-W1rFP/&F%6$F'\"#7$!IrHjo9mcD/f)[))>fzEOn3\"F+/&F%6$F'\"#8$!IEZd 5#)\\@)p4MMt/hKKA(\\C!#U/&F%6$F'\"#9$!I=E>i92/yrLJbZR/>\\Q)3\"F+/&F%6$ F'\"#:$\"IsSx.%>&3br31r0g6%ysO$QF+/&F%6$F'\"#;$\"I66Rm[(3F0/&F%6$F'\"#?F0" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "calculation of the inte rpolation coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order condition gives rise t o a group \{list) of equations to be satisfied by the \"d\" coefficie nts of the weight polynomials for a given stage (corresponding to an \+ \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO8_21 := SimpleOrderCon ditions(8,21,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "whch := [1,2,3,6,7,8,12,15,16,24,3 1,48,63,64,102,117,121,123,125,127,128]:\nordeqns9 := []:\nfor ct in w hch do\n eqn_group := convert(SO8_21[ct],'polynom_order_conditions', 8):\n ordeqns9 := [op(ordeqns9),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for \+ all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 " eqns9 := []:\nfor ct to nops(ordeqns9) do\n eqns9 := [op(eqns9),expa nd(subs(e9,ordeqns9[ct]))];\nend do:\nnops(eqns9);\nnops(indets(eqns9) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$o\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$o\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Solve the system of equations." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d9 := solve(\{op(eqns9)\}):\ninfolevel[solve]:=0 :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(map(rhs,d9));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions are satisfied ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 285 "nm := NULL:\nfor ct to nops(SO8_21) do\n eqn_group := convert(SO8_21[ct],'polynom_order_co nditions',8):\n tt := expand(subs(\{op(e9),op(d9)\},eqn_group));\n \+ tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op(i,tt),i=1. .nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6\\s\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"( \"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C \"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\" #U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"# f\"#g\"#h\"#i\"#j\"#k\"#l\"#m\"#n\"#o\"#p\"#q\"#r\"#s\"#t\"#u\"#v\"#w \"#x\"#y\"#z\"#!)\"#\")\"##)\"#$)\"#%)\"#&)\"#')\"#()\"#))\"#*)\"#!*\" #\"*\"##*\"#$*\"#%*\"#&*\"#'*\"#(*\"#)*\"#**\"$+\"\"$,\"\"$-\"\"$.\"\" $/\"\"$0\"\"$1\"\"$2\"\"$3\"\"$4\"\"$5\"\"$6\"\"$7\"\"$8\"\"$9\"\"$:\" \"$;\"\"$<\"\"$=\"\"$>\"\"$?\"\"$@\"\"$A\"\"$B\"\"$C\"\"$D\"\"$E\"\"$F \"\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$G\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "d9" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8255 "d9 := \{d[16,7] = 0, d [16,2] = 0, d[21,8] = 710593750000000000000000/1036980726192896927211, d[2,8] = 0, d[12,2] = 233117791391788013521866153/6901009155079139428 257358, d[16,5] = 0, d[16,3] = 0, d[10,8] = 48991818422888697226562500 00000000000/44271625997188767070871097196869, d[4,4] = 0, d[21,2] = 74 821468750000000000/1828890169652375533, d[17,2] = 32819343750000000000 /1137279729752564651, d[11,2] = -214290887746131583475076318750/653033 82014795057255539723, d[8,5] = -1932577201177215948160859687500000000/ 6605062219378124940802998572667, d[11,5] = 396850971856677825021041822 50000/92799542863129818205240659, d[11,6] = -4819028512153908607918703 7500000/92799542863129818205240659, d[20,6] = 346003736993600000/10322 5496496111, d[6,1] = 0, d[18,3] = -15282039538211200/72932136421221, d [3,2] = 0, d[12,4] = 180139925432341856271176240600/931636235935683822 81474333, d[14,4] = 0, d[14,8] = 0, d[14,2] = 0, d[3,7] = 0, d[6,4] = \+ -929589947385976234925689765240000000/32059114690366307911928929450308 9, d[8,2] = 10435496280648236371633664062500000/4648006746969791625009 517514099, d[9,1] = 0, d[6,7] = 11146272546697943137445248000000000000 /2244138028325641553835025061521623, d[3,3] = 0, d[18,7] = -3869901440 0000000/72932136421221, d[18,5] = -100367478809228800/72932136421221, \+ d[1,3] = 4866858017933283117300271828349158597/55243883829186794566798 320726324540, d[3,5] = 0, d[17,5] = -1072542351250000000000000/2149458 68923234719039, d[15,4] = 0, d[5,4] = 0, d[21,1] = 0, d[3,6] = 0, d[14 ,7] = 0, d[6,8] = -395819337595807639824050000000000000/32059114690366 3079119289294503089, d[15,8] = 0, d[13,2] = 429322515846/99234337169, \+ d[13,4] = 4735970186373275/18755289724941, d[5,1] = 0, d[8,1] = 0, d[2 ,2] = 0, d[20,4] = 128536039656876625/103225496496111, d[14,6] = 0, d[ 4,7] = 0, d[5,6] = 0, d[18,2] = 7156486020800/385884319689, d[17,4] = \+ 47709809968750000000000/23882874324803857671, d[8,3] = -39137594684014 4426363312640625000000/13944020240909374875028552542297, d[16,8] = 0, \+ d[7,4] = -4518259856247605211174943317800000000/2225181946162921914415 383231514627, d[7,2] = -78935145594617037712001693926500000/2225181946 162921914415383231514627, d[4,3] = 0, d[1,8] = -2689232137678259857410 86750412500000/2762194191459339728339916036316227, d[18,1] = 0, d[17,6 ] = 465344668750000000000000/71648622974411573013, d[4,6] = 0, d[21,6] = 1585663081250000000000000/345660242064298975737, d[11,4] = -1439926 8250109171701542017500000/76660491930411588952155327, d[19,1] = 0, d[1 9,4] = 1229671924018075/593800919631, d[7,7] = 54176312831435243301522 560000000000000/15576273623140453400907682620602389, d[17,7] = -303550 750000000000000000/71648622974411573013, d[2,1] = 0, d[8,7] = -1933819 35048160199990000000000000000000/878473275177290617126798810164711, d[ 9,4] = -15667142403343655099720182820611014359175961052867037229191078 920256569210788540203304995709158148558170/762753071306986071088908939 2429141554771738230605441669783505323591703147372156487764982355632884 7, d[13,3] = -16210396245523/297703011507, d[4,1] = 0, d[11,8] = -1410 17771591595420034375000000000/1763191314399466545899572521, d[21,4] = \+ 1850892689031250000000000/1036980726192896927211, d[15,5] = 0, d[3,4] \+ = 0, d[11,1] = 0, d[14,3] = 0, d[13,8] = 742871750000000/6251763241647 , d[10,6] = 859728026412220014410609375000000000/119653043235645316407 7597221537, d[5,7] = 0, d[2,3] = 0, d[4,5] = 0, d[3,8] = 0, d[12,3] = \+ -4371459386193026634140005453/10351513732618709142386037, d[21,7] = -2 894467750000000000000000/1036980726192896927211, d[20,5] = -2892354034 96838000/103225496496111, d[9,5] = 82040900300470742260461852574776437 2224223724394084266189248053955690001222851762340421926022883059509781 /175433206400606796350449056025870255759749979303925158405020622442609 1723895595992185945941795563481, d[6,6] = -365578740203487936141492580 0000000/456032925894257580539529579663, d[3,1] = 0, d[17,1] = 0, d[12, 1] = 0, d[2,4] = 0, d[20,3] = -28312886797639250/103225496496111, d[15 ,3] = 0, d[10,3] = -279212231479717716841214828125000000/4919069555243 196341207899688541, d[1,1] = 1, d[10,5] = -261957454127613375597467528 12500000000/44271625997188767070871097196869, d[13,7] = -8678595920000 000/18755289724941, d[15,1] = 0, d[1,2] = -607938376042463191847853201 5709717/454682171433636169274060252891560, d[4,2] = 0, d[1,4] = -53202 09802136186616924467002282055/17156485661238134958633018859107, d[21,5 ] = -4027267198750000000000000/1036980726192896927211, d[8,4] = 701212 785333580211537603125000000000/5456355746442798864141607516551, d[6,2] = -601487857049264481091906028100000/11873746181617151078492196092707 , d[8,8] = 6867256216198870738281250000000000000/125496182168184373875 256972880673, d[20,8] = 51264200000000000/103225496496111, d[1,6] = -1 904077280787552439313261030287324550/276219419145933972833991603631622 7, d[13,1] = 0, d[11,7] = 3971060448019327028168000000000000/123423392 00796265821297007647, d[16,4] = 0, d[12,5] = -215858506733409662503312 29340/4903348610187809593761807, d[2,5] = 0, d[1,7] = 1120309702569939 873097943499088000000/2762194191459339728339916036316227, d[9,2] = -25 1812235135954807138531425007359050800425322097898375355083945702755675 594859858932211000109670316351407/701732825602427185401796224103481023 03899991721570063362008248977043668955823839687437837671822539240, d[9 ,3] = 1574006788973167813056264745152031486866706215243187957601212205 591656364874346212117869477522654752149369/350866412801213592700898112 0517405115194999586078503168100412448852183447791191984371891883591126 9620, d[7,8] = -1923874745434490174059750000000000000/2225181946162921 914415383231514627, d[18,6] = 89548457093440000/72932136421221, d[15,6 ] = 0, d[5,3] = 0, d[7,6] = -12491541725629564727073943253000000000/22 25181946162921914415383231514627, d[12,6] = 26212063779824750637619049 000/4903348610187809593761807, d[8,6] = 234676120127407249513432812500 0000000/6605062219378124940802998572667, d[13,6] = 4542329168549000/62 51763241647, d[4,8] = 0, d[9,6] = -99623653650400398146906826387928089 0541226984806999625517214736420133271321037385264075396240496643200350 /175433206400606796350449056025870255759749979303925158405020622442609 1723895595992185945941795563481, d[19,2] = 1817271227725/51316128857, \+ d[14,5] = 0, d[9,7] = 392792495050022300109721985207700321610215496832 223564745422193759027722952774890122581902342628512000000/111639313164 0225067684675811073719809380227141024978280759222142816603824297197449 572874690233540397, d[21,3] = -6970865187500000000000/1646001152687137 9797, d[2,6] = 0, d[13,5] = -3657621338015740/6251763241647, d[20,7] = -210530626000000000/103225496496111, d[5,2] = 0, d[20,1] = 0, d[20,2] = 84257023851875/3823166536893, d[2,7] = 0, d[6,5] = 1113911296651951 01688561105836000000/16873218258087530479962594447531, d[19,6] = 82261 66105940000/1385535479139, d[17,3] = -3100437062500000000000/796095810 8267952557, d[10,1] = 0, d[9,8] = -15343456837891496098036015047175793 8128990428450087329978680544437120204278427691454133555602589262500000 /175433206400606796350449056025870255759749979303925158405020622442609 1723895595992185945941795563481, d[10,4] = 115058304595977301521484343 75000000000/44271625997188767070871097196869, d[7,1] = 0, d[6,3] = 225 58378943802581069026887276200000/35621238544851453235476588278121, d[1 2,7] = -196361134679665239521920000000/59285942286816243270029121, d[1 0,2] = 2481602175301950336095960937500000/5465632839159107045786555209 49, d[5,5] = 0, d[11,3] = 8036828995510655906812766387500/195910146044 385171766619169, d[18,4] = 57249819796068800/72932136421221, d[16,6] = 0, d[19,5] = -19937305826070400/4156606437417, d[5,8] = 0, d[10,7] = \+ -137960960678854571390000000000000000000/30990138198032136949609768037 8083, d[14,1] = 0, d[7,3] = 986802368736629447477139099051000000/22251 81946162921914415383231514627, d[16,1] = 0, d[18,8] = 6197680000000000 /72932136421221, d[12,8] = 76703568234244234188250000000/9316362359356 8382281474333, d[1,5] = 1700194041052351012381629182915954753/27621941 91459339728339916036316227, d[15,7] = 0, d[15,2] = 0, d[19,8] = 394616 5000000000/4156606437417, d[19,7] = -15588053200000000/4156606437417, \+ d[19,3] = -206022969924650/461845159713, d[17,8] = 2352812500000000000 00000/214945868923234719039, d[7,5] = 10286887619157566606218291775980 000000/2225181946162921914415383231514627\}: " }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "subs(d9,matrix([seq([seq(d[j,i],j=1..11)],i=1..8)])) :\nevalf[8](%);\nsubs(d9,matrix([seq([seq(d[j,i],j=11..21)],i=1..8)])) :\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7-$\" \"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+7-$!)B1P8!\"'F+F+F+F+$!)epl]F/$!)pNZNF /$\")^:XA!\"%$!)YV)e$F6$\")[PSXF6$!)pY\"G$F67-$\")!o(4))F/F+F+F+F+$\") _%GL'!\"&$\")UqMWFB$!)bw1G!\"$$\")s0'[%FG$!))=hn&FG$\")OI-TFG7-$!)/*45 $FBF+F+F+F+$!)?h**GF6$!)D^I?F6$\").8&G\"!\"#$!)b-a?FW$\")v\"*)f#FW$!)n Jy=FW7-$\")*H_:'FBF+F+F+F+$\")Hl,mF6$\")A%Hi%F6$!)J!f#HFW$\")]ZwYFW$!) 20&FW7-$\")(oe0%FBF+F+F+F+$\")\"Ro'\\F6$\")08yM F6$!)6M,AFW$\")mS=NFW$!))pGu#FD$!)1.NUFD7-$!)nJy=!\"#$ \")leL>F-$\")%Q^_#FDF(F(F($\")yl(*>F-$\")&Q(\\yFD$\"))[32#F-$\")m>X7F- $\")j)[y\"F-7-$\")CVwUFT$!)sE-WF-$!)Oa]eFDF(F(F($!)a#)*)\\F-$!)j!GF-$!)rk$)QF-7-$!)U%H>&FT$\")vuX`F-$\")pnlsFDF(F(F($\" )m\"[\\'F-$\")E$yA\"F-$\")YN$F-$\")fM(e%F-7-$\")%Hu@$FT$!) G57LF-$!)!zsi%FDF(F(F($!)&emB%F-$!)x;1`FD$!)u=]PF-$!);_R?F-$!)bC\"z#F- 7-$!)8(y*zFA$\"),@L#)FD$\")'f#)=\"FDF(F(F($\")qg%4\"F-$\")D(y\\)F0$\") #=P\\*FD$\")VBm\\FD$\")h__oFDQ)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "principal er ror graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "The interpolation scheme amounts to having a Runge-Kutta method fo r each value of the parameter " }{TEXT 269 1 "u" }{TEXT -1 8 " where \+ " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<1" "6#2%!G \"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "The nodes and l inking coefficients are ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eu9 := map(_U->lhs(_U)=rhs(_U)/u,e9):" }}}{PARA 0 "" 0 "" {TEXT -1 43 " ... and the weight polynomials (of degree " }{XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 33 " and re-using the weight symbol \+ " }{XPPEDIT 18 0 "b[j]" "6#&%\"bG6#%\"jG" }{TEXT -1 10 ") are ... " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "pols := [seq(b[j]=add(simpli fy(subs(d9,d[j,i]))*u^(i-1),i=1..8),j=1..21)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The whole scheme, includi ng the weights, is given by the set of equations:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eu := `union`(eu9,\{op(pols)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can calculate the principal error norm, that is, the root mean square of the residu es of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "errterms8_21 := PrincipalErrorTerms(8,21,'expanded'): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "nops(errterms8_21);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$'G" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "sm := 0:\nfor ct to 286 do\n sm := sm+expand(subs(eu,errterms8_21[ct]))^2;\nend do:\nssm := sqrt(sm)*u^9:\nevalf(eval(ssm,u=1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Mg'y#G!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(ssm,u=0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 508 295 295 {PLOTDATA 2 "6%-%'CURVESG6$7er7$$\"3`*****\\n5; \"o!#@$\"3?z#>[$*fA)p!#G7$$\"3#******\\8ABO\"!#?$\"3Go1JRe@lF!#F7$$\"3 3+++-K[V?F1$\"3Q,ok>J&*fhF47$$\"3#)******pUkCFF1$\"3W.;?C.@%3\"!#E7$$ \"3s*****\\Smp3%F1$\"3O#Q-<(o1\"R#F?7$$\"3k******R&)G\\aF1$\"3]g5@^%fh ;%F?7$$\"3Y******4G$R<)F1$\"3%=Q#))>d_-!*F?7$$\"3%******zqd)*3\"!#>$\" 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<5s#y*Faq$\"3uLl\\(Q1Z'HFdo7$$\"33+++!*3))4)*Faq$\"3tXrb'p6s!HFdo7$$\" 3C+++l2/P)*Faq$\"3;L(Qi$p=nGFdo7$$\"3U+++S1?k)*Faq$\"3oC=S^bGUGFdo7$$ \"3;+++50O\"*)*Faq$\"3d%G)er^QHGFdo7$$\"33+++b-oX**Faq$\"3(>, " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "abreviated calculation for stages 15 to 21" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6801 "e2 := \{a[9,3] = 0, a[13,7 ] = 3847749490868980348119500000/15517045062138271618141237517, a[13,8 ] = -13734512432397741476562500000/875132892924995907746928783, a[13,1 1] = 282035543183190840068750/12295407629873040425991, a[13,12] = -306 814272936976936753/1299331183183744997286, c[3] = 341/3200, a[13,6] = \+ 791638675191615279648100000/2235604725089973126411512319, a[14,2] = 0, c[2] = 1/20, a[5,3] = -3899844/2907025, c[4] = 1023/6400, a[8,1] = -1 221101821869329/690812928000000, c[10] = 909/1000, a[13,10] = -9798363 684577739445312500000/308722986341456031822630699, a[13,9] = 122747654 7031319687842881203774063505031923427600698639829444355496961634227421 5316330684448207141/48934514749371551765038583414351093488882928068660 9654482896526796523353052166757299452852166040, a[9,2] = 0, a[8,2] = 0 , a[11,2] = 0, a[11,3] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2 ), a[12,2] = 0, a[7,2] = 0, a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[12 ,3] = 0, a[10,3] = 0, a[14,3] = 0, a[14,5] = 0, a[13,5] = 0, a[8,3] = \+ 0, c[12] = 1, c[13] = 1, a[6,3] = 0, a[5,2] = 0, a[13,4] = 0, a[7,3] = 0, a[13,2] = 0, a[4,3] = 3069/25600, a[5,1] = 4202367/11628100, a[8,4 ] = -125/2, a[6,2] = 0, a[13,3] = 0, a[10,2] = 0, a[7,6] = 5611/283500 , a[12,10] = 40279545832706233433100438588458933210937500/889646084279 9482846916972126377338947215101, a[7,5] = -26782109/689364000, c[7] = \+ 31/200, c[8] = 943/1000, c[9] = 7067558016280/7837150160667, a[3,2] = \+ 116281/1024000, a[7,4] = 8602624/76559175, a[3,1] = -7161/1024000, c[5 ] = 39/100, a[14,12] = 11352128098668146659861/25466891190401401946805 6-5215842639928607924801/127334455952007009734028*7^(1/2), a[2,1] = 1/ 20, a[4,1] = 1023/25600, a[6,5] = 923521/5106400, a[7,1] = 21173/34320 0, c[11] = 47/50, a[6,4] = 31744/135025, a[14,1] = -942013/61397+28567 61/491176*7^(1/2), a[14,6] = -1090956177631910962162375596685/86058762 53572154615648617314+40421151553845087519011727532985/8433758728500711 52333564496772*7^(1/2), a[14,7] = 1274787529143761017157317260602825/2 6341935917422215554074218244182-958929531471161732045165837664925/5268 3871834844431108148436488364*7^(1/2), a[14,10] = -20796683614312164556 73395233734375/5957776266668670748330679562+22458241974476061273627650 0828125/1702221790476763070951622732*7^(1/2), a[14,8] = -8789797928960 067489629400558015625/16279513475811535751534066502+664447066255418404 5392548540578125/32559026951623071503068133004*7^(1/2), a[14,11] = 297 0221666030903358562614446315/4589064694536869684100726-224527748245299 6546801919407985/9178129389073739368201452*7^(1/2), a[10,5] = -1802692 59803172281163724663224981097/38100922558256871086579832832000000, a[1 0,7] = 318607235173649312405151265849660869927653414425413/67147167155 58965303132938072935465423910912000000, a[10,8] = 21208320243451908228 1842245535894/20022426044775672563822865371173879, a[9,6] = 7801251558 4389364132309055253043103656779559256849718270146067480312677011148162 5/18311042541273197219788987450715878685922610298086185950524144307362 9143100805376, a[9,8] = 1033284818445201560405683676728665685912400779 6970668046446015775000000/13127035500360336480738342487407279145379720 28638950165249582733679393783, a[9,7] = 664113122959911642134782135839 106469928140328160577035357155340392950009492511875/151784655985862481 36333023107295349175279765150089078301139943253016877823170816, a[9,4] = -517229431108566845837517565524698123003902533693369911413831527077 2319372469280000/12461938100480914589727863057121529836525707941023625 2921850936749076487132995191, c[6] = 93/200, a[9,5] = -120706792584692 54807978936441733187949484571516120469966534514296406891652614970375/2 7220311547616572217104781845311006994972840850483890150850769616734461 40398628096, a[8,5] = -1024030607959889/168929280000000, a[8,7] = 6070 139212132283/92502016000000, a[9,1] = -1472514264486215803881384708877 264246346044433307094207829051978044531801133057155/124689480162003200 1157059621643986024803301558393487900440453636168046069686436608, a[8, 6] = 1501408353528689/265697280000000, a[10,6] = 211276702141728028701 28286992003940810655221489/4679473877997892906145822697976708633673728 000, a[5,4] = 3982992/2907025, a[12,1] = -2866556991825663971778295329 101033887534912787724034363/868226711619262703011213925016143612030669 233795338240, a[11,5] = -26053085959256534152588089363841/437755280456 5683061011299942400, a[11,4] = -996286030132538159613930889652/1635306 8885996164905464325675, a[11,6] = 209808223450967602922240867949781053 12644533925634933539/3775889992007550803878727839115494641972212962174 156800, a[10,9] = -269840492940084251872116648508712979856226984822951 7793703413951226714583/46954567491393431507700044208087114188467603590 2717550325616728175875000000, a[10,1] = -29055573360337415088538618442 231036441314060511/22674759891089577691327962602370597632000000000, a[ 11,1] = -2342659845814086836951207140065609179073838476242943917/13584 80961351056777022231400139158760857532162795520000, a[10,4] = -2046274 9524591049105403365239069/454251913499893469596231268750, a[11,10] = - 31155237437111730665923206875/392862141594230515010338956291, a[13,1] \+ = 44901867737754616851973/1014046409980231013380680, a[12,7] = 1657121 559319846802171283690913610698586256573484808662625/134314804112551464 77259155104956093505361644432088109056, a[12,8] = 34568537955467705221 5495825476969226377187500/74771167436930077221667203179551347546362089 , a[12,9] = -320589096271707254279143431215272753400810277402321024057 1361570757249056167015230160352087048674542196011/94756954968396581478 3015124451273604984657747127257615372449205973192657306017239103491074 738324033259120, a[6,1] = 5611/114400, a[12,4] = -16957088714171468676 387054358954754000/143690415119654683326368228101570221, a[12,5] = -45 83493974484572912949314673356033540575/4519577036552507471573130342703 35135744, a[11,7] = 89072299375637918641892962209583383526432263578229 4899/13921242001395112657501941955594013822830119803764736, a[12,6] = \+ 2346305388553404258656258473446184419154740172519949575/25672671640789 5402892744978301151486254183185289662464, a[11,9] = 300760669768102517 8342324975654524349466722661958764963718742623926848522439253598648849 62513/4655443337501346455585065336604505603760824779615521285751892810 315680492364106674524398280000, a[11,8] = 1610214261431241783890751219 29246710833125/10997207722131034650667041364346422894371443, a[12,11] \+ = -6122933601070769591613093993993358877250/10505170015102355131982467 21302027675953, a[14,9] = 99874705782485733872972546676594228969633678 99428512772582082944283962013964900384895127596133/4129561398110764186 8485821081000930261227361072745605305202526150545875206586329586453004 -377489797344744275480292572132150892961344729378723536598312352713719 7728406982036112477728851/41295613981107641868485821081000930261227361 072745605305202526150545875206586329586453004*7^(1/2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "s et up order relations etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2114 "SO7_14 := SimpleOrderConditions(7 ,14,'expanded'):\nSO7_15 := SimpleOrderConditions(7,15,'expanded'):\nS O7_16 := SimpleOrderConditions(7,16,'expanded'):\nSO7_17 := SimpleOrde rConditions(7,17,'expanded'):\nSO7_18 := SimpleOrderConditions(7,18,'e xpanded'):\nSO7_19 := SimpleOrderConditions(7,19,'expanded'):\nSO7_20 \+ := SimpleOrderConditions(7,20,'expanded'):\nSO8_21 := SimpleOrderCondi tions(8,21,'expanded'):\nerrterms8_21 := PrincipalErrorTerms(8,21,'exp anded'):\nwhch := [1,2,3,6,7,8,12,15,16,21,27,31,32,64]:\nordeqns2 := \+ []:\nfor ct in whch do\n eqn_group := convert(SO7_14[ct],'polynom_or der_conditions',7):\n ordeqns2 := [op(ordeqns2),op(eqn_group)];\nend do:\nwhch := [1,2,3,6,7,8,12,15,16,21,27,31,32,36,64]:\nordeqns3 := [ ]:\nfor ct in whch do\n eqn_group := convert(SO7_15[ct],'polynom_ord er_conditions',7):\n ordeqns3 := [op(ordeqns3),op(eqn_group)];\nend \+ do:\nwhch := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nordeqns4 := []:\nfor ct in whch do\n eqn_group := convert(SO7_16[ct],'polynom_o rder_conditions',7):\n ordeqns4 := [op(ordeqns4),op(eqn_group)];\nen d do:\nwhch := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nordeqns5 \+ := []:\nfor ct in whch do\n eqn_group := convert(SO7_17[ct],'polynom _order_conditions',7):\n ordeqns5 := [op(ordeqns5),op(eqn_group)];\n end do:\nwhch := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nordeqns 6 := []:\nfor ct in whch do\n eqn_group := convert(SO7_18[ct],'polyn om_order_conditions',7):\n ordeqns6 := [op(ordeqns6),op(eqn_group)]; \nend do:\nwhch := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nordeq ns7 := []:\nfor ct in whch do\n eqn_group := convert(SO7_19[ct],'pol ynom_order_conditions',7):\n ordeqns7 := [op(ordeqns7),op(eqn_group) ];\nend do:\nwhch := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nord eqns8 := []:\nfor ct in whch do\n eqn_group := convert(SO7_20[ct],'p olynom_order_conditions',7):\n ordeqns8 := [op(ordeqns8),op(eqn_grou p)];\nend do:\nwhch := [1,2,3,6,7,8,12,15,16,24,31,48,63,64,102,117,12 1,123,125,127,128]:\nordeqns9 := []:\nfor ct in whch do\n eqn_group \+ := convert(SO8_21[ct],'polynom_order_conditions',8):\n ordeqns9 := [ op(ordeqns9),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4451 "calc_coeff s := proc()\n local eqns2,eqns3,eqns4,eqns5,eqns6,eqns7,eqns8,eqns9, \n eqs15,eqs16,eqs17,eqs18,eqs19,eqs20,eqs21,dd,sol,\n \+ d2,d3,d4,d5,d6,d7,d8,dd_5,dd_6,dd_7,dd_8,d_5,d_6,d_7,d_8,\n c t,eq,eq1,eq2,eq3,eq4,eu9,pols,eu;\n global e2,e3,e4,e5,e6,e7,e8,e9,d 9,sm;\n\n eqns2 := []:\n for ct to nops(ordeqns2) do\n eqns2 \+ := [op(eqns2),expand(subs(e2,ordeqns2[ct]))];\n end do:\n d2 := so lve(\{op(eqns2)\}):\n eqs15 := \{seq(a[15,j]=add(expand(subs(\{op(d2 ),c[15]=c_15\},d[j,i]*c[15]^i)),i=1..7),j=1..14)\}:\n e3 := `union`( eqs15,\{c[15]=c_15\},e2):\n eqns3 := []:\n for ct to nops(ordeqns3 ) do\n eqns3 := [op(eqns3),expand(subs(e3,ordeqns3[ct]))];\n en d do:\n d3 := solve(\{op(eqns3)\}):\n eqs16 := \{seq(a[16,j]=add(e xpand(subs(\{op(d3),c[16]=c_16\},d[j,i]*c[16]^i)),i=1..7),j=1..15)\}: \n e4 := `union`(eqs16,\{c[16]=c_16\},e3):\n eqns4 := []:\n for \+ ct to nops(ordeqns4) do\n eqns4 := [op(eqns4),expand(subs(e4,orde qns4[ct]))];\n end do:\n d4 := solve(\{op(eqns4)\}):\n eqs17 := \+ \{seq(a[17,j]=add(expand(subs(\{op(d4),c[17]=c_17\},d[j,i]*c[17]^i)),i =1..7),j=1..16)\}:\n e5 := `union`(eqs17,\{c[17]=c_17\},e4):\n eqn s5 := []:\n for ct to nops(ordeqns5) do\n eqns5 := [op(eqns5),e xpand(subs(e5,ordeqns5[ct]))];\n end do:\n d5 := solve(\{op(eqns5) \},indets(eqns5) minus \{seq(d[1,i],i=1..7)\}):\n eq := add(subs(\{o p(d5),c[18]=c_18\},d[17,i]*c[18]^i),i=1..7)=0:\n dd := \{d[1,1]=1,se q(d[1,i]=0,i=2..6)\}:\n sol := \{d[1,7]=expand(rationalize(solve(sub s(dd,eq))))\}:\n dd_5 := `union`(subs(sol,dd),sol):\n d_5 := `unio n`(subs(dd_5,d5),dd_5):\n eqs18 := \{seq(a[18,j]=add(expand(subs(\{o p(d_5),c[18]=c_18\},d[j,i]*c[18]^i)),i=1..7),j=1..17)\}:\n e6 := `un ion`(eqs18,\{c[18]=c_18\},e5):\n eqns6 := []:\n for ct to nops(ord eqns6) do\n eqns6 := [op(eqns6),expand(subs(e6,ordeqns6[ct]))];\n end do:\n d6 := solve(\{op(eqns6)\},indets(eqns6) minus \{seq(d[1 ,i],i=1..7),seq(d[9,i],i=1..7)\}):\n eq1 := add(subs(\{op(d6),c[19]= c_19\},d[17,i]*c[19]^i),i=1..7)=0:\n eq2 := add(subs(\{op(d6),c[19]= c_19\},d[18,i]*c[19]^i),i=1..7)=0:\n dd := \{d[1,1]=1,seq(d[1,i]=0,i =2..6),seq(d[9,i]=0,i=1..6)\}:\n sol := solve(subs(dd,\{eq1,eq2\})); \n dd_6 := `union`(subs(sol,dd),sol):\n d_6 := `union`(subs(dd_6,d 6),dd_6):\n eqs19 := \{seq(a[19,j]=add(expand(subs(\{op(d_6),c[19]=c _19\},d[j,i]*c[19]^i)),i=1..7),j=1..18)\}:\n e7 := `union`(eqs19,\{c [19]=c_19\},e6):\n eqns7 := []:\n for ct to nops(ordeqns7) do\n \+ eqns7 := [op(eqns7),expand(subs(e7,ordeqns7[ct]))];\n end do:\n \+ d7 := solve(\{op(eqns7)\},indets(eqns7) minus \{seq(seq(d[j,i],i=1..7 ),j=[1,9,13])\}):\n eq1 := add(subs(\{op(d7),c[20]=c_20\},d[17,i]*c[ 20]^i),i=1..7)=0:\n eq2 := add(subs(\{op(d7),c[20]=c_20\},d[18,i]*c[ 20]^i),i=1..7)=0:\n eq3 := add(subs(\{op(d7),c[20]=c_20\},d[19,i]*c[ 20]^i),i=1..7)=0:\n dd := \{d[1,1]=1,seq(d[1,i]=0,i=2..6),seq(d[9,i] =0,i=1..6),seq(d[13,i]=0,i=1..6)\}:\n sol := solve(subs(dd,\{eq1,eq2 ,eq3\}));\n dd_7 := `union`(subs(sol,dd),sol):\n d_7 := `union`(su bs(dd_7,d7),dd_7):\n eqs20 := \{seq(a[20,j]=add(expand(subs(\{op(d_7 ),c[20]=c_20\},d[j,i]*c[20]^i)),i=1..7),j=1..19)\}:\n e8 := `union`( eqs20,\{c[20]=c_20\},e7):\n eqns8 := []:\n for ct to nops(ordeqns8 ) do\n eqns8 := [op(eqns8),expand(subs(e8,ordeqns8[ct]))];\n en d do:\n d8 := solve(\{op(eqns8)\},indets(eqns8) minus \{seq(seq(d[j, i],i=1..7),j=[1,9,12,13])\}):\n eq1 := add(subs(\{op(d8),c[21]=c_21 \},d[17,i]*c[21]^i),i=1..7)=0:\n eq2 := add(subs(\{op(d8),c[21]=c_21 \},d[18,i]*c[21]^i),i=1..7)=0:\n eq3 := add(subs(\{op(d8),c[21]=c_21 \},d[19,i]*c[21]^i),i=1..7)=0:\n eq4 := add(subs(\{op(d8),c[21]=c_21 \},d[20,i]*c[21]^i),i=1..7)=0:\n dd := \{d[1,1]=1,seq(d[1,i]=0,i=2.. 6),seq(d[9,i]=0,i=1..6),seq(d[12,i]=0,i=1..6),seq(d[13,i]=0,i=1..6)\}: \n sol := solve(subs(dd,\{eq1,eq2,eq3,eq4\}));\n dd_8 := `union`(s ubs(sol,dd),sol):\n d_8 := expand(`union`(subs(dd_8,d8),dd_8)):\n \+ eqs21 := \{seq(a[21,j]=add(expand(subs(\{op(d_8),c[21]=c_21\},d[j,i]*c [21]^i)),i=1..7),j=1..20)\}:\n e9 := `union`(eqs21,\{c[21]=c_21\},e8 ):\n eqns9 := []:\n for ct to nops(ordeqns9) do\n eqns9 := [o p(eqns9),expand(subs(e9,ordeqns9[ct]))];\n end do:\n d9 := solve( \{op(eqns9)\}):\n eu9 := map(_U->lhs(_U)=rhs(_U)/u,e9):\n pols := \+ [seq(b[j]=add(simplify(subs(d9,d[j,i]))*u^(i-1),i=1..8),j=1..21)]:\n \+ eu := `union`(eu9,\{op(pols)\}):\n sm := 0:\n for ct to 286 do\n \+ sm := sm+expand(subs(eu,errterms8_21[ct]))^2;\n end do:\n ret urn(c[15]=c_15,c[16]=c_16,c[17]=c_17,c[18]=c_18,c[19]=c_19,c[20]=c_20, c[21]=c_21);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 259 17 "Sample comparison" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "c_15 : = 9/20:\nc_16 := 1/9:\nc_17 := 2/5:\nc_18 := 1/2:\nc_19 := 7/10:\nc_20 := 9/10:\nc_21 := 1/10:\ncalc_coeffs();\nssmA := sqrt(sm)*u^9:\nplot( ssmA,u=0..1,color=blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6# \"#:#\"\"*\"#?/&F%6#\"#;#\"\"\"F)/&F%6#\"#<#\"\"#\"\"&/&F%6#\"#=#F0F6/ &F%6#\"#>#\"\"(\"#5/&F%6#F*#F)FC/&F%6#\"#@#F0FC" }}{PARA 13 "" 1 "" {GLPLOT2D 551 322 322 {PLOTDATA 2 "6&-%'CURVESG6#7gq7$$\"3`*****\\n5; \"o!#@$\"3S*o&>?Vi%e(!#G7$$\"3#******\\8ABO\"!#?$\"3:i5S\"=sT+$!#F7$$ \"33+++-K[V?F1$\"3#e&Grsh:$p'F47$$\"3#)******pUkCFF1$\"3]T&z#[f@y6!#E7 $$\"3s*****\\Smp3%F1$\"3)p$Q<\"*[2*f#F?7$$\"3k******R&)G\\aF1$\"3Q9No( H(zHXF?7$$\"3Y******4G$R<)F1$\"3e!4utpBNz*F?7$$\"3%******zqd)*3\"!#>$ \"3]))=>N.as;!#D7$$\"3*)*****>c'yM;FR$\"3m#33o^N*pMFU7$$\"3')*****fT:( z@FR$\"3y4zxynl\"o&FU7$$\"3#*******zZ*z7$FR$\"3I$>tgF_o7$$\"3&** ****4z_\"4iFR$\"3>2AnjZ$*=CF_o7$$\"3y*****\\;hEG(FR$\"3%GLX4Qp3x#F_o7$ $\"3o******R&phN)FR$\"3#362nmDO-$F_o7$$\"3%*******HEP!*))FR$\"3#o@\\\" 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"6#/&%\"aG6$\"#:\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#:\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\" \"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"',&#\"C%e ([)[7EH\"=(y![E(po6%\"DvV['*Quov+3G3O[`)fE\"\"\"*(\"B%=(>Hu\"=N7$eBI)) [`X#F-\"Cv$feD(=)H>E+9-Kb/q)!\"\"\"\"(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"(,&#\"E)[nK%eQgU#ya'pfzhw9j\"FvVt_S!zo PB>eo,9z&fq#\"\"\"*(\"D3-FK^0WfX!HZ8tmuALF-\"Fvo/[#z*f'Q)\\B%*fgRQG#R! \"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\") ,&#\"?)[Z$H**)fT'y7N+L*)f\">Dc^GC,$3`)R9ps*G\"\"\"*(\"AkSYrcPNT>S/a'4^ Z\"F-\"@D\"GQ<1(*)fRvM:U>P\"!\"\"\"\"(#F-\"\"#F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"*,&#\"hq$QY%*)eM9e_:-=rn+\"yU,MTc(or#4 H:%3DL\"**Q\\2a8'GY&[I\\+S*4Fh'fk1nI\"gq+](oHH+G$*3;\\.T09\"*G^V,,#3GE :I%4H&yS*3aJa=(pYKl60T\"yF#=#>2Mt$!\"\"*(\"\\qPJxg_*)[T;Ioou=SXW4(eXb' [DYhKp%*f4!)Qk(f?%=CSp2%)>hZm\"=\"\"\"\"[qv=#\\a$)o$)*Gyeo5g\"R.Nk*fou 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"" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"&\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"',&#\"J&G>wgtF$z<%[5G OJ1s`vt=\"Jo4meY:hDe9#*R&>@lP!=kC'\"\"\"*(\"Ht!Q0m'R\"etLAtDsf4\"\\56% F-\"J?4k)ywJO*e\"*o.))oY53Wn:\"!\"\"\"\"(#F-\"\"#F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"(,&#\"KXo()\\TGT6NB5w'\\*)[Xo#*pi\" \"KCK=G-&R:cbA:G,vOoZebL%\"\"\"*(\"I\"p()4)R%yr*HPvGWeW8jg:9F-\"J[-')4 xX'RF/%>$[:@=#)\\/NL!\"\"F(#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"),&#\"Iv$4\"G]b#=!zg%R8,z\"=crQO\"I/P3-=)*=-ht!*yE OWqW$pG\"\"\"\"*(\"Fv$fu;G$3F[xCM7ReZ\")f'F-\"G_\\&y<^)[2\" \"*,&#\"\\r4y-P.(zlnenT(H1@FZBXhW:(3Y0!o\"4m&zdCIH%\\F(H\"p)*=`vIMIP?x F`&='4\\\"\\r++]alH5(H(RApV.i-Ij)zutDt[T'HK+G6y)f&oHeq5ygU*>zc$eGpl+hv Wj8#!\"\"*(\"jqn/VE2Xj%zLB<3\"=(*4\\\\`@>$4'\\!H%[KqK\\@B8Ns0>]Q&*)yC2 Ln;MWNA'4(HD\"\"\"\"\\r++gVsBoPy\"z`\\F'4-k!*Q)**eg)*=8Pe-C!\\-z%[Pmk& [i3afLao'GaD0)[!e24\"#5,&#\"Jvoasl*)40'z;A)F-\"GG&H8I#\\))GC\"oqk\"3=Y_dC!\" \"\"\"(#F-\"\"#F-" }}{PARA 11 "" 1 "" 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{XPPMATH 20 "6#/&%\"aG6$\"#@\"#?\"\"!" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "interpolation coefficients " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "cffs := [seq(seq( d[j,i]=subs(d9,d[j,i]),j=1..21),i=1..8)]:\nfor ct to nops(cffs) do if \+ rhs(cffs[ct])<>0 then print(cffs[ct]) end if end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"\"\"F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"\"\"\"\"##!C<(4d,K&y%=>jC/w$Qzg\"Bg:*GDgSFphjL9<#oa%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"\"'\"\"##!B++5Gg!>4\"[k#\\q &y[,'\"A2F4'>#\\y5:<;=YP(=\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"d G6$\"\"(\"\"##!D++]ERp,?rPqh%fX^$*y\"CFY^JKQ:W\">#H;Y>=DA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"\")\"\"##\"D++]iSmL;PO#[1G'\\N/\"\"@ *49v^4]i\"zppu1![Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"\"*\" 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G-qYCph'=O@!)4-K&\"A2\"f)=Ije\\8Q7m&[cr\"/&F&6$F,F5F-/&F&6$F1F5F-/&F&6 $F5F5F-/&F&6$F9F5F-/&F&6$F=F5#!E+++S_w*oD\\BwfQZ**eH*\"B*3.XH*G>\"zIm. p9\"f?$/&F&6$FAF5#!F++++yJV\\<6_gZi&)f#=XFar/&F&6$FEF5#\"E++++]7.w`6-e L`y77q\"@^l^2;9k))zUkubjX&/&F&6$FIF5#!cqq\"e&[\"e\"4d*\\I.-a)y5#plD?*y 5>Hs.nG0hf(4r32n( )=(*firU%/&F&6$FQF5#!A++]\\gm(/&F&6$FUF5# \"?+1Cw6Fc=MKa#*R,=\";LVZ\"G#QoNfBO;$*/&F&6$FYF5#\"1vKP'=qft%\"/T\\s*G b(=/&F&6$FgnF5F-/&F&6$F[oF5F-/&F&6$F_oF5F-/&F&6$FcoF5#\"8+++++vo*4)4x% \"5rw&Q![KuG)Q#/&F&6$FgoF5#\"2+)ogz>)\\s&Fb\\l/&F&6$F[pF5#\"1v!=S#>nH7 \"-J'>4!Qf/&F&6$F_pF5#\"3Dm(olRg`G\"F^]l/&F&6$FcpF5#\":+++++DJ!*o#*3&= \"76s#p*G>E2)p.\"/&F&6$F(F9#\"F`Z&f\"H=H;Q75N_5/%>+<\"CFiJOg\"*R$G(R$f 9>%>iF/&F&6$F,F9F-/&F&6$F1F9F-/&F&6$F5F9F-/&F&6$F9F9F-/&F&6$F=F9#\"E++ +Oe5h&)o,^>l'H6R6\"\"AJvW%fi*z/`(3e#=K(o\"/&F&6$FAF9#\"G+++!)fx\"H=igm v:>w)oG5Far/&F&6$FEF9#!F++++vof3;[f@x6?xD$>\"@nEd)*H!3%\\7y$>A10m/&F&6 $FIF9#\"dq\"y4&fI)G-E>USBw^GA,+pbR0[#*=mU3%RCPACAPkxuD&=YgAuq/I+4/#)\" _q\"[jbzTf%f=#*ff&*Qs\"4EWA1-0%e^#RIz*\\(fdDqe-c!\\/N'z11S1KVv\"/&F&6$ FMF9#!G++++D\"GvY(fvLhFTXd>EFc`l/&F&6$FQF9#\"A++D#=/@]#ync=(4&oR\";f1C 0#=)HJ'Ga*z#*/&F&6$FUF9#!>S$H7L]i'4Mt1&ee@\":2=w$f4y=5'[L!\\/&F&6$FYF9 #!1Sd,Q8idO\".Z;CjZBB*oe%\\@/&F&6$FgoF9#!3+)G#4)yuO+\"Fb\\l/&F&6 $F[pF9#!2+/2EeIP*>\".ns-%Fjcl/&F&6$F(F=#!F]XK(GI5E8$RCb(y!GxS!>F`dl/&F&6$F,F=F-/ &F&6$F1F=F-/&F&6$F5F=F-/&F&6$F9F=F-/&F&6$F=F=#!C++++e#\\Th$z[.-uybO\"? j'z&H&R0edU*e#H.c%/&F&6$FAF=#!G++++IDVR2FZcHcsT:\\7Far/&F&6$FEF=#\"F++ ++]7GV8&\\sSF,7wYBF]fl/&F&6$FIF=#!dq].?Vm\\SiRvSE&QP5KrK8?kt9s^D'**p![ )pAT0*3GzQEo!p9)R+/l`Oi**Fcfl/&F&6$FMF=#\"E++++]P41T9+A7k-G(f)\"@P:A(f xS;`kNK/`'>\"/&F&6$FQF=#!A++]Pq=zg3R:7&G!>[F^gl/&F&6$FUF=#\">+!\\!>wj] Z#)zP17i#Fdgl/&F&6$FYF=#\"1+!\\&o\"HBa%Fjgl/&F&6$FgnF=F-/&F&6$F[oF=F-/ &F&6$F_oF=F-/&F&6$FcoF=#\"9++++++](oYMl%\"58Id6W(Hi[;(/&F&6$FgoF=#\"2+ +W$4d%[&*)Fb\\l/&F&6$F[pF=#\"1++%f5mhA)\".R\"za`&Q\"/&F&6$F_pF=#\"3++g $*pt.gMF^]l/&F&6$FcpF=#\":++++++]73jce\"\"6Pd(*)Hk?CgcM/&F&6$F(FA#\"F+ ++)3*\\Vz4t)R*pDq4.7\"F`dl/&F&6$F,FAF-/&F&6$F1FAF-/&F&6$F5FAF-/&F&6$F9 FAF-/&F&6$F=FA#\"G++++++[_WPJ%zpYDFY6\"\"CB;_h]-NQbTcKG!QTC#/&F&6$FAFA #\"G++++++gD_,LCN9$GJwT&\"D*Q-1i#o24S`/9BOFwb\"/&F&6$FEFA#!H+++++++++! ***>g\"[]$>Q$>\"B6Z;5))zErh!Hx^Ft%y)/&F&6$FIFA#\"dq+++7&GEM->eA,*[x_Hs F!fP>AaukNAKo\\:-h@.q2_)>s4,IA+0&\\#z#R\"_q(RSNB!puGd\\u>(HCQg;G9A#f2G y\\-TrA!Q4)>P26en%on]AS;8$R;6/&F&6$FMFA#!H+++++++++!RrX&)y1'4'z8\"B$3y .o(4'\\p8K!)>Q,*4$/&F&6$FQFA#\"C++++++o\"GqK>![/1rR\">Zw+(H@eE'z+#RBM7 /&F&6$FUFA#!?+++?>_R_mzY8hj>\";@\"H+FVi\"oGUfGf/&F&6$FYFA#!1+++?ffy')F eal/&F&6$FgnFAF-/&F&6$F[oFAF-/&F&6$F_oFAF-/&F&6$FcoFA#!9++++++++v]NIFh ^m/&F&6$FgoFA#!2++++W,*pQFb\\l/&F&6$F[pFA#!2++++K0)e:Fdil/&F&6$F_pFA#! 3++++giI0@F^]l/&F&6$FcpFA#!:++++++++vnW*GFjcl/&F&6$F(FE#!E++]7/v'3Td)f #yw8K#*o#F`dl/&F&6$F,FEF-/&F&6$F1FEF-/&F&6$F5FEF-/&F&6$F9FEF-/&F&6$F=F E#!E++++++]S#)Rw!efP$>eRF\\_l/&F&6$FAFE#!F++++++](fSFar/&F& 6$FEFE#\"F++++++]7GQ2())>;iDno\"Bt1)G(pDvQP%=o@='\\D\"/&F&6$FIFE#!dq++ ]i#*e-cbLTX\"pF%yU??rVW0oy*Ht3]%G/**G\"Qzvr/:g.)4'\\\"*y$oXV`\"Fcfl/&F &6$FMFE#\"F++++++]ilA(p))GU==**[Fc`l/&F&6$FQFE#!B++++]PM+U&f\"frx,T\" \"=@Dd**eam%*R98>j+++]#)=MUCM#oNqwF_al/&F&6$FYFE#\"0+++ ]<(GuFjgl/&F&6$FgnFEF-/&F&6$F[oFEF-/&F&6$F_oFEF-/&F&6$FcoFE#\"9+++++++ +D\"GN#Fihl/&F&6$FgoFE#\"1+++++o(>'Fb\\l/&F&6$F[pFE#\"1++++];YRFdil/&F &6$F_pFE#\"2+++++?k7&F^]l/&F&6$FcpFE#\"9++++++++v$f5(Fjcl" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 97 "Stage by stage constr uction of the interpolation scheme C .. [8 stage scheme] .. (shorter m ethod)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 73 ": This is a slight variation of the scheme obtain ed by the longer method." }}{PARA 0 "" 0 "" {TEXT -1 35 " The linking coefficient " }{XPPEDIT 18 0 "a[14,1]" "6#&%\"aG6$\"#9\"\"\" " }{TEXT -1 78 " is used as a parameter, instead of a coefficient of \+ the \"weight polynomial\"." }}{PARA 0 "" 0 "" {TEXT -1 175 " \+ The initial value of this parameter and the nodes are based on the val ues obtained by trial and error using the longer method, but further i mprovements are obtained." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 125 "Start with linking coefficients using the weights of the 12 stage scheme as the linking coefficients for the first new \+ stage." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5614 "e1 := \{c[2] = 1/20,\nc[3] = 341/3200,\nc[4] = 1023/6400,\nc[5] = 39/100,\nc[6] = 93/200,\nc[7] \+ = 31/200,\nc[8] = 943/1000,\nc[9] = 7067558016280/7837150160667,\nc[10 ] = 909/1000,\nc[11] = 47/50,\nc[12] = 1,\nc[13] = 1,\na[2,1] = 1/20, \na[3,1] = -7161/1024000,\na[3,2] = 116281/1024000,\na[4,1] = 1023/256 00,\na[4,2] = 0,\na[4,3] = 3069/25600,\na[5,1] = 4202367/11628100,\na[ 5,2] = 0,\na[5,3] = -3899844/2907025,\na[5,4] = 3982992/2907025,\na[6, 1] = 5611/114400,\na[6,2] = 0,\na[6,3] = 0,\na[6,4] = 31744/135025,\na [6,5] = 923521/5106400,\na[7,1] = 21173/343200,\na[7,2] = 0,\na[7,3] = 0,\na[7,4] = 8602624/76559175,\na[7,5] = -26782109/689364000,\na[7,6] = 5611/283500,\na[8,1] = -1221101821869329/690812928000000,\na[8,2] = 0,\na[8,3] = 0,\na[8,4] = -125/2,\na[8,5] = -1024030607959889/1689292 80000000,\na[8,6] = 1501408353528689/265697280000000,\na[8,7] = 607013 9212132283/92502016000000,\na[9,1] = -14725142644862158038813847088772 64246346044433307094207829051978044531801133057155/\n 1246894 8016200320011570596216439860248033015583934879004404536361680460696864 36608,\na[9,2] = 0,\na[9,3] = 0,\na[9,4] = -51722943110856684583751756 55246981230039025336933699114138315270772319372469280000/\n 1 2461938100480914589727863057121529836525707941023625292185093674907648 7132995191,\na[9,5] = -12070679258469254807978936441733187949484571516 120469966534514296406891652614970375/\n 272203115476165722171 0478184531100699497284085048389015085076961673446140398628096,\na[9,6] = 7801251558438936413230905525304310365677955925684971827014606748031 26770111481625/\n 1831104254127319721978898745071587868592261 02980861859505241443073629143100805376,\na[9,7] = 66411312295991164213 4782135839106469928140328160577035357155340392950009492511875/\n \+ 151784655985862481363330231072953491752797651500890783011399432530 16877823170816,\na[9,8] = 10332848184452015604056836767286656859124007 796970668046446015775000000/\n 131270355003603364807383424874 0727914537972028638950165249582733679393783,\na[10,1] = -2905557336033 7415088538618442231036441314060511/\n 22674759891089577691327 962602370597632000000000,\na[10,2] = 0,\na[10,3] = 0,\na[10,4] = -2046 2749524591049105403365239069/454251913499893469596231268750,\na[10,5] \+ = -180269259803172281163724663224981097/381009225582568710865798328320 00000,\na[10,6] = 21127670214172802870128286992003940810655221489/\n \+ 4679473877997892906145822697976708633673728000,\na[10,7] = 31 8607235173649312405151265849660869927653414425413/\n 6714716 715558965303132938072935465423910912000000,\na[10,8] = 212083202434519 082281842245535894/20022426044775672563822865371173879,\na[10,9] = -26 9840492940084251872116648508712979856226984822951779370341395122671458 3/\n 4695456749139343150770004420808711418846760359027175503 25616728175875000000,\na[11,1] = -234265984581408683695120714006560917 9073838476242943917/\n 1358480961351056777022231400139158760 857532162795520000,\na[11,2] = 0,\na[11,3] = 0,\na[11,4] = -9962860301 32538159613930889652/16353068885996164905464325675,\na[11,5] = -260530 85959256534152588089363841/4377552804565683061011299942400,\na[11,6] = 20980822345096760292224086794978105312644533925634933539/\n \+ 3775889992007550803878727839115494641972212962174156800,\na[11,7] = 8 90722993756379186418929622095833835264322635782294899/\n 139 21242001395112657501941955594013822830119803764736,\na[11,8] = 1610214 26143124178389075121929246710833125/1099720772213103465066704136434642 2894371443,\na[11,9] = 30076066976810251783423249756545243494667226619 5876496371874262392684852243925359864884962513/\n 4655443337 5013464555850653366045056037608247796155212857518928103156804923641066 74524398280000,\na[11,10] = -31155237437111730665923206875/39286214159 4230515010338956291,\na[12,1] = -2866556991825663971778295329101033887 534912787724034363/\n 86822671161926270301121392501614361203 0669233795338240,\na[12,2] = 0,\na[12,3] = 0,\na[12,4] = -169570887141 71468676387054358954754000/143690415119654683326368228101570221,\na[12 ,5] = -4583493974484572912949314673356033540575/4519577036552507471573 13034270335135744,\na[12,6] = 2346305388553404258656258473446184419154 740172519949575/\n 25672671640789540289274497830115148625418 3185289662464,\na[12,7] = 16571215593198468021712836909136106985862565 73484808662625/\n 134314804112551464772591551049560935053616 44432088109056,\na[12,8] = 3456853795546770522154958254769692263771875 00/74771167436930077221667203179551347546362089,\na[12,9] = \n -32058 9096271707254279143431215272753400810277402321024057136157075724905616 7015230160352087048674542196011/\n 9475695496839658147830151244512736 0498465774712725761537244920597319265730601723910349107473832403325912 0,\na[12,10] = 40279545832706233433100438588458933210937500/8896460842 799482846916972126377338947215101,\na[12,11] = -6122933601070769591613 093993993358877250/1050517001510235513198246721302027675953,\na[13,1] \+ = 44901867737754616851973/1014046409980231013380680,\na[13,2] = 0,\na[ 13,3] = 0,\na[13,4] = 0,\na[13,5] = 0,\na[13,6] = 79163867519161527964 8100000/2235604725089973126411512319,\na[13,7] = 384774949086898034811 9500000/15517045062138271618141237517,\na[13,8] = -1373451243239774147 6562500000/875132892924995907746928783,\na[13,9] = 1227476547031319687 8428812037740635050319234276006986398294443554969616342274215316330684 448207141/\n 489345147493715517650385834143510934888829280686609 654482896526796523353052166757299452852166040,\na[13,10] = -9798363684 577739445312500000/308722986341456031822630699,\na[13,11] = 2820355431 83190840068750/12295407629873040425991,\na[13,12] = -30681427293697693 6753/1299331183183744997286\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "subs(e1,matrix([se q([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..13)])):\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7/$\"+++++]!#6F(%!GF+F+F +F+F+F+F+F+F+F+7/$\"++]il5!#5$!+iS;$*p!#7$\"+TmbN6F/F+F+F+F+F+F+F+F+F+ F+7/$\"++vV)f\"F/$\"+]P4'*RF*$\"\"!F;$\"+D\"G))>\"F/F+F+F+F+F+F+F+F+F+ 7/$\"+++++RF/$\"+Gc(Rh$F/F:$!+nS_T8!\"*$\"+/l7q8FEF+F+F+F+F+F+F+F+7/$ \"++++]YF/$\"+!G?Z!\\F*F:F:$\"+U?(4N#F/$\"+Ifb3=F/F+F+F+F+F+F+F+7/$\"+ +++]:F/$\"+W!*GphF*F:F:$\"+JolB6F/$!+rg/&)QF*$\"+8()=z>F*F+F+F+F+F+F+7 /$\"++++I%*F/$!+S-jnB3l&FE$\"+U'p @c'F]oF+F+F+F+F+7/$\"+$QWV%FE$ \"+*=3/E%FE$\"+ASOvVF]o$\"+!\\D9(yF2F+F+F+F+7/$\"++++!4*F/$!+**fS\"G\" FEF:F:$!+'*Rr/XF]o$!+p?OJZFE$\"+,$)R'F]o$\"+DG?k9F*$\"+s(3/Y'F*$!+pJKIzF*F+F+7/$\"\"\"F;$ !+oEi,LFEF:F:$!+CF6!=\"!\"($!+RA995F]o$\"+K8JR\"*FE$\"+G%fPB\"F^s$\"+z VCBYFE$!+QxF$Q$FE$\"++@fFXFE$!+'[&\\GeFEF+7/Fhr$\"+>%*)zU%F*F:F:F:F:$ \"+#R\\5a$F/$\"+b@pzCF/$!+/-Up:F]o$\"+(\\1%3DF]o$!+zn$Q<$F]o$\"+F$GQH# F]o$!+LYKhBF/Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "convert(ListTools[Enumerate](Simpl eOrderConditions(7)),matrix):\nlinalg[augment](linalg[delcols](%,2..2) ,matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7\\o7%\"\"\"%#~~G/*&%\"bGF (%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/7%\"\"$F)/*&F,F(-%!G6#*&%\"aGF(F 2F(F(#F(\"\"'7%\"\"%F)/*&F,F()F2F/F(#F(F57%\"\"&F)/*&F,F(-F96#*&FF)/*(F,F(F2F(F8F(#F(\"\")7%\"\"(F)/*&F,F(-F96#*&FCF(FF)/*&F,F(-F96#*(F< F(F2F(FIF(F(#F(\"$!=7%F]rF)/*(F,F(F2F(F_oF(#F(\"$W\"7%\"#@F)/*&F,F(-F9 6#*&FF(#F(FTQ)pprin t236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "c[15] = 57/125, c[16] = 1163/10000, c[17] = 233/625, c[18] = 13/25, c[19] = 177/250, c[20] = 2249/2500, c[21] = 979/10000 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 14" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order conditions can be adapted to give a method of stage by stage construction for an interpolation sch eme that avoids dealing with the weight polynomials for a given stage \+ (corresponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO 6_13 := SimpleOrderConditions(6,13,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "whch := [1, 2,4,8,16,27,31,32]:\ninterp_order_eqns14 := []:\nfor ct in whch do\n \+ temp_eqn := convert(SO6_13[ct],'interpolation_order_condition'):\n \+ interp_order_eqns14 := [op(interp_order_eqns14),temp_eqn];\nend do:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alterna tively, the order conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "interp_order_eqns14 := [a dd(a[14,i],i=1..13)=c[14],seq(op(StageOrderConditions(i,14..14,'expand ed')),i=2..6),\n add(a[14,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1) ,j=2..i-1),i=2..13)=c[14]^6/120, #27\n add(a[14,i]*add(a[i,j]*c[j]^ 4,j=2..i-1),i=2..13)=c[14]^6/30]: #31" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "e2 := `union`(e1,\{seq( a[14,i]=0,i=2..5)\}):\neqs_14 := expand(subs(e2,interp_order_eqns14)): \nnops(eqs_14);\nindets(eqs_14);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<,&%\"aG6$\"#9 \"#6&F%6$F'\"#5&F%6$F'\"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$F' \"\"\"&F%6$F'\"#7&F%6$F'\"#8&%\"cG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We solve for the linking coefficients in terms of " }{XPPEDIT 18 0 "c[14];" "6#&%\"cG6#\"#9" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=0:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 88 "sol_14 := solve(\{op(eqs_14)\},indets(eqs_14) minus \{c[14],a[14,1]\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "indets(map(rhs,sol _14));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$&%\"aG6$\"#9\"\"\"&%\"cG6# F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "We choose the node " }{XPPEDIT 18 0 "c[14];" "6#&%\"cG6#\"#9" }{TEXT -1 63 " so that an additional (adapted) order condition is satisfied. " }}{PARA 0 "" 0 "" {TEXT -1 6 "EITHER" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "extra_eqn := add(a[14,i]*add(a[i,j]*add(a[j,k]*add(a [k,l]*c[l]^2,l=2..k-1),\n k=2..j-1),j=2..i-1),i=2..13)=c[ 14]^6/360:" }}}{PARA 0 "" 0 "" {TEXT -1 6 "OR ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "extra_eqn := convert(SO6_13[21],'interpolatio n_order_condition'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "expand(subs(e2,extra_eqn)):\neq_14 := sub s(sol_14,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eq_14G/,,*&#\"Af0+# )o:nUBnN$=>R:\"\"E+$pGD3p>Z)>:9u;cie^\"\"\"*$)&%\"cG6#\"#9\"\"#F+F+!\" \"*&#\"@`omg*=dvWd&yF(RYQ\"DlMk7a%)fB*f22P3GJzDF+*$)F.\"\"$F+F+F+*&#\" Ax;+Y1Z,Gq,2]vvhM\"EgQd];QR%pRIG[L7D<.\"F+*$)F.\"\"%F+F+F3*&#\"A8R+u\" )4q)R1(\\$GMu2)\"E]YVETX)fB*f22P3GJzDF+*$)F.\"\"&F+F+F+*&#\"D4lfCtm=RJ Guopf$Q#[#\"F+uk^&[Va\\sNZX85h_&G*F+*$)F.\"\"'F+F+F+,$*&#F+\"$g$F+FMF+ F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "No te" }{TEXT -1 26 ": This is independent of " }{XPPEDIT 18 0 "a[14,1] " "6#&%\"aG6$\"#9\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lhs(eq_14)-rhs(eq_14); \nfactor(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*&#\"Af0+#)o:nUBnN$= >R:\"\"E+$pGD3p>Z)>:9u;cie^\"\"\"*$)&%\"cG6#\"#9\"\"#F(F(!\"\"*&#\"@`o mg*=dvWd&yF(RYQ\"DlMk7a%)fB*f22P3GJzDF(*$)F+\"\"$F(F(F(*&#\"Ax;+Y1Z,Gq ,2]vvhM\"EgQd];QR%pRIG[L7D<.\"F(*$)F+\"\"%F(F(F0*&#\"A8R+u\")4q)R1(\\$ GMu2)\"E]YVETX)fB*f22P3GJzDF(*$)F+\"\"&F(F(F(*&#\"ArzmCF.!H8-*\\%4yCp# FBF(*$)F+\"\"'F(F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"@`omg*= dvWd&yF(RYQ\"E+$pGD3p>Z)>:9u;cie^\"\"\"*()&%\"cG6#\"#9\"\"#F(,(*&F.F(F *F(F(*&F.F(F+F(!\"\"\"\"$F(F(),&F+F(F(F3F/F(F(F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "14*c[14]^2-1 4*c[14]+3;\nsolve(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,( *&\"#9\"\"\")&%\"cG6#F%\"\"#F&F&*&F%F&F(F&!\"\"\"\"$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"\"\"\"\"#F%*&\"#9!\"\"\"\"(F$F%,&F$F%*&F(F)F* F$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+lB#)*)o!#5$\"+Nw<5JF%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "We take \+ " }{XPPEDIT 18 0 "c[14]=1/2-sqrt(7)/14" "6#/&%\"cG6#\"#9,&*&\"\"\"F*\" \"#!\"\"F**&-%%sqrtG6#\"\"(F*F'F,F," }{TEXT -1 50 " and obtain values for the linking coefficients " }{XPPEDIT 18 0 "a[14,6]" "6#&%\"aG6$ \"#9\"\"'" }{TEXT -1 6 " to " }{XPPEDIT 18 0 "a[14,13]" "6#&%\"aG6$ \"#9\"#8" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "a[14,1]" "6#& %\"aG6$\"#9\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sol := solve(\{op(subs(c[14] =1/2-1/14*7^(1/2),eqs_14))\},\{seq(a[14,i],i=6..13)\});" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%$solG<*/&%\"aG6$\"#9\"\"*,(*&#\"fo$GcsZ)*>%H$) )Q#G(y]oO[rnTKv+vLP:Fu8WHJJ2$\\\"\"ao$Gm8ak**Ri0H_0`?tC=rvL#p8h:AuN%yq rl'*\\*\"\"\"&F(6$F*F1F1!\"\"#\"epB6Ck_#[uVPp>jg]1'e2M)*o,qyaTiKbXT:K! *zr<\"[\"o\"4t!G;\"bpw'3Jd8T:,]iY=e=[Y=!Qsb(oAF1]Z_L]Ub@!RB+W3avYVR=Y&*3q W=X/j&\"?eA'\\Fv9^c*pJlhr9F1*(\"S,/>\"*oYDF1*(\"7,[#z gG*RE%e@&F1\"9GSt4q+_fXMt7F4F;F.M`\"5BC6Oj4/>?:F1F2F1F1#\"?v= USQ(4/rY0U')RV\"\";QV^)>/];eE+Z4*F4*(\"F1F2F1F4#\"; &='Q@x&p1F(GwW[\"8%>aOdt.$HBPc#F1*(\"9&o!fD\" GG\\P]#*[!y?[)*Q$\";yk))oSQQUHmOGLF4*(\" " 0 "" {MPLTEXT 1 0 84 "sm14 := simplify(subs(`uni on`(sol,\{seq(a[14,i]=0,i=2..5)\}),add(a[14,i]^2,i=1..13)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%sm14G,,#\"exhq&pAdn%zIQxN5G'z#G*\\$GcKvvj \\Jn)3K\"))))G9$*)\\f%GqhezhhLH0\"\\\"eI\\@Dv/3NeO%3'p.`v3&[d)[&zYRH;B Y!4WpD\\f<(fE(3\\*=_qkpV_^`Y?]%oJ^\"^x/jv^n2RJ#>&=+Jy='RxgBt-4V%e@y2:[ G))e>V+&ol\"RG>U2suz^bWHeRKALDXUUR=ezc\\f[Fa_=mvG%GvVYtv!p$pz MD)e!)>qLPten\"\"\"*&#\"_w,^3mlw!\\L#p!)Q/-+;Qogq\"41'oF4;2[@ROYa2i3gI A/3!*zD?zrq$p(>.01xmw\"[&G))HeYuY\")ey^mriYWI1B`O.pMvw`YOVVk,da%Q(H\"f v1QwO&e(zzVkiJ(f,*o9\\K[%=.8$GG,\"4h`8@O?lcGd$psB6o;p.ZG/ya+7)z)F)&%\"aG6$\"#9F)F)!\"\"*(\"fx,Jj !>e`DhmD\"p]n,u(*>zM'eCE8Ei(4Ef2_Uy)[N>l%4xrbC$*y%z%ol@Zuc*p=7K;VMb(H. V/)\\1w*)=]sE]nxA4P1%Q7L*yk;#zlm#y%\\QIbkr==\\Aps7c#F)\"axO&4H%oQGvYl/ Zclc9lCFp\\CC3j$*R\"QJV>(\\JBw&zBqWZ_3quFy4m.[Og,ePf,UjeR'>Gz;V>zt@cOs ciE?H6H8'R#HmZ,IqMQ6>9IZ5U%e2!3:;!*F2\"\"(#F)\"\"#F)*&#\"juQ2tx+-#p<^# 4eHes!fn+IHz/`Kb3wq6(*HWa.TC5fKQ9c?XEC!o$*GW4r3o@i)Q:ZW)4m/x#R&p!>!*G# *4&4Z'>II\"\"`up`VJMwz.Jk[IEJhE(3&o>#\\rh#*p,L>Cwg!f$f/z.!3` >2$f?mI'=0;gu#41i_@O4p)RL\\k:cI'[*z5-*[Rc+d$yOY$F)*$)F.F8F)F)F)*&#\"]w LnL-$*QW(yOw,:UJ\"p`VWOlU0O%)=xL&)e2No.6J3N'yyd!R_41Iu<7\"e[fF%fg\"pWy =@UuTMr'=%[@pYA4^:hZ\\:F#fa]5p@t]w]jt76\"gv7w_tq^ff()GDj%>.y$H)\\m*ojg icc-#=A2FUsSI8d9(QXZAOLQNYSSMiMV^=>AjNc*GFk3*zPkOo?=KWqvxX\\8D\"R1%p&3 c4,C'f " 0 "" {MPLTEXT 1 0 26 "plot(sm14,a[14,1]=0..0.1);" }} {PARA 13 "" 1 "" {GLPLOT2D 357 358 358 {PLOTDATA 2 "6%-%'CURVESG6$7S7$ $\"\"!F)$\"3gTAFn4$ym(!#67$$\"3[mmm;arz@!#?$\"3-(G#*3)QHXpF,7$$\"3mLL$ e9ui2%F0$\"3]\">#e&[,dM'F,7$$\"3-nmm\"z_\"4iF0$\"3]R*3q[GPq&F,7$$\"3[m mmT&phN)F0$\"3s95$4po?4&F,7$$\"3KLLe*=)H\\5!#>$\"3V)*o?Y>vI2g!>EF,7$ $\"3gLLL3En$4#FE$\"3Szgcy\\.0AF,7$$\"3wmm;/RE&G#FE$\"3%p>v%=%[)p=F,7$$ \"3A+++D.&4]#FE$\"33@mUYVbD:F,7$$\"3!)*****\\PAvr#FE$\"3(p_J^8k]@\"F,7 $$\"3)******\\nHi#HFE$\"3)y)>d-(*R#\\*!#77$$\"3jmm\"z*ev:JFE$\"3-nxIow RitF_p7$$\"31LLL347TLFE$\"3s>5zr#p8=&F_p7$$\"3cLLLLY.KNFE$\"3G/H=tqsKO F_p7$$\"3!****\\7o7Tv$FE$\"3sc2X%)3Kw@F_p7$$\"3sKLL$Q*o]RFE$\"3S4vV-]v '>\"F_p7$$\"33++D\"=lj;%FE$\"3%HV[d6@]c%!#87$$\"33++vV&RB PN8p(!#97$$\"3gLL$e9Ege%FE$\"3s0k`!Q?c\">Fdr7$$\"3ILLeR\"3Gy%FE$\"3fsS Ti*[Uq#F^r7$$\"3smm;/T1&*\\FE$\"3+$Rm&esn!o)F^r7$$\"3Smm\"zRQb@&FE$\"3 #QWD8?jx%=F_p7$$\"3!****\\(=>Y2aFE$\"39T'3\\u \"F,7$$\"3k+++v.I%)oFE$\"3FacZM*HB6#F,7$$\"3Amm\"zpe*zqFE$\"316xx0IcvC F,7$$\"37+++D\\'QH(FE$\"3'Hh$ep0j0HF,7$$\"3GKLe9S8&\\(FE$\"3Wvv8]isTLF ,7$$\"3]++D1#=bq(FE$\"3W!)\\rLLL3s?6zFE$\"3$QXi;qt)RVF,7 $$\"3)*)**\\7`Wl7)FE$\"3\"*zJbD:h2\\F,7$$\"3[nmmm*RRL)FE$\"3qwj[UER([& F,7$$\"3Smm;a<.Y&)FE$\"3xEY>'fsP6'F,7$$\"3-MLe9tOc()FE$\"3]&[\")\\K\"R onF,7$$\"3u******\\Qk\\*)FE$\"3i$*GuxWE*R(F,7$$\"3!QLL3dg6<*FE$\"3Mb]Y -D)o:)F,7$$\"3-mmmmxGp$*FE$\"3!H+S 9:a'*F,7$$\"3'****\\(=5s#y*FE$\"3F4-[En,W5!#57$$\"3/+++++++5!#=$\"3]ae `fN\">8\"Fgz-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q(a[14,1]6 \"Q!Fi[l-%%VIEWG6$;F($\"\"\"Fd[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "minimize(sm14,loca tion):\nevalf[20](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"16k< " 0 "" {MPLTEXT 1 0 55 "convert(.4 5147062884921438293e-1,rational,7);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$(G\"%dj" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'>3Z ^%!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "convert(.45147062884921438293e-1,rational,8);\nevalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$2$\"%+o" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+#)eq9X!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "a[14,1] = 307/68 00;" "6#/&%\"aG6$\"#9\"\"\"*&\"$2$F(\"%+o!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "e3 := \{c[14]=1/2-1/14*7^(1/2),a[14,1] = 307/6800\};\ne4 := subs(e 3,sol);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G<$/&%\"aG6$\"#9\" \"\"#\"$2$\"%+o/&%\"cG6#F*,&#F+\"\"#F+*&F*!\"\"\"\"(F4F7" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%#e4G<*/&%\"aG6$\"#9\"\"(,&#\">&)\\UCn\";QV^)>/];eE+Z4*F 0*(\"'>]i#R \"*e\">!RJYIZ)o&Gm5v?*pbAUmLwR![V>)H>#RF4*(\"bpf#ym()fG0pkanv9[rb`Beh& 3!\\Y'*oqm%*eKQH\"*)*y<=#*GI**zF0\"bp_t@YrA3B+]KpjrjHp.wW6v`WD,]\\q1]3 J/yY+)o\"3NSh%F4F+F5F0/&F(6$F*\"#6,&#\"9`E[XLItp!zIO#\"8%>aOdt.$HBPc#F 4*(\"9&o!fS,/>\"*oYDF0*(\"7,[#zgG*RE%e@&F0\"9GSt4q+_fXMt7F4F+F5F 4/&F(6$F*\"#8,&#\"\"$\"$#RF0*(FipF0FjpF4F+F5F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "e5 := `unio n`(e2,e3,e4):\n[c[14]=subs(e5,c[14]),seq(a[14,i]=subs(e5,a[14,i]),i=1. .13)]:\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&%\"cG6#\" #9$\"I@fp*4H)=tTFR'Q'Q&\\jx,6$!#S/&%\"aG6$F(\"\"\"$\"Ifqkmj$HOF+/&F.6$F(\"\"($\"Iz\"f!ys \\?\"fpW!*e_X!pX)*pBF+/&F.6$F(\"\")$\"Hk6&3mOd'))ye\"RJ!)4&[clZ'!#R/&F .6$F(\"\"*$\"H\\eu(\\Zk$e9E?(4q&p!*G?5(FX/&F.6$F(\"#5$!I8aq)>JX50Yu?Ro nl!>7;5FX/&F.6$F(\"#6$!H)yN`ZAj=\"QC#\\]Gf#)RkFFFX/&F.6$F(\"#7$!IB8yS, L-u14&4SWZ'o(e)zjF3/&F.6$F(\"#8$!ILD;U)>XYK5Coi;'QaN]f7F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#---------------- -----------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check which of the (adapted) simple order condi tions are satisfied at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "recd := []:\nfor ct to nops (SO6_13) do\n tt := convert(SO6_13[ct],'interpolation_order_conditio n'):\n if expand(subs(e5,lhs(tt)=rhs(tt))) then recd := [op(recd),ct ] end if: \nend do:\nop(recd);\nnops(recd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6B\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6 \"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\" #H\"#I\"#J\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#K" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#------------------- --------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6681 "e5 := \{a[10,2] = 0, a[11, 2] = 0, a[11,3] = 0, a[12,7] = 165712155931984680217128369091361069858 6256573484808662625/13431480411255146477259155104956093505361644432088 109056, a[12,8] = 345685379554677052215495825476969226377187500/747711 67436930077221667203179551347546362089, a[14,4] = 0, a[14,5] = 0, a[11 ,10] = -31155237437111730665923206875/392862141594230515010338956291, \+ a[14,2] = 0, a[14,3] = 0, a[9,1] = -1472514264486215803881384708877264 246346044433307094207829051978044531801133057155/124689480162003200115 7059621643986024803301558393487900440453636168046069686436608, c[4] = \+ 1023/6400, c[11] = 47/50, c[12] = 1, c[13] = 1, a[11,1] = -23426598458 14086836951207140065609179073838476242943917/1358480961351056777022231 400139158760857532162795520000, c[2] = 1/20, c[3] = 341/3200, a[7,2] = 0, a[7,3] = 0, c[8] = 943/1000, c[9] = 7067558016280/7837150160667, c [10] = 909/1000, a[9,2] = 0, a[9,3] = 0, c[5] = 39/100, c[6] = 93/200, c[7] = 31/200, a[5,3] = -3899844/2907025, a[5,4] = 3982992/2907025, a [8,1] = -1221101821869329/690812928000000, a[8,2] = 0, a[8,6] = 150140 8353528689/265697280000000, a[8,7] = 6070139212132283/92502016000000, \+ a[2,1] = 1/20, a[3,1] = -7161/1024000, a[7,6] = 5611/283500, a[6,1] = \+ 5611/114400, a[6,2] = 0, a[4,2] = 0, a[4,3] = 3069/25600, a[3,2] = 116 281/1024000, a[4,1] = 1023/25600, a[5,1] = 4202367/11628100, a[5,2] = \+ 0, a[6,5] = 923521/5106400, a[7,1] = 21173/343200, a[9,4] = -517229431 1085668458375175655246981230039025336933699114138315270772319372469280 000/124619381004809145897278630571215298365257079410236252921850936749 076487132995191, a[9,5] = -1207067925846925480797893644173318794948457 1516120469966534514296406891652614970375/27220311547616572217104781845 31100699497284085048389015085076961673446140398628096, a[8,3] = 0, a[8 ,4] = -125/2, a[6,3] = 0, a[6,4] = 31744/135025, a[8,5] = -10240306079 59889/168929280000000, a[9,6] = 78012515584389364132309055253043103656 7795592568497182701460674803126770111481625/18311042541273197219788987 4507158786859226102980861859505241443073629143100805376, a[7,4] = 8602 624/76559175, a[7,5] = -26782109/689364000, a[10,8] = 2120832024345190 82281842245535894/20022426044775672563822865371173879, a[10,6] = 21127 670214172802870128286992003940810655221489/467947387799789290614582269 7976708633673728000, a[10,9] = -26984049294008425187211664850871297985 62269848229517793703413951226714583/4695456749139343150770004420808711 41884676035902717550325616728175875000000, a[13,11] = 2820355431831908 40068750/12295407629873040425991, a[13,12] = -306814272936976936753/12 99331183183744997286, a[11,4] = -996286030132538159613930889652/163530 68885996164905464325675, a[13,1] = 44901867737754616851973/10140464099 80231013380680, a[10,5] = -180269259803172281163724663224981097/381009 22558256871086579832832000000, a[10,4] = -2046274952459104910540336523 9069/454251913499893469596231268750, a[13,5] = 0, a[13,6] = 7916386751 91615279648100000/2235604725089973126411512319, a[12,5] = -45834939744 84572912949314673356033540575/451957703655250747157313034270335135744, a[12,6] = 2346305388553404258656258473446184419154740172519949575/256 726716407895402892744978301151486254183185289662464, a[12,10] = 402795 45832706233433100438588458933210937500/8896460842799482846916972126377 338947215101, a[12,4] = -16957088714171468676387054358954754000/143690 415119654683326368228101570221, a[10,7] = 3186072351736493124051512658 49660869927653414425413/6714716715558965303132938072935465423910912000 000, a[12,9] = -320589096271707254279143431215272753400810277402321024 0571361570757249056167015230160352087048674542196011/94756954968396581 4783015124451273604984657747127257615372449205973192657306017239103491 074738324033259120, a[12,1] = -286655699182566397177829532910103388753 4912787724034363/86822671161926270301121392501614361203066923379533824 0, a[12,2] = 0, a[12,3] = 0, a[13,3] = 0, a[13,4] = 0, a[13,2] = 0, a[ 13,7] = 3847749490868980348119500000/15517045062138271618141237517, a[ 13,8] = -13734512432397741476562500000/875132892924995907746928783, a[ 13,9] = 12274765470313196878428812037740635050319234276006986398294443 554969616342274215316330684448207141/489345147493715517650385834143510 934888829280686609654482896526796523353052166757299452852166040, a[13, 10] = -9798363684577739445312500000/308722986341456031822630699, a[12, 11] = -6122933601070769591613093993993358877250/1050517001510235513198 246721302027675953, a[11,9] = 3007606697681025178342324975654524349466 72266195876496371874262392684852243925359864884962513/4655443337501346 4555850653366045056037608247796155212857518928103156804923641066745243 98280000, a[14,1] = 307/6800, a[11,7] = 890722993756379186418929622095 833835264322635782294899/139212420013951126575019419555940138228301198 03764736, a[11,8] = 161021426143124178389075121929246710833125/1099720 7722131034650667041364346422894371443, a[11,6] = 209808223450967602922 24086794978105312644533925634933539/3775889992007550803878727839115494 641972212962174156800, a[11,5] = -26053085959256534152588089363841/437 7552804565683061011299942400, a[10,3] = 0, a[9,7] = 664113122959911642 134782135839106469928140328160577035357155340392950009492511875/151784 6559858624813633302310729534917527976515008907830113994325301687782317 0816, a[9,8] = 1033284818445201560405683676728665685912400779697066804 6446015775000000/13127035500360336480738342487407279145379720286389501 65249582733679393783, a[10,1] = -2905557336033741508853861844223103644 1314060511/22674759891089577691327962602370597632000000000, a[14,7] = \+ 35514316969207250641724424985/147161653169956511475274962258-481702304 929173090974271575/294323306339913022950549924516*7^(1/2), a[14,8] = 6 71994922390044434145634375/90947002658165004198514338-4634544296279707 22580265625/181894005316330008397028676*7^(1/2), a[14,6] = 46045486303 1265521343678403/2355798527514165229982023734-283438859697640884437065 685/4711597055028330459964047468*7^(1/2), a[14,10] = 18875353273971576 7256965625/33283666294238384068886478-168249974410951846618609375/6656 7332588476768137772956*7^(1/2), a[14,9] = -152041753025355398038004360 2085925886494262345387178427997982057187814442042273423080071783/39219 2981943480397633664222556992075106628568847304631390191589139262501961 993074284749200+799930289218177898912938325894667068964649008561582353 557148147567546469052859876678259/461403508168800467804310850067049500 125445375114476036929637163693250002308227146217352*7^(1/2), a[14,11] \+ = -236307906973303345482653/25637232930373573654194+173345833072833817 590685/51274465860747147308388*7^(1/2), c[14] = 1/2-1/14*7^(1/2), a[14 ,12] = 11352128098668146659861/254668911904014019468056-52158426399286 07924801/127334455952007009734028*7^(1/2), a[14,13] = 3/392-3/392*7^(1 /2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 24 "calculation for stage 15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order co nditions can be adapted to give a method of stage by stage constructio n for an interpolation scheme that avoids dealing with the weight poly nomials for a given stage (corresponding to an \"approximate\" interpo lation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "SO7_14 := SimpleOrderConditions(7,14,'expanded '):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "whch := [1,2,4,8,16,21,27,31,32,64]:\ninterp_order_e qns15 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_14[ct],'in terpolation_order_condition'):\n interp_order_eqns15 := [op(interp_o rder_eqns15),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can b e specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 410 "interp_order_eqns15 := [add(a[15,i],i=1..14)=c[15],s eq(op(StageOrderConditions(i,15..15,'expanded')),i=2..7),\n add(a[1 5,i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*c[l]^2,l=2..k-1),k=2..j-1),\n \+ j=2..i-1),i=2..14)=c[15]^6/360, #21\n add(a [15,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..14)=c[15] ^6/120, #27\n add(a[15,i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..14)=c[15 ]^6/30]: #31" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We specify the node for this stage immediately, namely \+ " }{XPPEDIT 18 0 "c[15] = 57/125;" "6#/&%\"cG6#\"#:*&\"#d\"\"\"\"$D\"! \"\"" }{TEXT -1 80 ", and have enough equations to determine the corre sponding linking coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "e6 := `union`(e5,\{c[15]=57/125,seq(a[15,i]=0,i=2..5)\}):\neqs_ 15 := expand(subs(e6,interp_order_eqns15)):\nnops(eqs_15);\nindets(eqs _15);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<,&%\"aG6$\"#:\"#9&F%6$F'\"\"\"&F%6$F'\"#8&F%6$F '\"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$F'\"#5&F%6$F'\"#6&F%6$F '\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[so lve]:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "e7 := solve(\{o p(eqs_15)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "e8 := `union`(e6,e7):\n[se q(a[15,i]=subs(e8,a[15,i]),i=1..14)]:\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&%\"aG6$\"#:\"\"\"$\"I)o3s%3O&=G=&Q7?Oe)Gze$\\!#T /&F&6$F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F& 6$F(\"\"'$\"I'f*yk4*QhPa&>lOo%=t^7,)F,/&F&6$F(\"\"($\"Ib0;!*[0UZn*)*4$ )R_mr]&4@!#S/&F&6$F(\"\")$!H4Gqh=n'>?r@y1Y\"z8SZx(!#Q/&F&6$F(\"\"*$\"I 0$f#GJ8kmI)*zIl#4C?Zz5$!#R/&F&6$F(\"#5$!H_zbpUN6_/NO8&y*HL\"*zh%FR/&F& 6$F(\"#6$\"Hnc9XZW&)H`k\\Px]'GiR(G*FR/&F&6$F(\"#7$!I$=x/?)*Rh7c!QQh,.= E_#)p!#U/&F&6$F(\"#8$\"HnA+k\\-Cml;nR%))*)yL+)z\"Ffo/&F&6$F(\"#9$\"IN1 #**)psaQm!3\"R\\,j-Qk(>\"FK" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#---------------------------------------------- -----------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check w hich of the (adapted) simple order conditions are satisfied at this st age." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "recd := []:\nfor ct to nops(SO7_14) do\n tt := con vert(SO7_14[ct],'interpolation_order_condition'):\n if expand(subs(e 8,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop(r ecd);\nnops(recd);\nop(\{seq(i,i=1..nops(SO7_14))\} minus \{op(recd)\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6J\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"' \"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"# B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#Z\"#^\"#`\"#d\"#f\"#g\"#i\"#k " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6:\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#[\"# \\\"#]\"#_\"#a\"#b\"#c\"#e\"#h\"#j" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 70 "#--------------------------------------- ------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8300 "e8 := \{a[8,3] = 0, a[11,4] = -9962860301325 38159613930889652/16353068885996164905464325675, c[14] = 1/2-1/14*7^(1 /2), a[12,5] = -4583493974484572912949314673356033540575/4519577036552 50747157313034270335135744, a[11,3] = 0, a[8,4] = -125/2, a[6,3] = 0, \+ a[11,2] = 0, a[10,4] = -20462749524591049105403365239069/4542519134998 93469596231268750, a[13,5] = 0, a[12,7] = 1657121559319846802171283690 913610698586256573484808662625/134314804112551464772591551049560935053 61644432088109056, a[12,8] = 34568537955467705221549582547696922637718 7500/74771167436930077221667203179551347546362089, a[10,2] = 0, a[14,1 2] = 11352128098668146659861/254668911904014019468056-5215842639928607 924801/127334455952007009734028*7^(1/2), a[14,13] = 3/392-3/392*7^(1/2 ), a[14,5] = 0, a[11,10] = -31155237437111730665923206875/392862141594 230515010338956291, a[8,6] = 1501408353528689/265697280000000, a[8,7] \+ = 6070139212132283/92502016000000, a[7,2] = 0, a[7,3] = 0, c[8] = 943/ 1000, c[9] = 7067558016280/7837150160667, c[10] = 909/1000, a[9,2] = 0 , a[9,1] = -1472514264486215803881384708877264246346044433307094207829 051978044531801133057155/124689480162003200115705962164398602480330155 8393487900440453636168046069686436608, c[4] = 1023/6400, a[6,2] = 0, a [4,2] = 0, a[9,3] = 0, c[5] = 39/100, c[6] = 93/200, c[7] = 31/200, a[ 5,3] = -3899844/2907025, a[14,4] = 0, c[11] = 47/50, c[12] = 1, c[13] \+ = 1, a[2,1] = 1/20, a[3,1] = -7161/1024000, a[5,4] = 3982992/2907025, \+ a[8,1] = -1221101821869329/690812928000000, a[8,2] = 0, a[11,1] = -234 2659845814086836951207140065609179073838476242943917/13584809613510567 77022231400139158760857532162795520000, c[2] = 1/20, c[3] = 341/3200, \+ a[7,6] = 5611/283500, a[6,1] = 5611/114400, a[7,1] = 21173/343200, a[9 ,4] = -517229431108566845837517565524698123003902533693369911413831527 0772319372469280000/12461938100480914589727863057121529836525707941023 6252921850936749076487132995191, a[4,1] = 1023/25600, a[5,1] = 4202367 /11628100, a[14,2] = 0, a[14,3] = 0, a[4,3] = 3069/25600, a[3,2] = 116 281/1024000, a[5,2] = 0, a[6,5] = 923521/5106400, a[9,5] = -1207067925 8469254807978936441733187949484571516120469966534514296406891652614970 375/272203115476165722171047818453110069949728408504838901508507696167 3446140398628096, a[12,4] = -16957088714171468676387054358954754000/14 3690415119654683326368228101570221, a[10,7] = 318607235173649312405151 265849660869927653414425413/671471671555896530313293807293546542391091 2000000, a[12,9] = -32058909627170725427914343121527275340081027740232 10240571361570757249056167015230160352087048674542196011/9475695496839 6581478301512445127360498465774712725761537244920597319265730601723910 3491074738324033259120, a[13,4] = 0, a[13,2] = 0, a[13,7] = 3847749490 868980348119500000/15517045062138271618141237517, a[14,6] = 4604548630 31265521343678403/2355798527514165229982023734-28343885969764088443706 5685/4711597055028330459964047468*7^(1/2), a[14,10] = 1887535327397157 67256965625/33283666294238384068886478-168249974410951846618609375/665 67332588476768137772956*7^(1/2), a[12,3] = 0, a[10,3] = 0, a[14,8] = 6 71994922390044434145634375/90947002658165004198514338-4634544296279707 22580265625/181894005316330008397028676*7^(1/2), a[10,5] = -1802692598 03172281163724663224981097/38100922558256871086579832832000000, a[13,1 ] = 44901867737754616851973/1014046409980231013380680, a[11,9] = 30076 0669768102517834232497565452434946672266195876496371874262392684852243 925359864884962513/465544333750134645558506533660450560376082477961552 1285751892810315680492364106674524398280000, a[12,11] = -6122933601070 769591613093993993358877250/1050517001510235513198246721302027675953, \+ a[11,6] = 20980822345096760292224086794978105312644533925634933539/377 5889992007550803878727839115494641972212962174156800, a[11,5] = -26053 085959256534152588089363841/4377552804565683061011299942400, a[11,8] = 161021426143124178389075121929246710833125/10997207722131034650667041 364346422894371443, a[7,4] = 8602624/76559175, a[7,5] = -26782109/6893 64000, a[9,6] = 780125155843893641323090552530431036567795592568497182 701460674803126770111481625/183110425412731972197889874507158786859226 102980861859505241443073629143100805376, a[13,9] = 1227476547031319687 8428812037740635050319234276006986398294443554969616342274215316330684 448207141/489345147493715517650385834143510934888829280686609654482896 526796523353052166757299452852166040, a[13,8] = -137345124323977414765 62500000/875132892924995907746928783, a[9,8] = 10332848184452015604056 836767286656859124007796970668046446015775000000/131270355003603364807 3834248740727914537972028638950165249582733679393783, a[10,1] = -29055 573360337415088538618442231036441314060511/226747598910895776913279626 02370597632000000000, a[14,9] = -1520417530253553980380043602085925886 494262345387178427997982057187814442042273423080071783/392192981943480 3976336642225569920751066285688473046313901915891392625019619930742847 49200+7999302892181778989129383258946670689646490085615823535571481475 67546469052859876678259/4614035081688004678043108500670495001254453751 14476036929637163693250002308227146217352*7^(1/2), a[14,11] = -2363079 06973303345482653/25637232930373573654194+173345833072833817590685/512 74465860747147308388*7^(1/2), a[14,1] = 307/6800, a[11,7] = 8907229937 56379186418929622095833835264322635782294899/1392124200139511265750194 1955594013822830119803764736, a[13,3] = 0, a[12,1] = -2866556991825663 971778295329101033887534912787724034363/868226711619262703011213925016 143612030669233795338240, a[12,2] = 0, a[10,9] = -26984049294008425187 21166485087129798562269848229517793703413951226714583/4695456749139343 15077000442080871141884676035902717550325616728175875000000, a[13,10] \+ = -9798363684577739445312500000/308722986341456031822630699, a[13,6] = 791638675191615279648100000/2235604725089973126411512319, a[14,7] = 3 5514316969207250641724424985/147161653169956511475274962258-4817023049 29173090974271575/294323306339913022950549924516*7^(1/2), a[10,8] = 21 2083202434519082281842245535894/20022426044775672563822865371173879, a [10,6] = 21127670214172802870128286992003940810655221489/4679473877997 892906145822697976708633673728000, a[12,6] = 2346305388553404258656258 473446184419154740172519949575/256726716407895402892744978301151486254 183185289662464, a[12,10] = 402795458327062334331004385884589332109375 00/8896460842799482846916972126377338947215101, a[13,11] = 28203554318 3190840068750/12295407629873040425991, a[13,12] = -3068142729369769367 53/1299331183183744997286, a[9,7] = 6641131229599116421347821358391064 69928140328160577035357155340392950009492511875/1517846559858624813633 3023107295349175279765150089078301139943253016877823170816, a[6,4] = 3 1744/135025, a[8,5] = -1024030607959889/168929280000000, a[15,10] = 54 365264025507653851977480738208/4466478390356713423359819140625-1496294 528311131416682934929792/235786943683989396406626953125*7^(1/2), a[15, 9] = -3067066459661270994000493048546286135407493899133250841529092716 875641340142781006771180215525814345889446383/373340719218227781410511 6532466971854315408940785290943015262808201014351289114054103491608932 80029296875000+1816647611984076940241842059764388009599469326146254865 545587094445401874686830164148895260773137/424475654667905797096751747 915993683544031508184382874685996435033916010685878289836883544921875* 7^(1/2), a[15,14] = 172678683744/3814697265625*7^(1/2), a[15,1] = 5719 6651428918572875631366996149/1245822888041363915313720703125000+108917 02347425878099023621466959/8357395207277482931896209716796875*7^(1/2), a[15,6] = -245534888302358312351817429197184/870045532021400261929818 7255859375*7^(1/2)+4116869726480787181292612488487584/2659853483608280 8007568743896484375, a[15,13] = -15630718068/3814697265625+6167095848/ 3814697265625*7^(1/2), a[15,7] = 631476617959696547824260385843267488/ 2705957914016858192337687904052734375-33227466731347290455944055132270 208/3922838396059942349838659979248046875*7^(1/2), a[15,11] = -3598066 799092816058284144620546/146286964102678984417724609375+65570869112872 951387443794088672/5120043743593764454620361328125*7^(1/2), a[15,8] = \+ 598933003512786415989929347488/28972691439853083012428515625-147510965 40440194135375671464064/1371942153475395989706173828125*7^(1/2), a[15, 12] = -1927555453883917676542797699/276054308640062825012207031250, a[ 15,4] = 0, a[15,5] = 0, c[15] = 57/125, a[15,2] = 0, a[15,3] = 0\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 16" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order conditio ns can be adapted to give a method of stage by stage construction for \+ an interpolation scheme that avoids dealing with the weight polynomial s for a given stage (corresponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_15 := SimpleOrderConditions(7,15,'expanded'):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "whch := [1,2,4,8,16,21,27,31,32,63,64]:\ninterp_order_eqns16 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_15[ct],'interpolat ion_order_condition'):\n interp_order_eqns16 := [op(interp_order_eqn s16),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specifi ed explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 481 "interp_order_eqns16 := [add(a[16,i],i=1..15)=c[16],seq(op(StageOr derConditions(i,16..16,'expanded')),i=2..7),\n add(a[16,i]*add(a[i, j]*add(a[j,k]*add(a[k,l]*c[l]^2,l=2..k-1),k=2..j-1),\n \+ j=2..i-1),i=2..15)=c[16]^6/360, #21\n add(a[16,i]*add(a[ i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..15)=c[16]^6/120, #27 \n add(a[16,i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..15)=c[16]^6/30, #31 \n add(a[16,i]*add(a[i,j]*c[j]^5,j=2..i-1),i=2..15)=c[16]^7/42]: # 63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the node " }{XPPEDIT 18 0 "c[16] = 1163/10000;" "6#/&%\"cG6# \"#;*&\"%j6\"\"\"\"&++\"!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "e9 := `union`(e8,\{c[16]=1163/10000,seq(a[16,i] =0,i=2..5)\}):\neqs_16 := expand(subs(e9,interp_order_eqns16)):\nnops( eqs_16);\nindets(eqs_16);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<-&%\"aG6$\"#;\"\"\"&F%6$F'\" #7&F%6$F'\"\"*&F%6$F'\"\")&F%6$F'\"\"(&F%6$F'\"\"'&F%6$F'\"#6&F%6$F'\" #8&F%6$F'\"#5&F%6$F'\"#9&F%6$F'\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e10 := solve(\{op(eqs_16)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "e11 := `union`(e9,e10):\nseq(a[16,i]=subs(e11,a[16,i]),i=1..15 ):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "61/&%\"aG6$\"#;\"\" \"$\"IA*p\\C)G=*=T\"3ga<`$H1\\%\\!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$ F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"I1NK=KuN\\dQ!=Wt]fYm'f >F+/&F%6$F'\"\"($\"IK-m\\\"pqLdGnC:e')*)>\\y5\"!#S/&F%6$F'\"\")$!I/(>+ \\0N\\'p(=?td\"eYE*G:)FJ/&F%6$F'\"\"*$\"IJk\\lG)o%pkuG)G(oC1[$o6'FJ/&F %6$F'\"#5$!I)4>[A4'))\\E9yEADq&=)G#=*FJ/&F%6$F'\"#6$\"IYS!yyT\"Q]Jf[2B U--rYX6!#R/&F%6$F'\"#7$!I0b\"G8SW4,,))e=D!)4MWVS#F+/&F%6$F'\"#8$!IL!)H 9UW?dX/%fa7-\"QGRQT!#U/&F%6$F'\"#9$!Iky)\\)HH@dk\"[RbZ)*y/x9P(F+/&F%6$ F'\"#:$\"IclN#H%>,nciSC&f!H\")fMt9F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "#------------------------------------ ---------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We c an check which of the (adapted) simple order conditions are satisfied \+ at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "recd := []:\nfor ct to nops(SO7_15) do\n tt : = convert(SO7_15[ct],'interpolation_order_condition'):\n if expand(s ubs(e11,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do: \nop(recd);\nnops(recd);\nop(\{seq(i,i=1..nops(SO7_15))\} minus \{op(r ecd)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6R\"\"\"\"\"#\"\"$\"\"%\"\"& \"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\" #A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#O\"#R\"#V\"#Y\"#Z\"#\\\"# ^\"#`\"#a\"#c\"#d\"#f\"#g\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"#L\"#M\"#N\"#P\"#Q\"#S\"# T\"#U\"#W\"#X\"#[\"#]\"#_\"#b\"#e\"#h" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "#------------------------------------ ---------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "e11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13437 "e11 := \{a[8,3] = 0, a[11,4] = -9962860301 32538159613930889652/16353068885996164905464325675, c[14] = 1/2-1/14*7 ^(1/2), a[12,5] = -4583493974484572912949314673356033540575/4519577036 55250747157313034270335135744, a[11,3] = 0, a[8,4] = -125/2, a[6,3] = \+ 0, a[11,2] = 0, a[10,4] = -20462749524591049105403365239069/4542519134 99893469596231268750, a[13,5] = 0, a[12,7] = 1657121559319846802171283 690913610698586256573484808662625/134314804112551464772591551049560935 05361644432088109056, a[12,8] = 34568537955467705221549582547696922637 7187500/74771167436930077221667203179551347546362089, a[10,2] = 0, a[1 4,12] = 11352128098668146659861/254668911904014019468056-5215842639928 607924801/127334455952007009734028*7^(1/2), a[14,13] = 3/392-3/392*7^( 1/2), a[14,5] = 0, a[11,10] = -31155237437111730665923206875/392862141 594230515010338956291, a[8,6] = 1501408353528689/265697280000000, a[8, 7] = 6070139212132283/92502016000000, a[7,2] = 0, a[7,3] = 0, c[8] = 9 43/1000, c[9] = 7067558016280/7837150160667, c[10] = 909/1000, a[9,2] \+ = 0, a[9,1] = -1472514264486215803881384708877264246346044433307094207 829051978044531801133057155/124689480162003200115705962164398602480330 1558393487900440453636168046069686436608, c[4] = 1023/6400, a[6,2] = 0 , a[4,2] = 0, a[9,3] = 0, c[5] = 39/100, c[6] = 93/200, c[7] = 31/200, a[5,3] = -3899844/2907025, a[14,4] = 0, c[11] = 47/50, c[12] = 1, c[1 3] = 1, a[2,1] = 1/20, a[3,1] = -7161/1024000, a[5,4] = 3982992/290702 5, a[8,1] = -1221101821869329/690812928000000, a[8,2] = 0, a[11,1] = - 2342659845814086836951207140065609179073838476242943917/13584809613510 56777022231400139158760857532162795520000, c[2] = 1/20, c[3] = 341/320 0, a[7,6] = 5611/283500, a[6,1] = 5611/114400, a[7,1] = 21173/343200, \+ a[9,4] = -517229431108566845837517565524698123003902533693369911413831 5270772319372469280000/12461938100480914589727863057121529836525707941 0236252921850936749076487132995191, a[4,1] = 1023/25600, a[5,1] = 4202 367/11628100, a[14,2] = 0, a[14,3] = 0, a[4,3] = 3069/25600, a[3,2] = \+ 116281/1024000, a[5,2] = 0, a[6,5] = 923521/5106400, a[9,5] = -1207067 9258469254807978936441733187949484571516120469966534514296406891652614 970375/272203115476165722171047818453110069949728408504838901508507696 1673446140398628096, a[16,7] = 125183725987006091358284807772832081560 4583364155823495764005237660669814915991480157717229809815277573980104 7/12917725976661339897426511964385702783984431335430335690360713394329 7973977319598535515764891840000000000000000+17615228617354110108508911 7905026501113264968494721227239621147981802702218634797919435901693608 9217819331021/33586087539319483733308931107402827238359521472118872794 9378548252574732341030956192340988718784000000000000000*7^(1/2), a[16, 9] = 68743614876734800062859317305274277780650783519032025828979300088 5234889881117832016344667077769765702709805954192296963021552895479234 993257819821106998915574193829101860119823330279/112536814843150050913 6765423497373044075977108341847582170921264810018468142400631425900826 9298745284780477171202703304426061241025574183284196896086932221863180 96000000000000000000000+2963232731987952356517772319258878191238255598 1099887894167361193662448225807977952199471123771928706835338605328112 3249786227048915757888049930193669040074737641797319943395161/94568751 9690336562299802876888548776534434544825082001824303583873965099279328 2617024376696889701919983594261514876726408454824391238809482518400073 05228727998400000000000000000000*7^(1/2), a[16,1] = 571262365974722592 1475087020856744131230749797119548738128074641162573025768022263130281 64589259868449219745217/1107721982975045860773527525379031925113390736 5668324124318115180733432062987800250708692452000000000000000000000-41 8059397241678150958395224952493598697047777024498277797593332867900350 838856267902560305003793843931777113/521280933164727463893424717825426 7882886544642667446646737936555639262147288376588568796448000000000000 00000000*7^(1/2), a[12,4] = -16957088714171468676387054358954754000/14 3690415119654683326368228101570221, a[10,7] = 318607235173649312405151 265849660869927653414425413/671471671555896530313293807293546542391091 2000000, a[12,9] = -32058909627170725427914343121527275340081027740232 10240571361570757249056167015230160352087048674542196011/9475695496839 6581478301512445127360498465774712725761537244920597319265730601723910 3491074738324033259120, a[13,4] = 0, a[13,2] = 0, a[13,7] = 3847749490 868980348119500000/15517045062138271618141237517, a[14,6] = 4604548630 31265521343678403/2355798527514165229982023734-28343885969764088443706 5685/4711597055028330459964047468*7^(1/2), a[14,10] = 1887535327397157 67256965625/33283666294238384068886478-168249974410951846618609375/665 67332588476768137772956*7^(1/2), a[12,3] = 0, a[10,3] = 0, a[14,8] = 6 71994922390044434145634375/90947002658165004198514338-4634544296279707 22580265625/181894005316330008397028676*7^(1/2), a[10,5] = -1802692598 03172281163724663224981097/38100922558256871086579832832000000, a[13,1 ] = 44901867737754616851973/1014046409980231013380680, a[11,9] = 30076 0669768102517834232497565452434946672266195876496371874262392684852243 925359864884962513/465544333750134645558506533660450560376082477961552 1285751892810315680492364106674524398280000, a[12,11] = -6122933601070 769591613093993993358877250/1050517001510235513198246721302027675953, \+ a[11,6] = 20980822345096760292224086794978105312644533925634933539/377 5889992007550803878727839115494641972212962174156800, a[11,5] = -26053 085959256534152588089363841/4377552804565683061011299942400, a[11,8] = 161021426143124178389075121929246710833125/10997207722131034650667041 364346422894371443, a[7,4] = 8602624/76559175, a[7,5] = -26782109/6893 64000, a[9,6] = 780125155843893641323090552530431036567795592568497182 701460674803126770111481625/183110425412731972197889874507158786859226 102980861859505241443073629143100805376, a[13,9] = 1227476547031319687 8428812037740635050319234276006986398294443554969616342274215316330684 448207141/489345147493715517650385834143510934888829280686609654482896 526796523353052166757299452852166040, a[13,8] = -137345124323977414765 62500000/875132892924995907746928783, a[9,8] = 10332848184452015604056 836767286656859124007796970668046446015775000000/131270355003603364807 3834248740727914537972028638950165249582733679393783, a[10,1] = -29055 573360337415088538618442231036441314060511/226747598910895776913279626 02370597632000000000, a[14,9] = -1520417530253553980380043602085925886 494262345387178427997982057187814442042273423080071783/392192981943480 3976336642225569920751066285688473046313901915891392625019619930742847 49200+7999302892181778989129383258946670689646490085615823535571481475 67546469052859876678259/4614035081688004678043108500670495001254453751 14476036929637163693250002308227146217352*7^(1/2), a[14,11] = -2363079 06973303345482653/25637232930373573654194+173345833072833817590685/512 74465860747147308388*7^(1/2), a[14,1] = 307/6800, a[11,7] = 8907229937 56379186418929622095833835264322635782294899/1392124200139511265750194 1955594013822830119803764736, a[13,3] = 0, a[12,1] = -2866556991825663 971778295329101033887534912787724034363/868226711619262703011213925016 143612030669233795338240, a[12,2] = 0, a[10,9] = -26984049294008425187 21166485087129798562269848229517793703413951226714583/4695456749139343 15077000442080871141884676035902717550325616728175875000000, a[13,10] \+ = -9798363684577739445312500000/308722986341456031822630699, a[13,6] = 791638675191615279648100000/2235604725089973126411512319, a[14,7] = 3 5514316969207250641724424985/147161653169956511475274962258-4817023049 29173090974271575/294323306339913022950549924516*7^(1/2), a[10,8] = 21 2083202434519082281842245535894/20022426044775672563822865371173879, a [10,6] = 21127670214172802870128286992003940810655221489/4679473877997 892906145822697976708633673728000, a[12,6] = 2346305388553404258656258 473446184419154740172519949575/256726716407895402892744978301151486254 183185289662464, a[12,10] = 402795458327062334331004385884589332109375 00/8896460842799482846916972126377338947215101, a[13,11] = 28203554318 3190840068750/12295407629873040425991, a[13,12] = -3068142729369769367 53/1299331183183744997286, a[9,7] = 6641131229599116421347821358391064 69928140328160577035357155340392950009492511875/1517846559858624813633 3023107295349175279765150089078301139943253016877823170816, a[6,4] = 3 1744/135025, a[8,5] = -1024030607959889/168929280000000, a[16,13] = -5 6937797859785409375631098707008813998527540294648257184713456009912422 18048279830357689/6623260257269323853870635275237719261692995498645959 12922410705613811200000000000000000000+1116058978133290227473643648991 308872297960011076080647100654462877250873303116055359689/662326025726 9323853870635275237719261692995498645959129224107056138112000000000000 00000000*7^(1/2), a[16,15] = 12566723979477860207267692556004160956197 2597583209979258314845487667433090419263759799/19043248758339535773056 04826984240424690363045991229128931126504895049076310016000000000-5902 8946739025681742626285442331126491223596724290616786809339665195402731 038729879/304691980133432572368896772317478467950458087358596660628980 2407832078522096025600000*7^(1/2), a[16,6] = -997452187886603466500121 5093798908273380361965165365845681748561480350415538556069039355436033 6092065584815633/12097215626349617187624215881471948530761194836093010 88106017767355971757142551992626941965147200000000000000000+9332073356 0040239498721445105436146592080370238787178794613177818778456707219181 79975766545310520465203275977/2419443125269923437524843176294389706152 2389672186021762120355347119435142851039852538839302944000000000000000 0*7^(1/2), a[16,8] = -547598389828861834598181333883759080061418585501 53990686846978348863143039759306003458396414251841877395553/3030710018 0597417208049096613010446805012644086401478209258240042904260490962503 761623721182105600000000000+132843617879870751739882462233453607294562 68579730365891407251773114191650397356739750286947316947113707/3544690 0796020371003566194869018066438611279633218103168723087769478667240891 817265056983838720000000000*7^(1/2), a[16,10] = -103426333826764901372 6626367070353641944586812274059324287684035797195504387677241159024633 249191823996323/822424408623777187146783153818399345197713720767960756 194595910492784789119961559213755380633600000000000+274263425446302069 9447644274499181977315922821414432551248566017099173343536444217982959 50196247775939471/2138303462421820686581636199927838297514055673996697 966105949367281240451711900053955763989647360000000000*7^(1/2), a[16,1 2] = -1700668024857780947303857318649956084860573315650681108553264313 641197383910289904206751497096329440010629/270418821851798197921577276 0946666451031013390150181735548114903923055225824021756875680000000000 0000000000+18110923375969700228455846935056478436359098753435265103392 713976023265962815946771736872563829234342887/123348936283276370981772 0908501988205733444704279030265337736622842095366165343257522240000000 000000000000*7^(1/2), a[16,11] = 6623909106040336836026262070216648184 0242021269701059559317336313240648392339911854838860941565539670913883 /266129688568183402539243009457627860959737366888672888383931360688306 74315050279167258323200000000000000000-9009365467871791955400608463323 7473317647606687760893597983724578633330182810231678808929007209261130 4509/17741979237878893502616200630508524063982491125911525892262090712 55378287670018611150554880000000000000000*7^(1/2), a[16,14] = 36611364 2577897032178883959289263402896100850925956136650174323553270486082847 776282814967/179923786915305271897978397237983483914152734336378647080 28281999192834400000000000000000000-1023478027503079403148682342138509 935107943980471305000079396396912342658754866704389719/287878059064488 4350367654355807735742626443749382058353284525119870853504000000000000 0000*7^(1/2), a[16,3] = 0, a[16,4] = 0, a[16,5] = 0, a[15,10] = 543652 64025507653851977480738208/4466478390356713423359819140625-14962945283 11131416682934929792/235786943683989396406626953125*7^(1/2), a[15,9] = -30670664596612709940004930485462861354074938991332508415290927168756 41340142781006771180215525814345889446383/3733407192182277814105116532 4669718543154089407852909430152628082010143512891140541034916089328002 9296875000+18166476119840769402418420597643880095994693261462548655455 87094445401874686830164148895260773137/4244756546679057970967517479159 93683544031508184382874685996435033916010685878289836883544921875*7^(1 /2), a[15,14] = 172678683744/3814697265625*7^(1/2), a[15,1] = 57196651 428918572875631366996149/1245822888041363915313720703125000+1089170234 7425878099023621466959/8357395207277482931896209716796875*7^(1/2), a[1 5,6] = -245534888302358312351817429197184/8700455320214002619298187255 859375*7^(1/2)+4116869726480787181292612488487584/26598534836082808007 568743896484375, a[15,13] = -15630718068/3814697265625+6167095848/3814 697265625*7^(1/2), a[15,7] = 631476617959696547824260385843267488/2705 957914016858192337687904052734375-33227466731347290455944055132270208/ 3922838396059942349838659979248046875*7^(1/2), a[15,11] = -35980667990 92816058284144620546/146286964102678984417724609375+655708691128729513 87443794088672/5120043743593764454620361328125*7^(1/2), a[15,8] = 5989 33003512786415989929347488/28972691439853083012428515625-1475109654044 0194135375671464064/1371942153475395989706173828125*7^(1/2), a[16,2] = 0, a[15,12] = -1927555453883917676542797699/2760543086400628250122070 31250, a[15,4] = 0, a[15,5] = 0, c[15] = 57/125, a[15,2] = 0, a[15,3] \+ = 0, c[16] = 1163/10000\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 17" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "The stan dard (simple) order conditions can be adapted to give a method of stag e by stage construction for an interpolation scheme that avoids dealin g with the weight polynomials for a given stage (corresponding to an \+ \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_16 := SimpleOrderCon ditions(7,16,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,21,27,31,32,61 ,63,64]:\ninterp_order_eqns17 := []:\nfor ct in whch do\n temp_eqn : = convert(SO7_16[ct],'interpolation_order_condition'):\n interp_orde r_eqns17 := [op(interp_order_eqns17),temp_eqn];\nend do:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 555 "interp_order_eqns17 := [add(a[17,i ],i=1..16)=c[17],seq(op(StageOrderConditions(i,17..17,'expanded')),i=2 ..7),\n add(a[17,i]*add(a[i,j]*add(a[j,k]*add(a[k,l]*c[l]^2,l=2..k- 1),k=2..j-1),\n j=2..i-1),i=2..16)=c[17]^6/36 0, #21\n add(a[17,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i -1),i=2..16)=c[17]^6/120, #27\n add(a[17,i]*add(a[i,j]*c[j]^4,j=2.. i-1),i=2..16)=c[17]^6/30, #31\n add(a[17,i]*c[i]*add(a[i,j]*c[j]^4, j=2..i-1),i=2..16)=c[17]^7/35, #61\n add(a[17,i]*add(a[i,j]*c[j]^5 ,j=2..i-1),i=2..16)=c[17]^7/42]: #63" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the node " }{XPPEDIT 18 0 "c[17] = 233/625;" "6#/&%\"cG6#\"#<*&\"$L#\"\"\"\"$D'!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "e12 := `union`( e11,\{c[17]=233/625,seq(a[17,i]=0,i=2..5)\}):\neqs_17 := expand(subs(e 12,interp_order_eqns17)):\nnops(eqs_17);\nindets(eqs_17);\nnops(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#<\"\")&F%6$F'\"\"*&F%6$F'\"\"\"&F%6$F'\"\"'&F%6$F'\"\" (&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\"#7&F%6$F'\"#8&F%6$F'\"#9&F%6$F'\"#:&F% 6$F'\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[so lve]:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e13 := solve(\{ op(eqs_17)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e14 := `union`(e12,e13): \nseq(a[17,i]=subs(e14,a[17,i]),i=1..16):\nevalf[40](%);" }}{PARA 12 " " 1 "" {XPPMATH 20 "62/&%\"aG6$\"#<\"\"\"$\"I>+%>/E!)p([dV_U66wp`$[$!# T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F %6$F'\"\"'$\"Grabi9*=c8I$*z4_3&[vu)*!#S/&F%6$F'\"\"($\"He9g@@@>Yw.yV** =sU#))y7FD/&F%6$F'\"\")$\"I1$)z!R#pcxEaZsO*e!e/tb8FD/&F%6$F'\"\"*$!Ip* H3(p4jl^x*3-**fz([@x8FD/&F%6$F'\"#5$\"I/Pd#yaWWp**\\alE1)yo7<>FD/&F%6$ F'\"#6$!I0]nh<,&G0*faxP?0D:#*G>FD/&F%6$F'\"#7$\"H^!GOWw[5iWd)HjB@k\"Go IF+/&F%6$F'\"#8$\"H)yZVV/%4.Zcd?1$[:F+/&F%6$F'\"#;$ \"I*zwOD&p6QWl*z*3dY*)p\"FD" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 70 "#---------------------------------------- -----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can c heck which of the (adapted) simple order conditions are satisfied at t his stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 244 "recd := []:\nfor ct to nops(SO7_16) do\n tt := c onvert(SO7_16[ct],'interpolation_order_condition'):\n if expand(subs (e14,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\no p(recd);\nnops(recd);\nop(\{seq(i,i=1..64)\} minus \{op(recd)\});" }} {PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"( \"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C \"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\" #U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"# f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#-------- -------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e14" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16588 "e14 := \{ a[8,3] = 0, a[11,4] = -996286030132538159613930889652/1635306888599616 4905464325675, c[14] = 1/2-1/14*7^(1/2), a[12,5] = -458349397448457291 2949314673356033540575/451957703655250747157313034270335135744, a[11,3 ] = 0, a[8,4] = -125/2, a[6,3] = 0, a[11,2] = 0, a[10,4] = -2046274952 4591049105403365239069/454251913499893469596231268750, a[13,5] = 0, a[ 12,7] = 1657121559319846802171283690913610698586256573484808662625/134 31480411255146477259155104956093505361644432088109056, a[12,8] = 34568 5379554677052215495825476969226377187500/74771167436930077221667203179 551347546362089, a[17,12] = 428770827233452252269463007586818356571604 44961937467671/1923535017412709386208278631598776372836079978942871093 750-3502578350717681024540012102744371580394344491444736/4820889767951 65259701322965312976534545383453369140625*7^(1/2), a[17,14] = 28766694 0234183647228741967485029371867251084288/76846790963048987238659333568 29603862762451171875+572017064437929229652116249893368639587841024/122 95486554087837958185493370927366180419921875*7^(1/2), a[17,13] = 57522 20750522348143924684511521705800928/2001596446569898054839108908557891 845703125-1698861030059055060173707215296857930816/2001596446569898054 839108908557891845703125*7^(1/2), a[10,2] = 0, a[14,12] = 113521280986 68146659861/254668911904014019468056-5215842639928607924801/1273344559 52007009734028*7^(1/2), a[14,13] = 3/392-3/392*7^(1/2), a[14,5] = 0, a [11,10] = -31155237437111730665923206875/39286214159423051501033895629 1, a[8,6] = 1501408353528689/265697280000000, a[8,7] = 607013921213228 3/92502016000000, a[7,2] = 0, a[7,3] = 0, c[8] = 943/1000, c[9] = 7067 558016280/7837150160667, c[10] = 909/1000, a[9,2] = 0, a[9,1] = -14725 1426448621580388138470887726424634604443330709420782905197804453180113 3057155/12468948016200320011570596216439860248033015583934879004404536 36168046069686436608, c[4] = 1023/6400, a[6,2] = 0, a[4,2] = 0, a[9,3] = 0, c[5] = 39/100, c[6] = 93/200, c[7] = 31/200, a[5,3] = -3899844/2 907025, a[14,4] = 0, c[11] = 47/50, c[12] = 1, c[13] = 1, a[2,1] = 1/2 0, a[3,1] = -7161/1024000, a[5,4] = 3982992/2907025, a[8,1] = -1221101 821869329/690812928000000, a[8,2] = 0, a[11,1] = -23426598458140868369 51207140065609179073838476242943917/1358480961351056777022231400139158 760857532162795520000, c[2] = 1/20, c[3] = 341/3200, a[7,6] = 5611/283 500, a[6,1] = 5611/114400, a[7,1] = 21173/343200, a[9,4] = -5172294311 0856684583751756552469812300390253369336991141383152707723193724692800 00/1246193810048091458972786305712152983652570794102362529218509367490 76487132995191, a[4,1] = 1023/25600, a[5,1] = 4202367/11628100, a[14,2 ] = 0, a[14,3] = 0, a[4,3] = 3069/25600, a[3,2] = 116281/1024000, a[5, 2] = 0, a[6,5] = 923521/5106400, a[9,5] = -120706792584692548079789364 41733187949484571516120469966534514296406891652614970375/2722031154761 657221710478184531100699497284085048389015085076961673446140398628096, a[16,7] = 12518372598700609135828480777283208156045833641558234957640 052376606698149159914801577172298098152775739801047/129177259766613398 9742651196438570278398443133543033569036071339432979739773195985355157 64891840000000000000000+1761522861735411010850891179050265011132649684 947212272396211479818027022186347979194359016936089217819331021/335860 8753931948373330893110740282723835952147211887279493785482525747323410 30956192340988718784000000000000000*7^(1/2), a[16,9] = 687436148767348 0006285931730527427778065078351903202582897930008852348898811178320163 4466707776976570270980595419229696302155289547923499325781982110699891 5574193829101860119823330279/11253681484315005091367654234973730440759 7710834184758217092126481001846814240063142590082692987452847804771712 0270330442606124102557418328419689608693222186318096000000000000000000 000+296323273198795235651777231925887819123825559810998878941673611936 6244822580797795219947112377192870683533860532811232497862270489157578 88049930193669040074737641797319943395161/9456875196903365622998028768 8854877653443454482508200182430358387396509927932826170243766968897019 1998359426151487672640845482439123880948251840007305228727998400000000 000000000000*7^(1/2), a[16,1] = 57126236597472259214750870208567441312 3074979711954873812807464116257302576802226313028164589259868449219745 217/110772198297504586077352752537903192511339073656683241243181151807 33432062987800250708692452000000000000000000000-4180593972416781509583 9522495249359869704777702449827779759333286790035083885626790256030500 3793843931777113/52128093316472746389342471782542678828865446426674466 4673793655563926214728837658856879644800000000000000000000*7^(1/2), a[ 12,4] = -16957088714171468676387054358954754000/1436904151196546833263 68228101570221, a[10,7] = 31860723517364931240515126584966086992765341 4425413/6714716715558965303132938072935465423910912000000, a[12,9] = - 3205890962717072542791434312152727534008102774023210240571361570757249 056167015230160352087048674542196011/947569549683965814783015124451273 6049846577471272576153724492059731926573060172391034910747383240332591 20, a[13,4] = 0, a[13,2] = 0, a[13,7] = 3847749490868980348119500000/1 5517045062138271618141237517, a[14,6] = 460454863031265521343678403/23 55798527514165229982023734-283438859697640884437065685/471159705502833 0459964047468*7^(1/2), a[14,10] = 188753532739715767256965625/33283666 294238384068886478-168249974410951846618609375/66567332588476768137772 956*7^(1/2), a[12,3] = 0, a[10,3] = 0, a[14,8] = 671994922390044434145 634375/90947002658165004198514338-463454429627970722580265625/18189400 5316330008397028676*7^(1/2), a[10,5] = -180269259803172281163724663224 981097/38100922558256871086579832832000000, a[13,1] = 4490186773775461 6851973/1014046409980231013380680, a[11,9] = 3007606697681025178342324 97565452434946672266195876496371874262392684852243925359864884962513/4 6554433375013464555850653366045056037608247796155212857518928103156804 92364106674524398280000, a[12,11] = -612293360107076959161309399399335 8877250/1050517001510235513198246721302027675953, a[11,6] = 2098082234 5096760292224086794978105312644533925634933539/37758899920075508038787 27839115494641972212962174156800, a[11,5] = -2605308595925653415258808 9363841/4377552804565683061011299942400, a[11,8] = 1610214261431241783 89075121929246710833125/10997207722131034650667041364346422894371443, \+ a[7,4] = 8602624/76559175, a[7,5] = -26782109/689364000, a[9,6] = 7801 2515584389364132309055253043103656779559256849718270146067480312677011 1481625/18311042541273197219788987450715878685922610298086185950524144 3073629143100805376, a[13,9] = 122747654703131968784288120377406350503 19234276006986398294443554969616342274215316330684448207141/4893451474 9371551765038583414351093488882928068660965448289652679652335305216675 7299452852166040, a[13,8] = -13734512432397741476562500000/87513289292 4995907746928783, a[9,8] = 1033284818445201560405683676728665685912400 7796970668046446015775000000/13127035500360336480738342487407279145379 72028638950165249582733679393783, a[10,1] = -2905557336033741508853861 8442231036441314060511/22674759891089577691327962602370597632000000000 , a[14,9] = -152041753025355398038004360208592588649426234538717842799 7982057187814442042273423080071783/39219298194348039763366422255699207 5106628568847304631390191589139262501961993074284749200+79993028921817 7898912938325894667068964649008561582353557148147567546469052859876678 259/461403508168800467804310850067049500125445375114476036929637163693 250002308227146217352*7^(1/2), a[14,11] = -236307906973303345482653/25 637232930373573654194+173345833072833817590685/51274465860747147308388 *7^(1/2), a[14,1] = 307/6800, a[11,7] = 890722993756379186418929622095 833835264322635782294899/139212420013951126575019419555940138228301198 03764736, a[13,3] = 0, a[12,1] = -286655699182566397177829532910103388 7534912787724034363/86822671161926270301121392501614361203066923379533 8240, a[12,2] = 0, a[10,9] = -2698404929400842518721166485087129798562 269848229517793703413951226714583/469545674913934315077000442080871141 884676035902717550325616728175875000000, a[13,10] = -97983636845777394 45312500000/308722986341456031822630699, a[13,6] = 7916386751916152796 48100000/2235604725089973126411512319, a[14,7] = 355143169692072506417 24424985/147161653169956511475274962258-481702304929173090974271575/29 4323306339913022950549924516*7^(1/2), a[10,8] = 2120832024345190822818 42245535894/20022426044775672563822865371173879, a[10,6] = 21127670214 172802870128286992003940810655221489/467947387799789290614582269797670 8633673728000, a[12,6] = 234630538855340425865625847344618441915474017 2519949575/256726716407895402892744978301151486254183185289662464, a[1 2,10] = 40279545832706233433100438588458933210937500/88964608427994828 46916972126377338947215101, a[13,11] = 282035543183190840068750/122954 07629873040425991, a[13,12] = -306814272936976936753/12993311831837449 97286, a[9,7] = 664113122959911642134782135839106469928140328160577035 357155340392950009492511875/151784655985862481363330231072953491752797 65150089078301139943253016877823170816, a[6,4] = 31744/135025, a[8,5] \+ = -1024030607959889/168929280000000, a[17,2] = 0, a[17,3] = 0, a[16,13 ] = -56937797859785409375631098707008813998527540294648257184713456009 91242218048279830357689/6623260257269323853870635275237719261692995498 64595912922410705613811200000000000000000000+1116058978133290227473643 648991308872297960011076080647100654462877250873303116055359689/662326 0257269323853870635275237719261692995498645959129224107056138112000000 00000000000000*7^(1/2), a[16,15] = 12566723979477860207267692556004160 9561972597583209979258314845487667433090419263759799/19043248758339535 7730560482698424042469036304599122912893112650489504907631001600000000 0-59028946739025681742626285442331126491223596724290616786809339665195 402731038729879/304691980133432572368896772317478467950458087358596660 6289802407832078522096025600000*7^(1/2), a[16,6] = -997452187886603466 5001215093798908273380361965165365845681748561480350415538556069039355 4360336092065584815633/12097215626349617187624215881471948530761194836 09301088106017767355971757142551992626941965147200000000000000000+9332 0733560040239498721445105436146592080370238787178794613177818778456707 21918179975766545310520465203275977/2419443125269923437524843176294389 7061522389672186021762120355347119435142851039852538839302944000000000 0000000*7^(1/2), a[16,8] = -547598389828861834598181333883759080061418 58550153990686846978348863143039759306003458396414251841877395553/3030 7100180597417208049096613010446805012644086401478209258240042904260490 962503761623721182105600000000000+132843617879870751739882462233453607 29456268579730365891407251773114191650397356739750286947316947113707/3 5446900796020371003566194869018066438611279633218103168723087769478667 240891817265056983838720000000000*7^(1/2), a[16,10] = -103426333826764 9013726626367070353641944586812274059324287684035797195504387677241159 024633249191823996323/822424408623777187146783153818399345197713720767 960756194595910492784789119961559213755380633600000000000+274263425446 3020699447644274499181977315922821414432551248566017099173343536444217 98295950196247775939471/2138303462421820686581636199927838297514055673 996697966105949367281240451711900053955763989647360000000000*7^(1/2), \+ a[16,12] = -1700668024857780947303857318649956084860573315650681108553 264313641197383910289904206751497096329440010629/270418821851798197921 5772760946666451031013390150181735548114903923055225824021756875680000 0000000000000000+18110923375969700228455846935056478436359098753435265 103392713976023265962815946771736872563829234342887/123348936283276370 9817720908501988205733444704279030265337736622842095366165343257522240 000000000000000000*7^(1/2), a[16,11] = 6623909106040336836026262070216 6481840242021269701059559317336313240648392339911854838860941565539670 913883/266129688568183402539243009457627860959737366888672888383931360 68830674315050279167258323200000000000000000-9009365467871791955400608 4633237473317647606687760893597983724578633330182810231678808929007209 2611304509/17741979237878893502616200630508524063982491125911525892262 09071255378287670018611150554880000000000000000*7^(1/2), a[16,14] = 36 6113642577897032178883959289263402896100850925956136650174323553270486 082847776282814967/179923786915305271897978397237983483914152734336378 64708028281999192834400000000000000000000-1023478027503079403148682342 138509935107943980471305000079396396912342658754866704389719/287878059 0644884350367654355807735742626443749382058353284525119870853504000000 0000000000*7^(1/2), a[16,3] = 0, a[16,4] = 0, a[16,5] = 0, a[15,10] = \+ 54365264025507653851977480738208/4466478390356713423359819140625-14962 94528311131416682934929792/235786943683989396406626953125*7^(1/2), a[1 5,9] = -30670664596612709940004930485462861354074938991332508415290927 16875641340142781006771180215525814345889446383/3733407192182277814105 1165324669718543154089407852909430152628082010143512891140541034916089 3280029296875000+18166476119840769402418420597643880095994693261462548 65545587094445401874686830164148895260773137/4244756546679057970967517 4791599368354403150818438287468599643503391601068587828983688354492187 5*7^(1/2), a[15,14] = 172678683744/3814697265625*7^(1/2), a[15,1] = 57 196651428918572875631366996149/1245822888041363915313720703125000+1089 1702347425878099023621466959/8357395207277482931896209716796875*7^(1/2 ), a[15,6] = -245534888302358312351817429197184/8700455320214002619298 187255859375*7^(1/2)+4116869726480787181292612488487584/26598534836082 808007568743896484375, a[15,13] = -15630718068/3814697265625+616709584 8/3814697265625*7^(1/2), a[15,7] = 63147661795969654782426038584326748 8/2705957914016858192337687904052734375-332274667313472904559440551322 70208/3922838396059942349838659979248046875*7^(1/2), a[15,11] = -35980 66799092816058284144620546/146286964102678984417724609375+655708691128 72951387443794088672/5120043743593764454620361328125*7^(1/2), a[15,8] \+ = 598933003512786415989929347488/28972691439853083012428515625-1475109 6540440194135375671464064/1371942153475395989706173828125*7^(1/2), a[1 7,1] = 141015240429350030100965417504818793218163712680892499503527761 3/33171979099485416659192424319640924155867419822925567626953125000-34 3669929529154146426950913103012707061848467742113726473984/11847135392 6733630925687229713003300556669356510448455810546875*7^(1/2), a[17,6] \+ = 144317504741273108436770639890093130241213191928121002485536/1059071 381820782115429096951498359519812365804784881591796875-361386504350893 73461278335139553986413666973704247679234048/7564795584434157967350692 51070256799865975574846343994140625*7^(1/2), a[17,16] = 14133833477396 98940972600039030481649401856/2643976603550095246929966505286480846769 3125+1861757224737799874854447250922400645120/423036256568015239508794 64084583693548309*7^(1/2), a[17,9] = -11701055582659954072431612713805 8007798756657360547279654237886672876967817530471332239610851127136258 213798655554999195607667534787/724428489816394108820303136848490351295 2289858145246070883723225205123853962964474784685814260973652691157934 12394100189208984375000+2327341405171539284799921417004174552247521703 2773573657285744432473650838946288163911020446707168311579553151269834 892177196544/258724460648712181721536834588746554034010352076615931102 9901151858972804986773026708816362236062018818270690758550357818603515 625*7^(1/2), a[17,11] = -543182560757541564307152147803467679236313157 5507490074/5824694411529015933948896260840005226582958953857421875+430 78828428937580969195523720831420116933769072147456/1540924447494448659 77484028064550402819654998779296875*7^(1/2), a[17,7] = 241521473964570 429990475882210375964811719239806626681198304/147017566847133423378821 2618705488919434060375826495361328125-20734019037472044011987770732548 36154760997520307226750976/3621122336136291216227124676614504727670099 4478485107421875*7^(1/2), a[17,8] = 3609629607031343047012837230915901 844578627451417318752/530657358827813080090445988623672196949600655675 2734375-68445701579780131347383082192778657908542785254600704/33249207 9466048295796018789864456263752882616337890625*7^(1/2), a[17,15] = -18 0219433009153795201274490416832044056/22122613556709877116980102673676 93359375+88272455565304907970131443124874176/3539618169073580338716816 427788309375*7^(1/2), a[17,10] = 7656605713605905045224108110805637110 90654609706246624/1872014249102541079804637393782950555096740829301171 875-18934143677045047338847621384679376883588265306034176/230543626736 766142833083422879673713681864634150390625*7^(1/2), a[16,2] = 0, c[17] = 233/625, a[15,12] = -1927555453883917676542797699/27605430864006282 5012207031250, a[15,4] = 0, a[15,5] = 0, c[15] = 57/125, a[15,2] = 0, \+ a[15,3] = 0, c[16] = 1163/10000, a[17,5] = 0, a[17,4] = 0\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 " calculation for stage 18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 253 "The standard (simple) order conditions can be ada pted to give a method of stage by stage construction for an interpolat ion scheme that avoids dealing with the weight polynomials for a given stage (corresponding to an \"approximate\" interpolation scheme)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_17 := SimpleOrderConditions(7,17,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch : = [1,2,4,8,16,21,27,31,32,61,63,64]:\ninterp_order_eqns18 := []:\nfor \+ ct in whch do\n temp_eqn := convert(SO7_17[ct],'interpolation_order_ condition'):\n interp_order_eqns18 := [op(interp_order_eqns18),temp_ eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified expl icitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 555 "inter p_order_eqns18 := [add(a[18,i],i=1..17)=c[18],seq(op(StageOrderConditi ons(i,18..18,'expanded')),i=2..7),\n add(a[18,i]*add(a[i,j]*add(a[j ,k]*add(a[k,l]*c[l]^2,l=2..k-1),k=2..j-1),\n \+ j=2..i-1),i=2..17)=c[18]^6/360, #21\n add(a[18,i]*add(a[i,j]*add(a [j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..17)=c[18]^6/120, #27\n add(a [18,i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..17)=c[18]^6/30, #31\n add(a [18,i]*c[i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..17)=c[18]^7/35, #61\n \+ add(a[18,i]*add(a[i,j]*c[j]^5,j=2..i-1),i=2..17)=c[18]^7/42]: #63" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We speci fy " }{XPPEDIT 18 0 "c[18] = 13/25;" "6#/&%\"cG6#\"#=*&\"#8\"\"\"\"#D !\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[18,17]=0" "6#/&%\"aG6$ \"#=\"#<\"\"!" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "e15 := `union`(e14,\{c[18]=13/25,seq(a[18,i]=0,i=2..5),a[18,17] =0\}):\neqs_18 := expand(subs(e15,interp_order_eqns18)):\nnops(eqs_18) ;\nindets(eqs_18);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#=\"#;&F%6$F'\"#7&F%6$F' \"#8&F%6$F'\"#9&F%6$F'\"#:&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\"\"\"&F%6$F'\" \"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e16 := solve(\{op(eqs_18)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e17 := `union`(e15,e16):\nseq(a[18,i]=subs(e17,a[18,i]),i=1..1 7):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "63/&%\"aG6$\"#=\" \"\"$\"ID!R.NKp%HQQ()[N6if@jQN!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/ &F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"HZZu2g\"z;/o_lMeg(zavc\"! #S/&F%6$F'\"\"($\"Hodb)[W3j#G=&4Wq(eNF\")*>FD/&F%6$F'\"\")$\"I&*)ycG;X F#4\"Rs2UQ'*ov'R=FD/&F%6$F'\"\"*$!IF!eFW;I&*eH>@OljZkFM2H\"3Z1\"R6FD/&F%6$F'\"#;$\"Ih])*[8u(=3X\"\\af7w%3.Ij\"FD/ &F%6$F'\"# " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_17) do\n tt := convert(SO7_17[c t],'interpolation_order_condition'):\n if expand(subs(e17,lhs(tt)=rh s(tt))) then recd := [op(recd),ct] end if: \nend do:\nop(recd);\nnops( recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"& \"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\" #A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R \"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c \"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#- --------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e17" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19383 "e17 := \{ a[8,3] = 0, a[11,4] = -996286030132538159613930889652/1635306888599616 4905464325675, c[14] = 1/2-1/14*7^(1/2), a[12,5] = -458349397448457291 2949314673356033540575/451957703655250747157313034270335135744, a[11,3 ] = 0, a[8,4] = -125/2, a[6,3] = 0, a[11,2] = 0, a[10,4] = -2046274952 4591049105403365239069/454251913499893469596231268750, a[13,5] = 0, a[ 12,7] = 1657121559319846802171283690913610698586256573484808662625/134 31480411255146477259155104956093505361644432088109056, a[12,8] = 34568 5379554677052215495825476969226377187500/74771167436930077221667203179 551347546362089, a[17,12] = 428770827233452252269463007586818356571604 44961937467671/1923535017412709386208278631598776372836079978942871093 750-3502578350717681024540012102744371580394344491444736/4820889767951 65259701322965312976534545383453369140625*7^(1/2), a[17,14] = 28766694 0234183647228741967485029371867251084288/76846790963048987238659333568 29603862762451171875+572017064437929229652116249893368639587841024/122 95486554087837958185493370927366180419921875*7^(1/2), a[17,13] = 57522 20750522348143924684511521705800928/2001596446569898054839108908557891 845703125-1698861030059055060173707215296857930816/2001596446569898054 839108908557891845703125*7^(1/2), a[10,2] = 0, a[14,12] = 113521280986 68146659861/254668911904014019468056-5215842639928607924801/1273344559 52007009734028*7^(1/2), a[14,13] = 3/392-3/392*7^(1/2), a[14,5] = 0, a [11,10] = -31155237437111730665923206875/39286214159423051501033895629 1, a[8,6] = 1501408353528689/265697280000000, a[8,7] = 607013921213228 3/92502016000000, a[7,2] = 0, a[7,3] = 0, c[8] = 943/1000, c[9] = 7067 558016280/7837150160667, c[10] = 909/1000, a[9,2] = 0, a[9,1] = -14725 1426448621580388138470887726424634604443330709420782905197804453180113 3057155/12468948016200320011570596216439860248033015583934879004404536 36168046069686436608, c[4] = 1023/6400, a[6,2] = 0, a[4,2] = 0, a[9,3] = 0, c[5] = 39/100, c[6] = 93/200, c[7] = 31/200, a[5,3] = -3899844/2 907025, a[14,4] = 0, c[11] = 47/50, c[12] = 1, c[13] = 1, a[2,1] = 1/2 0, a[3,1] = -7161/1024000, a[5,4] = 3982992/2907025, a[8,1] = -1221101 821869329/690812928000000, a[8,2] = 0, a[11,1] = -23426598458140868369 51207140065609179073838476242943917/1358480961351056777022231400139158 760857532162795520000, c[2] = 1/20, c[3] = 341/3200, a[7,6] = 5611/283 500, a[6,1] = 5611/114400, a[7,1] = 21173/343200, a[9,4] = -5172294311 0856684583751756552469812300390253369336991141383152707723193724692800 00/1246193810048091458972786305712152983652570794102362529218509367490 76487132995191, a[4,1] = 1023/25600, a[5,1] = 4202367/11628100, a[14,2 ] = 0, a[14,3] = 0, a[4,3] = 3069/25600, a[3,2] = 116281/1024000, a[5, 2] = 0, a[6,5] = 923521/5106400, a[9,5] = -120706792584692548079789364 41733187949484571516120469966534514296406891652614970375/2722031154761 657221710478184531100699497284085048389015085076961673446140398628096, a[16,7] = 12518372598700609135828480777283208156045833641558234957640 052376606698149159914801577172298098152775739801047/129177259766613398 9742651196438570278398443133543033569036071339432979739773195985355157 64891840000000000000000+1761522861735411010850891179050265011132649684 947212272396211479818027022186347979194359016936089217819331021/335860 8753931948373330893110740282723835952147211887279493785482525747323410 30956192340988718784000000000000000*7^(1/2), a[16,9] = 687436148767348 0006285931730527427778065078351903202582897930008852348898811178320163 4466707776976570270980595419229696302155289547923499325781982110699891 5574193829101860119823330279/11253681484315005091367654234973730440759 7710834184758217092126481001846814240063142590082692987452847804771712 0270330442606124102557418328419689608693222186318096000000000000000000 000+296323273198795235651777231925887819123825559810998878941673611936 6244822580797795219947112377192870683533860532811232497862270489157578 88049930193669040074737641797319943395161/9456875196903365622998028768 8854877653443454482508200182430358387396509927932826170243766968897019 1998359426151487672640845482439123880948251840007305228727998400000000 000000000000*7^(1/2), a[16,1] = 57126236597472259214750870208567441312 3074979711954873812807464116257302576802226313028164589259868449219745 217/110772198297504586077352752537903192511339073656683241243181151807 33432062987800250708692452000000000000000000000-4180593972416781509583 9522495249359869704777702449827779759333286790035083885626790256030500 3793843931777113/52128093316472746389342471782542678828865446426674466 4673793655563926214728837658856879644800000000000000000000*7^(1/2), a[ 12,4] = -16957088714171468676387054358954754000/1436904151196546833263 68228101570221, a[10,7] = 31860723517364931240515126584966086992765341 4425413/6714716715558965303132938072935465423910912000000, a[12,9] = - 3205890962717072542791434312152727534008102774023210240571361570757249 056167015230160352087048674542196011/947569549683965814783015124451273 6049846577471272576153724492059731926573060172391034910747383240332591 20, a[13,4] = 0, a[13,2] = 0, a[13,7] = 3847749490868980348119500000/1 5517045062138271618141237517, a[14,6] = 460454863031265521343678403/23 55798527514165229982023734-283438859697640884437065685/471159705502833 0459964047468*7^(1/2), a[14,10] = 188753532739715767256965625/33283666 294238384068886478-168249974410951846618609375/66567332588476768137772 956*7^(1/2), a[12,3] = 0, a[10,3] = 0, a[14,8] = 671994922390044434145 634375/90947002658165004198514338-463454429627970722580265625/18189400 5316330008397028676*7^(1/2), a[10,5] = -180269259803172281163724663224 981097/38100922558256871086579832832000000, a[13,1] = 4490186773775461 6851973/1014046409980231013380680, a[11,9] = 3007606697681025178342324 97565452434946672266195876496371874262392684852243925359864884962513/4 6554433375013464555850653366045056037608247796155212857518928103156804 92364106674524398280000, a[12,11] = -612293360107076959161309399399335 8877250/1050517001510235513198246721302027675953, a[11,6] = 2098082234 5096760292224086794978105312644533925634933539/37758899920075508038787 27839115494641972212962174156800, a[11,5] = -2605308595925653415258808 9363841/4377552804565683061011299942400, a[11,8] = 1610214261431241783 89075121929246710833125/10997207722131034650667041364346422894371443, \+ a[7,4] = 8602624/76559175, a[7,5] = -26782109/689364000, a[9,6] = 7801 2515584389364132309055253043103656779559256849718270146067480312677011 1481625/18311042541273197219788987450715878685922610298086185950524144 3073629143100805376, a[13,9] = 122747654703131968784288120377406350503 19234276006986398294443554969616342274215316330684448207141/4893451474 9371551765038583414351093488882928068660965448289652679652335305216675 7299452852166040, a[13,8] = -13734512432397741476562500000/87513289292 4995907746928783, a[9,8] = 1033284818445201560405683676728665685912400 7796970668046446015775000000/13127035500360336480738342487407279145379 72028638950165249582733679393783, a[10,1] = -2905557336033741508853861 8442231036441314060511/22674759891089577691327962602370597632000000000 , a[14,9] = -152041753025355398038004360208592588649426234538717842799 7982057187814442042273423080071783/39219298194348039763366422255699207 5106628568847304631390191589139262501961993074284749200+79993028921817 7898912938325894667068964649008561582353557148147567546469052859876678 259/461403508168800467804310850067049500125445375114476036929637163693 250002308227146217352*7^(1/2), a[14,11] = -236307906973303345482653/25 637232930373573654194+173345833072833817590685/51274465860747147308388 *7^(1/2), a[14,1] = 307/6800, a[11,7] = 890722993756379186418929622095 833835264322635782294899/139212420013951126575019419555940138228301198 03764736, a[13,3] = 0, a[12,1] = -286655699182566397177829532910103388 7534912787724034363/86822671161926270301121392501614361203066923379533 8240, a[12,2] = 0, a[10,9] = -2698404929400842518721166485087129798562 269848229517793703413951226714583/469545674913934315077000442080871141 884676035902717550325616728175875000000, a[13,10] = -97983636845777394 45312500000/308722986341456031822630699, a[13,6] = 7916386751916152796 48100000/2235604725089973126411512319, a[14,7] = 355143169692072506417 24424985/147161653169956511475274962258-481702304929173090974271575/29 4323306339913022950549924516*7^(1/2), a[10,8] = 2120832024345190822818 42245535894/20022426044775672563822865371173879, a[10,6] = 21127670214 172802870128286992003940810655221489/467947387799789290614582269797670 8633673728000, a[12,6] = 234630538855340425865625847344618441915474017 2519949575/256726716407895402892744978301151486254183185289662464, a[1 2,10] = 40279545832706233433100438588458933210937500/88964608427994828 46916972126377338947215101, a[13,11] = 282035543183190840068750/122954 07629873040425991, a[13,12] = -306814272936976936753/12993311831837449 97286, a[9,7] = 664113122959911642134782135839106469928140328160577035 357155340392950009492511875/151784655985862481363330231072953491752797 65150089078301139943253016877823170816, a[6,4] = 31744/135025, a[8,5] \+ = -1024030607959889/168929280000000, a[17,2] = 0, a[17,3] = 0, a[16,13 ] = -56937797859785409375631098707008813998527540294648257184713456009 91242218048279830357689/6623260257269323853870635275237719261692995498 64595912922410705613811200000000000000000000+1116058978133290227473643 648991308872297960011076080647100654462877250873303116055359689/662326 0257269323853870635275237719261692995498645959129224107056138112000000 00000000000000*7^(1/2), a[16,15] = 12566723979477860207267692556004160 9561972597583209979258314845487667433090419263759799/19043248758339535 7730560482698424042469036304599122912893112650489504907631001600000000 0-59028946739025681742626285442331126491223596724290616786809339665195 402731038729879/304691980133432572368896772317478467950458087358596660 6289802407832078522096025600000*7^(1/2), a[16,6] = -997452187886603466 5001215093798908273380361965165365845681748561480350415538556069039355 4360336092065584815633/12097215626349617187624215881471948530761194836 09301088106017767355971757142551992626941965147200000000000000000+9332 0733560040239498721445105436146592080370238787178794613177818778456707 21918179975766545310520465203275977/2419443125269923437524843176294389 7061522389672186021762120355347119435142851039852538839302944000000000 0000000*7^(1/2), a[16,8] = -547598389828861834598181333883759080061418 58550153990686846978348863143039759306003458396414251841877395553/3030 7100180597417208049096613010446805012644086401478209258240042904260490 962503761623721182105600000000000+132843617879870751739882462233453607 29456268579730365891407251773114191650397356739750286947316947113707/3 5446900796020371003566194869018066438611279633218103168723087769478667 240891817265056983838720000000000*7^(1/2), a[16,10] = -103426333826764 9013726626367070353641944586812274059324287684035797195504387677241159 024633249191823996323/822424408623777187146783153818399345197713720767 960756194595910492784789119961559213755380633600000000000+274263425446 3020699447644274499181977315922821414432551248566017099173343536444217 98295950196247775939471/2138303462421820686581636199927838297514055673 996697966105949367281240451711900053955763989647360000000000*7^(1/2), \+ a[16,12] = -1700668024857780947303857318649956084860573315650681108553 264313641197383910289904206751497096329440010629/270418821851798197921 5772760946666451031013390150181735548114903923055225824021756875680000 0000000000000000+18110923375969700228455846935056478436359098753435265 103392713976023265962815946771736872563829234342887/123348936283276370 9817720908501988205733444704279030265337736622842095366165343257522240 000000000000000000*7^(1/2), a[16,11] = 6623909106040336836026262070216 6481840242021269701059559317336313240648392339911854838860941565539670 913883/266129688568183402539243009457627860959737366888672888383931360 68830674315050279167258323200000000000000000-9009365467871791955400608 4633237473317647606687760893597983724578633330182810231678808929007209 2611304509/17741979237878893502616200630508524063982491125911525892262 09071255378287670018611150554880000000000000000*7^(1/2), a[16,14] = 36 6113642577897032178883959289263402896100850925956136650174323553270486 082847776282814967/179923786915305271897978397237983483914152734336378 64708028281999192834400000000000000000000-1023478027503079403148682342 138509935107943980471305000079396396912342658754866704389719/287878059 0644884350367654355807735742626443749382058353284525119870853504000000 0000000000*7^(1/2), a[16,3] = 0, a[16,4] = 0, a[16,5] = 0, a[15,10] = \+ 54365264025507653851977480738208/4466478390356713423359819140625-14962 94528311131416682934929792/235786943683989396406626953125*7^(1/2), a[1 5,9] = -30670664596612709940004930485462861354074938991332508415290927 16875641340142781006771180215525814345889446383/3733407192182277814105 1165324669718543154089407852909430152628082010143512891140541034916089 3280029296875000+18166476119840769402418420597643880095994693261462548 65545587094445401874686830164148895260773137/4244756546679057970967517 4791599368354403150818438287468599643503391601068587828983688354492187 5*7^(1/2), a[15,14] = 172678683744/3814697265625*7^(1/2), a[15,1] = 57 196651428918572875631366996149/1245822888041363915313720703125000+1089 1702347425878099023621466959/8357395207277482931896209716796875*7^(1/2 ), a[15,6] = -245534888302358312351817429197184/8700455320214002619298 187255859375*7^(1/2)+4116869726480787181292612488487584/26598534836082 808007568743896484375, a[15,13] = -15630718068/3814697265625+616709584 8/3814697265625*7^(1/2), a[15,7] = 63147661795969654782426038584326748 8/2705957914016858192337687904052734375-332274667313472904559440551322 70208/3922838396059942349838659979248046875*7^(1/2), a[15,11] = -35980 66799092816058284144620546/146286964102678984417724609375+655708691128 72951387443794088672/5120043743593764454620361328125*7^(1/2), a[15,8] \+ = 598933003512786415989929347488/28972691439853083012428515625-1475109 6540440194135375671464064/1371942153475395989706173828125*7^(1/2), a[1 8,4] = 0, a[18,5] = 0, a[17,1] = 1410152404293500301009654175048187932 181637126808924995035277613/331719790994854166591924243196409241558674 19822925567626953125000-3436699295291541464269509131030127070618484677 42113726473984/1184713539267336309256872297130033005566693565104484558 10546875*7^(1/2), a[17,6] = 144317504741273108436770639890093130241213 191928121002485536/105907138182078211542909695149835951981236580478488 1591796875-36138650435089373461278335139553986413666973704247679234048 /756479558443415796735069251070256799865975574846343994140625*7^(1/2), a[17,16] = 1413383347739698940972600039030481649401856/26439766035500 952469299665052864808467693125+186175722473779987485444725092240064512 0/42303625656801523950879464084583693548309*7^(1/2), a[17,9] = -117010 5558265995407243161271380580077987566573605472796542378866728769678175 30471332239610851127136258213798655554999195607667534787/7244284898163 9410882030313684849035129522898581452460708837232252051238539629644747 8468581426097365269115793412394100189208984375000+23273414051715392847 9992141700417455224752170327735736572857444324736508389462881639110204 46707168311579553151269834892177196544/2587244606487121817215368345887 4655403401035207661593110299011518589728049867730267088163622360620188 18270690758550357818603515625*7^(1/2), a[17,11] = -5431825607575415643 071521478034676792363131575507490074/582469441152901593394889626084000 5226582958953857421875+43078828428937580969195523720831420116933769072 147456/154092444749444865977484028064550402819654998779296875*7^(1/2), a[17,7] = 24152147396457042999047588221037596481171923980662668119830 4/1470175668471334233788212618705488919434060375826495361328125-207340 1903747204401198777073254836154760997520307226750976/36211223361362912 162271246766145047276700994478485107421875*7^(1/2), a[17,8] = 36096296 07031343047012837230915901844578627451417318752/5306573588278130800904 459886236721969496006556752734375-684457015797801313473830821927786579 08542785254600704/3324920794660482957960187898644562637528826163378906 25*7^(1/2), a[17,15] = -180219433009153795201274490416832044056/221226 1355670987711698010267367693359375+88272455565304907970131443124874176 /3539618169073580338716816427788309375*7^(1/2), a[17,10] = 76566057136 0590504522410811080563711090654609706246624/18720142491025410798046373 93782950555096740829301171875-1893414367704504733884762138467937688358 8265306034176/230543626736766142833083422879673713681864634150390625*7 ^(1/2), c[18] = 13/25, a[16,2] = 0, c[17] = 233/625, a[15,12] = -19275 55453883917676542797699/276054308640062825012207031250, a[18,17] = 0, \+ a[18,10] = 13521134712758181369289831786617583384500556000/28357138921 316598605324684781564221426317494219-886401905266983644635343568980406 6485248/101445417221405134333310741139852901561249*7^(1/2), a[18,11] = -624931687431641426621118270642785259964248638186/5964487077405712316 36366977110016535202094996875+4693205177799375282949632884666545966117 964544/15779066342343154276094364473809961248732671875*7^(1/2), a[18,7 ] = 161373973469595107260482187200001399198950768983264/89080466539328 1807928479125180130564201466168548125-13366030437010218653253989683337 01077347089079296/21941001610671965712524116383747058231563206121875*7 ^(1/2), a[18,8] = 10370962436573077008766029273604620733739288898400/1 3584828385992014850315417308766008241909776785287-48737181047905230621 987375887798081383160832/222530646649172192741910614915819094171863921 *7^(1/2), a[18,12] = 28899294106750011031701193669610820543459670391/1 165502874455984858862294271453933139517246093750-250878995631377260872 470039095670096659584/32456220396992059561745872220939380103515625*7^( 1/2), a[18,16] = 1667634263017147072374514384896000000000/423036256568 01523950879464084583693548309+1980745822360913756216308531200000000000 /42303625656801523950879464084583693548309*7^(1/2), a[18,6] = 16283256 908017943404871165834200503438819098307881504/108448909498448088619939 527833432014828786258409971875-145818901909625960229141275905926424005 344756683776/2869018769800213984654484863318307270602811069046875*7^(1 /2), a[18,15] = 611266295965159733830102317250/13983676717327724794930 632801139+21824632869622508258865350000/822569218666336752642978400067 *7^(1/2), a[18,9] = -1471082580019238344036493559401261937554185467295 5854090580175358951645433099691448186629741883086125225426868034484205 191963/741814773571987567431990412132854119726314481474073197658493258 26100468264580756221795182738032370203557457245429155859375000+2535512 4852747346663348168643778819286288841281929340338134883360043823044851 28389976568844110774595925130633317370691456/2649338477042812740828537 1861887647133082660052645471344946187795035881523064555793498279549297 2750726990918733675556640625*7^(1/2), a[18,13] = 732260192217525894872 79696352758/22773719569861951201724972470703125-1209684402178421037903 496517328/1339630562933055953042645439453125*7^(1/2), a[18,1] = 147937 299499519182239019177721288159839355546541856055637/339681065978730666 5901304250331230633560823789867578125000-46223430353153843317212066793 6726432477695301843136/14977119311231510872580706571125355527164126057 6171875*7^(1/2), a[18,14] = 386910568376965091074907779867460087552/97 14952339032365794121871305424091796875+7693600360987978823534433818915 43296/15543923742451785270594994088678546875*7^(1/2), a[15,4] = 0, a[1 5,5] = 0, c[15] = 57/125, a[15,2] = 0, a[15,3] = 0, c[16] = 1163/10000 , a[18,2] = 0, a[18,3] = 0, a[17,5] = 0, a[17,4] = 0\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 " calculation for stage 19" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 253 "The standard (simple) order conditions can be ada pted to give a method of stage by stage construction for an interpolat ion scheme that avoids dealing with the weight polynomials for a given stage (corresponding to an \"approximate\" interpolation scheme)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_18 := SimpleOrderConditions(7,18,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch : = [1,2,4,8,16,21,27,31,32,61,63,64]:\ninterp_order_eqns19 := []:\nfor \+ ct in whch do\n temp_eqn := convert(SO7_18[ct],'interpolation_order_ condition'):\n interp_order_eqns19 := [op(interp_order_eqns19),temp_ eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified expl icitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 558 "inter p_order_eqns19 := [add(a[19,i],i=1..18)=c[19],seq(op(StageOrderConditi ons(i,19..19,'expanded')),i=2..7),\n add(a[19,i]*add(a[i,j]*add(a[j ,k]*add(a[k,l]*c[l]^2,l=2..k-1),k=2..j-1),\n \+ j=2..i-1),i=2..18)=c[19]^6/360, #21\n add(a[19,i]*add(a[i,j]*add(a [j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..18)=c[19]^6/120, #27\n add(a [19,i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..18)=c[19]^6/30, #31\n add(a [19,i]*c[i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..18)=c[19]^7/35, ##61\n \+ add(a[19,i]*add(a[i,j]*c[j]^5,j=2..i-1),i=2..18)=c[19]^7/42]: ##63 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We s pecify " }{XPPEDIT 18 0 "c[19] = 177/250;" "6#/&%\"cG6#\"#>*&\"$x\"\" \"\"\"$]#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[19,17] = 0" "6#/&% \"aG6$\"#>\"#<\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[19,18] = \+ 0;" "6#/&%\"aG6$\"#>\"#=\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "e18 := `union`(e17,\{c[19]=177/250,seq(a[19,i]= 0,i=2..5),a[19,17]=0,a[19,18]=0\}):\neqs_19 := expand(subs(e18,interp_ order_eqns19)):\nnops(eqs_19);\nindets(eqs_19);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"a G6$\"#>\"\")&F%6$F'\"\"*&F%6$F'\"#:&F%6$F'\"#;&F%6$F'\"#5&F%6$F'\"#6&F %6$F'\"#8&F%6$F'\"#9&F%6$F'\"\"'&F%6$F'\"\"\"&F%6$F'\"#7&F%6$F'\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e19 := solve(\{op(eqs_19) \}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e20 := `union`(e18,e19):\nseq(a[19, i]=subs(e20,a[19,i]),i=1..18):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "64/&%\"aG6$\"#>\"\"\"$\"I:&=G#=R`!>^=_Z$[`7!3_%e!#T/&F%6$ F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\" \"'$\"IJZpVLUT,vyVc&zo!yzVYU!#S/&F%6$F'\"\"($\"IlG\\zXo%ea'3g!f?h$e9aZ _FD/&F%6$F'\"\")$\"I)ol2ap=BZ\"ek#R1Hi&3WqL!#R/&F%6$F'\"\"*$!I%3)*fw6i Lvf$Hu'[!H&Ho+L#FQ/&F%6$F'\"#5$\"IkScH2/@nHp*z$Qu/[d'Ri$FQ/&F%6$F'\"#6 $!Id&zhG@W)3]H!)G`!)ol*)fAZFQ/&F%6$F'\"#7$\"Iq@X8!4VM]L`X$H*RY#*[D0*F+ /&F%6$F'\"#8$\"ILMWn!\\S_!Quw8T7

Pw9F+/&F%6$F'\"#9$!I!*z,-W$pS<*))== [TNC9-w8FD/&F%6$F'\"#:$\"H&[DSEt8D@\"*R?%*QG+/XI\\F+/&F%6$F'\"#;$!I#ok %>!e#f " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops (SO7_18) do\n tt := convert(SO7_18[ct],'interpolation_order_conditio n'):\n if expand(subs(e20,lhs(tt)=rhs(tt))) then recd := [op(recd),c t] end if: \nend do:\nop(recd);\nnops(recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"# 6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G \"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\" #Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"# j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#------------------------ ---------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22416 "e20 := \{a[8,3] = 0, a[11 ,4] = -996286030132538159613930889652/16353068885996164905464325675, c [14] = 1/2-1/14*7^(1/2), a[12,5] = -4583493974484572912949314673356033 540575/451957703655250747157313034270335135744, a[11,3] = 0, a[8,4] = \+ -125/2, a[6,3] = 0, a[11,2] = 0, a[10,4] = -20462749524591049105403365 239069/454251913499893469596231268750, a[13,5] = 0, a[12,7] = 16571215 59319846802171283690913610698586256573484808662625/1343148041125514647 7259155104956093505361644432088109056, a[12,8] = 345685379554677052215 495825476969226377187500/74771167436930077221667203179551347546362089, a[17,12] = 42877082723345225226946300758681835657160444961937467671/1 923535017412709386208278631598776372836079978942871093750-350257835071 7681024540012102744371580394344491444736/48208897679516525970132296531 2976534545383453369140625*7^(1/2), a[17,14] = 287666940234183647228741 967485029371867251084288/768467909630489872386593335682960386276245117 1875+572017064437929229652116249893368639587841024/1229548655408783795 8185493370927366180419921875*7^(1/2), a[17,13] = 575222075052234814392 4684511521705800928/2001596446569898054839108908557891845703125-169886 1030059055060173707215296857930816/20015964465698980548391089085578918 45703125*7^(1/2), a[10,2] = 0, a[14,12] = 11352128098668146659861/2546 68911904014019468056-5215842639928607924801/127334455952007009734028*7 ^(1/2), a[14,13] = 3/392-3/392*7^(1/2), a[14,5] = 0, a[11,10] = -31155 237437111730665923206875/392862141594230515010338956291, a[8,6] = 1501 408353528689/265697280000000, a[8,7] = 6070139212132283/92502016000000 , a[7,2] = 0, a[7,3] = 0, c[8] = 943/1000, c[9] = 7067558016280/783715 0160667, c[10] = 909/1000, a[9,2] = 0, a[9,1] = -147251426448621580388 1384708877264246346044433307094207829051978044531801133057155/12468948 0162003200115705962164398602480330155839348790044045363616804606968643 6608, c[4] = 1023/6400, a[6,2] = 0, a[4,2] = 0, a[9,3] = 0, c[5] = 39/ 100, c[6] = 93/200, c[7] = 31/200, a[5,3] = -3899844/2907025, a[14,4] \+ = 0, c[11] = 47/50, c[12] = 1, c[13] = 1, a[2,1] = 1/20, a[3,1] = -716 1/1024000, a[5,4] = 3982992/2907025, a[8,1] = -1221101821869329/690812 928000000, a[8,2] = 0, a[11,1] = -234265984581408683695120714006560917 9073838476242943917/13584809613510567770222314001391587608575321627955 20000, c[2] = 1/20, c[3] = 341/3200, a[7,6] = 5611/283500, a[6,1] = 56 11/114400, a[7,1] = 21173/343200, a[9,4] = -51722943110856684583751756 55246981230039025336933699114138315270772319372469280000/1246193810048 09145897278630571215298365257079410236252921850936749076487132995191, \+ a[4,1] = 1023/25600, a[5,1] = 4202367/11628100, a[14,2] = 0, a[14,3] = 0, a[4,3] = 3069/25600, a[3,2] = 116281/1024000, a[5,2] = 0, a[6,5] = 923521/5106400, a[9,5] = -1207067925846925480797893644173318794948457 1516120469966534514296406891652614970375/27220311547616572217104781845 31100699497284085048389015085076961673446140398628096, a[16,7] = 12518 3725987006091358284807772832081560458336415582349576400523766066981491 59914801577172298098152775739801047/1291772597666133989742651196438570 2783984431335430335690360713394329797397731959853551576489184000000000 0000000+17615228617354110108508911790502650111326496849472122723962114 79818027022186347979194359016936089217819331021/3358608753931948373330 8931107402827238359521472118872794937854825257473234103095619234098871 8784000000000000000*7^(1/2), a[16,9] = 6874361487673480006285931730527 4277780650783519032025828979300088523488988111783201634466707776976570 2709805954192296963021552895479234993257819821106998915574193829101860 119823330279/112536814843150050913676542349737304407597710834184758217 0921264810018468142400631425900826929874528478047717120270330442606124 102557418328419689608693222186318096000000000000000000000+296323273198 7952356517772319258878191238255598109988789416736119366244822580797795 2199471123771928706835338605328112324978622704891575788804993019366904 0074737641797319943395161/94568751969033656229980287688854877653443454 4825082001824303583873965099279328261702437669688970191998359426151487 672640845482439123880948251840007305228727998400000000000000000000*7^( 1/2), a[16,1] = 571262365974722592147508702085674413123074979711954873 812807464116257302576802226313028164589259868449219745217/110772198297 5045860773527525379031925113390736566832412431811518073343206298780025 0708692452000000000000000000000-41805939724167815095839522495249359869 7047777024498277797593332867900350838856267902560305003793843931777113 /521280933164727463893424717825426788288654464266744664673793655563926 214728837658856879644800000000000000000000*7^(1/2), a[12,4] = -1695708 8714171468676387054358954754000/143690415119654683326368228101570221, \+ a[10,7] = 318607235173649312405151265849660869927653414425413/67147167 15558965303132938072935465423910912000000, a[12,9] = -3205890962717072 5427914343121527275340081027740232102405713615707572490561670152301603 52087048674542196011/9475695496839658147830151244512736049846577471272 57615372449205973192657306017239103491074738324033259120, a[13,4] = 0, a[13,2] = 0, a[13,7] = 3847749490868980348119500000/15517045062138271 618141237517, a[14,6] = 460454863031265521343678403/235579852751416522 9982023734-283438859697640884437065685/4711597055028330459964047468*7^ (1/2), a[14,10] = 188753532739715767256965625/332836662942383840688864 78-168249974410951846618609375/66567332588476768137772956*7^(1/2), a[1 2,3] = 0, a[10,3] = 0, a[14,8] = 671994922390044434145634375/909470026 58165004198514338-463454429627970722580265625/181894005316330008397028 676*7^(1/2), a[10,5] = -180269259803172281163724663224981097/381009225 58256871086579832832000000, a[13,1] = 44901867737754616851973/10140464 09980231013380680, a[11,9] = 30076066976810251783423249756545243494667 2266195876496371874262392684852243925359864884962513/46554433375013464 5558506533660450560376082477961552128575189281031568049236410667452439 8280000, a[12,11] = -6122933601070769591613093993993358877250/10505170 01510235513198246721302027675953, a[11,6] = 20980822345096760292224086 794978105312644533925634933539/377588999200755080387872783911549464197 2212962174156800, a[11,5] = -26053085959256534152588089363841/43775528 04565683061011299942400, a[11,8] = 16102142614312417838907512192924671 0833125/10997207722131034650667041364346422894371443, a[7,4] = 8602624 /76559175, a[7,5] = -26782109/689364000, a[9,6] = 78012515584389364132 3090552530431036567795592568497182701460674803126770111481625/18311042 5412731972197889874507158786859226102980861859505241443073629143100805 376, a[13,9] = 1227476547031319687842881203774063505031923427600698639 8294443554969616342274215316330684448207141/48934514749371551765038583 4143510934888829280686609654482896526796523353052166757299452852166040 , a[13,8] = -13734512432397741476562500000/875132892924995907746928783 , a[9,8] = 10332848184452015604056836767286656859124007796970668046446 015775000000/131270355003603364807383424874072791453797202863895016524 9582733679393783, a[10,1] = -29055573360337415088538618442231036441314 060511/22674759891089577691327962602370597632000000000, a[14,9] = -152 0417530253553980380043602085925886494262345387178427997982057187814442 042273423080071783/392192981943480397633664222556992075106628568847304 631390191589139262501961993074284749200+799930289218177898912938325894 667068964649008561582353557148147567546469052859876678259/461403508168 8004678043108500670495001254453751144760369296371636932500023082271462 17352*7^(1/2), a[14,11] = -236307906973303345482653/256372329303735736 54194+173345833072833817590685/51274465860747147308388*7^(1/2), a[14,1 ] = 307/6800, a[11,7] = 8907229937563791864189296220958338352643226357 82294899/13921242001395112657501941955594013822830119803764736, a[13,3 ] = 0, a[12,1] = -2866556991825663971778295329101033887534912787724034 363/868226711619262703011213925016143612030669233795338240, a[12,2] = \+ 0, a[10,9] = -26984049294008425187211664850871297985622698482295177937 03413951226714583/4695456749139343150770004420808711418846760359027175 50325616728175875000000, a[13,10] = -9798363684577739445312500000/3087 22986341456031822630699, a[13,6] = 791638675191615279648100000/2235604 725089973126411512319, a[14,7] = 35514316969207250641724424985/1471616 53169956511475274962258-481702304929173090974271575/294323306339913022 950549924516*7^(1/2), a[10,8] = 212083202434519082281842245535894/2002 2426044775672563822865371173879, a[10,6] = 211276702141728028701282869 92003940810655221489/4679473877997892906145822697976708633673728000, a [12,6] = 2346305388553404258656258473446184419154740172519949575/25672 6716407895402892744978301151486254183185289662464, a[12,10] = 40279545 832706233433100438588458933210937500/889646084279948284691697212637733 8947215101, a[13,11] = 282035543183190840068750/1229540762987304042599 1, a[13,12] = -306814272936976936753/1299331183183744997286, a[9,7] = \+ 6641131229599116421347821358391064699281403281605770353571553403929500 09492511875/1517846559858624813633302310729534917527976515008907830113 9943253016877823170816, a[6,4] = 31744/135025, a[8,5] = -1024030607959 889/168929280000000, a[17,2] = 0, a[17,3] = 0, a[16,13] = -56937797859 7854093756310987070088139985275402946482571847134560099124221804827983 0357689/66232602572693238538706352752377192616929954986459591292241070 5613811200000000000000000000+11160589781332902274736436489913088722979 60011076080647100654462877250873303116055359689/6623260257269323853870 63527523771926169299549864595912922410705613811200000000000000000000*7 ^(1/2), a[16,15] = 125667239794778602072676925560041609561972597583209 979258314845487667433090419263759799/190432487583395357730560482698424 0424690363045991229128931126504895049076310016000000000-59028946739025 681742626285442331126491223596724290616786809339665195402731038729879/ 3046919801334325723688967723174784679504580873585966606289802407832078 522096025600000*7^(1/2), a[16,6] = -9974521878866034665001215093798908 2733803619651653658456817485614803504155385560690393554360336092065584 815633/120972156263496171876242158814719485307611948360930108810601776 7355971757142551992626941965147200000000000000000+93320733560040239498 7214451054361465920803702387871787946131778187784567072191817997576654 5310520465203275977/24194431252699234375248431762943897061522389672186 0217621203553471194351428510398525388393029440000000000000000*7^(1/2), a[16,8] = -5475983898288618345981813338837590800614185855015399068684 6978348863143039759306003458396414251841877395553/30307100180597417208 0490966130104468050126440864014782092582400429042604909625037616237211 82105600000000000+1328436178798707517398824622334536072945626857973036 5891407251773114191650397356739750286947316947113707/35446900796020371 0035661948690180664386112796332181031687230877694786672408918172650569 83838720000000000*7^(1/2), a[16,10] = -1034263338267649013726626367070 3536419445868122740593242876840357971955043876772411590246332491918239 96323/8224244086237771871467831538183993451977137207679607561945959104 92784789119961559213755380633600000000000+2742634254463020699447644274 4991819773159228214144325512485660170991733435364442179829595019624777 5939471/21383034624218206865816361999278382975140556739966979661059493 67281240451711900053955763989647360000000000*7^(1/2), a[16,12] = -1700 6680248577809473038573186499560848605733156506811085532643136411973839 10289904206751497096329440010629/2704188218517981979215772760946666451 0310133901501817355481149039230552258240217568756800000000000000000000 +181109233759697002284558469350564784363590987534352651033927139760232 65962815946771736872563829234342887/1233489362832763709817720908501988 2057334447042790302653377366228420953661653432575222400000000000000000 00*7^(1/2), a[16,11] = 66239091060403368360262620702166481840242021269 701059559317336313240648392339911854838860941565539670913883/266129688 5681834025392430094576278609597373668886728883839313606883067431505027 9167258323200000000000000000-90093654678717919554006084633237473317647 6066877608935979837245786333301828102316788089290072092611304509/17741 9792378788935026162006305085240639824911259115258922620907125537828767 0018611150554880000000000000000*7^(1/2), a[16,14] = 366113642577897032 1788839592892634028961008509259561366501743235532704860828477762828149 67/1799237869153052718979783972379834839141527343363786470802828199919 2834400000000000000000000-10234780275030794031486823421385099351079439 80471305000079396396912342658754866704389719/2878780590644884350367654 3558077357426264437493820583532845251198708535040000000000000000*7^(1/ 2), a[16,3] = 0, a[16,4] = 0, a[16,5] = 0, a[15,10] = 5436526402550765 3851977480738208/4466478390356713423359819140625-149629452831113141668 2934929792/235786943683989396406626953125*7^(1/2), a[15,9] = -30670664 5966127099400049304854628613540749389913325084152909271687564134014278 1006771180215525814345889446383/37334071921822778141051165324669718543 1540894078529094301526280820101435128911405410349160893280029296875000 +181664761198407694024184205976438800959946932614625486554558709444540 1874686830164148895260773137/42447565466790579709675174791599368354403 1508184382874685996435033916010685878289836883544921875*7^(1/2), a[15, 14] = 172678683744/3814697265625*7^(1/2), a[15,1] = 571966514289185728 75631366996149/1245822888041363915313720703125000+10891702347425878099 023621466959/8357395207277482931896209716796875*7^(1/2), a[15,6] = -24 5534888302358312351817429197184/8700455320214002619298187255859375*7^( 1/2)+4116869726480787181292612488487584/265985348360828080075687438964 84375, a[15,13] = -15630718068/3814697265625+6167095848/3814697265625* 7^(1/2), a[15,7] = 631476617959696547824260385843267488/27059579140168 58192337687904052734375-33227466731347290455944055132270208/3922838396 059942349838659979248046875*7^(1/2), a[15,11] = -359806679909281605828 4144620546/146286964102678984417724609375+6557086911287295138744379408 8672/5120043743593764454620361328125*7^(1/2), a[15,8] = 59893300351278 6415989929347488/28972691439853083012428515625-14751096540440194135375 671464064/1371942153475395989706173828125*7^(1/2), a[18,4] = 0, a[18,5 ] = 0, a[17,1] = 14101524042935003010096541750481879321816371268089249 95035277613/3317197909948541665919242431964092415586741982292556762695 3125000-343669929529154146426950913103012707061848467742113726473984/1 18471353926733630925687229713003300556669356510448455810546875*7^(1/2) , a[17,6] = 1443175047412731084367706398900931302412131919281210024855 36/1059071381820782115429096951498359519812365804784881591796875-36138 650435089373461278335139553986413666973704247679234048/756479558443415 796735069251070256799865975574846343994140625*7^(1/2), a[17,16] = 1413 383347739698940972600039030481649401856/264397660355009524692996650528 64808467693125+1861757224737799874854447250922400645120/42303625656801 523950879464084583693548309*7^(1/2), a[17,9] = -1170105558265995407243 1612713805800779875665736054727965423788667287696781753047133223961085 1127136258213798655554999195607667534787/72442848981639410882030313684 8490351295228985814524607088372322520512385396296447478468581426097365 269115793412394100189208984375000+232734140517153928479992141700417455 2247521703277357365728574443247365083894628816391102044670716831157955 3151269834892177196544/25872446064871218172153683458874655403401035207 6615931102990115185897280498677302670881636223606201881827069075855035 7818603515625*7^(1/2), a[17,11] = -54318256075754156430715214780346767 92363131575507490074/5824694411529015933948896260840005226582958953857 421875+43078828428937580969195523720831420116933769072147456/154092444 749444865977484028064550402819654998779296875*7^(1/2), a[17,7] = 24152 1473964570429990475882210375964811719239806626681198304/14701756684713 34233788212618705488919434060375826495361328125-2073401903747204401198 777073254836154760997520307226750976/362112233613629121622712467661450 47276700994478485107421875*7^(1/2), a[17,8] = 360962960703134304701283 7230915901844578627451417318752/53065735882781308009044598862367219694 96006556752734375-6844570157978013134738308219277865790854278525460070 4/332492079466048295796018789864456263752882616337890625*7^(1/2), a[17 ,15] = -180219433009153795201274490416832044056/2212261355670987711698 010267367693359375+88272455565304907970131443124874176/353961816907358 0338716816427788309375*7^(1/2), a[17,10] = 765660571360590504522410811 080563711090654609706246624/187201424910254107980463739378295055509674 0829301171875-18934143677045047338847621384679376883588265306034176/23 0543626736766142833083422879673713681864634150390625*7^(1/2), c[18] = \+ 13/25, a[16,2] = 0, c[17] = 233/625, a[15,12] = -192755545388391767654 2797699/276054308640062825012207031250, c[19] = 177/250, a[19,18] = 0, a[19,17] = 0, a[18,17] = 0, a[18,10] = 135211347127581813692898317866 17583384500556000/28357138921316598605324684781564221426317494219-8864 019052669836446353435689804066485248/101445417221405134333310741139852 901561249*7^(1/2), a[18,11] = -624931687431641426621118270642785259964 248638186/596448707740571231636366977110016535202094996875+46932051777 99375282949632884666545966117964544/1577906634234315427609436447380996 1248732671875*7^(1/2), a[18,7] = 1613739734695951072604821872000013991 98950768983264/890804665393281807928479125180130564201466168548125-133 6603043701021865325398968333701077347089079296/21941001610671965712524 116383747058231563206121875*7^(1/2), a[18,8] = 10370962436573077008766 029273604620733739288898400/135848283859920148503154173087660082419097 76785287-48737181047905230621987375887798081383160832/2225306466491721 92741910614915819094171863921*7^(1/2), a[18,12] = 28899294106750011031 701193669610820543459670391/116550287445598485886229427145393313951724 6093750-250878995631377260872470039095670096659584/3245622039699205956 1745872220939380103515625*7^(1/2), a[18,16] = 166763426301714707237451 4384896000000000/42303625656801523950879464084583693548309+19807458223 60913756216308531200000000000/4230362565680152395087946408458369354830 9*7^(1/2), a[18,6] = 1628325690801794340487116583420050343881909830788 1504/108448909498448088619939527833432014828786258409971875-1458189019 09625960229141275905926424005344756683776/2869018769800213984654484863 318307270602811069046875*7^(1/2), a[18,15] = 6112662959651597338301023 17250/13983676717327724794930632801139+21824632869622508258865350000/8 22569218666336752642978400067*7^(1/2), a[18,9] = -14710825800192383440 3649355940126193755418546729558540905801753589516454330996914481866297 41883086125225426868034484205191963/7418147735719875674319904121328541 1972631448147407319765849325826100468264580756221795182738032370203557 457245429155859375000+253551248527473466633481686437788192862888412819 2934033813488336004382304485128389976568844110774595925130633317370691 456/264933847704281274082853718618876471330826600526454713449461877950 358815230645557934982795492972750726990918733675556640625*7^(1/2), a[1 8,13] = 73226019221752589487279696352758/22773719569861951201724972470 703125-1209684402178421037903496517328/1339630562933055953042645439453 125*7^(1/2), a[18,1] = 14793729949951918223901917772128815983935554654 1856055637/3396810659787306665901304250331230633560823789867578125000- 462234303531538433172120667936726432477695301843136/149771193112315108 725807065711253555271641260576171875*7^(1/2), a[18,14] = 3869105683769 65091074907779867460087552/9714952339032365794121871305424091796875+76 9360036098797882353443381891543296/15543923742451785270594994088678546 875*7^(1/2), a[15,4] = 0, a[15,5] = 0, c[15] = 57/125, a[15,2] = 0, a[ 15,3] = 0, a[19,4] = 0, a[19,5] = 0, c[16] = 1163/10000, a[19,3] = 0, \+ a[19,2] = 0, a[18,2] = 0, a[18,3] = 0, a[17,5] = 0, a[17,4] = 0, a[19, 16] = -4839265312222765202258896404987936768000/4230362565680152395087 9464084583693548309-1595940816178932602370969394040832000000/423036256 56801523950879464084583693548309*7^(1/2), a[19,1] = 553052300452019173 6960195542348698154104110118263839314563/10661678153758024688955757220 1231344430659880410156250000000+53024609659096171207519665161701297338 237007144082879671/213233563075160493779115144402462688861319760820312 50000000*7^(1/2), a[19,15] = 165094365518528096370316638519506567/2684 865929726923160626681497818688000-459171703963343898329722995722387/21 478927437815385285013451982549504*7^(1/2), a[19,14] = -935233124784966 29327249133347563042523424791/2914485701709709738236561391627227539062 500000-185968295894238188887694028978095988881993/46631771227355355811 78498226603564062500000*7^(1/2), a[19,10] = 61017330127910419916270883 8720280036930719997064817/17749468435935204312221747141053160818695024 1593000+828648390770158536669530237046152912694947771687/1177020453311 3530711022378740751432903643915225000*7^(1/2), a[19,13] = 467817704897 204714306167183851765640617/36437951311779121922759955953125000000000+ 26511177545934515439957752429061942951/3643795131177912192275995595312 5000000000*7^(1/2), a[19,7] = 1486234627304027290885603728717958757029 589614659889902501/376364971128661563849782430388605163375119456211582 8125000+2450032695809623249306525774682192560340266795349154999/499157 78664278721995992364773024557476806293927265625000*7^(1/2), a[19,11] = -9031882193968573190100226290191444897756098873792197/220906928792804 1598653211026333394574822574062500000-26470101979653635988884791935781 85849983594257475863/1104534643964020799326605513166697287411287031250 0000*7^(1/2), a[19,9] = -181498393767461703325802288869617554427730954 7301192139393251180251003230842991271357378863411390292946236146310179 1056402129653/78582073471608852482202374166615902513380771342592499751 9590315954454112972253773535965918835088667410566284379546142578125000 0-48476338655627429633159249648132448704713025320691295853125059645919 927178833757198048236974746322271158408663982057250777761/628656587772 8708198576189933329272201070461707407399980156722527635632903778030188 287727350680709339284530275036369140625000000*7^(1/2), a[19,12] = 5401 9040833900918183496545826453513602367636600622753/72951846586319052276 9361969910054446586720703125000000+36789486295752866742517550896118750 84920448820393/590703211225255484023774874421096717883984375000000*7^( 1/2), a[19,8] = 1460901536507797911430143644112950654968542708040647/5 03141792073778327789459900324666971922584325381000+1168248977101226311 81617536954552563387143135344547/6620286737812872734071840793745618051 61295164975000*7^(1/2), a[19,6] = 317611357960908802165139257838389057 38809726937453499027/1004156569430074894629069702161407544710983874166 40625000+2937256445284077425545167027958391452146185679178908559/71725 469245005349616362121582957681765070276726171875000*7^(1/2)\}: " } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 20" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order conditio ns can be adapted to give a method of stage by stage construction for \+ an interpolation scheme that avoids dealing with the weight polynomial s for a given stage (corresponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_19 := SimpleOrderConditions(7,19,'expanded'):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,21,27,31,32,61,63,64]:\ninterp_order_eqns20 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_19[ct],'interpo lation_order_condition'):\n interp_order_eqns20 := [op(interp_order_ eqns20),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be spec ified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 558 "interp_order_eqns20 := [add(a[20,i],i=1..19)=c[20],seq(op(StageOr derConditions(i,20..20,'expanded')),i=2..7),\n add(a[20,i]*add(a[i, j]*add(a[j,k]*add(a[k,l]*c[l]^2,l=2..k-1),k=2..j-1),\n \+ j=2..i-1),i=2..19)=c[20]^6/360, #21\n add(a[20,i]*add(a[ i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..19)=c[20]^6/120, #27 \n add(a[20,i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..19)=c[20]^6/30, #31 \n add(a[20,i]*c[i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..19)=c[20]^7/35 , ##61\n add(a[20,i]*add(a[i,j]*c[j]^5,j=2..i-1),i=2..19)=c[20]^7/ 42]: ##63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We specify " }{XPPEDIT 18 0 "c[20] = 2249/2500;" "6#/&%\"cG6# \"#?*&\"%\\A\"\"\"\"%+D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[20,1 7] = 0" "6#/&%\"aG6$\"#?\"#<\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a [20,18] = 0;" "6#/&%\"aG6$\"#?\"#=\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[20,19] = 0;" "6#/&%\"aG6$\"#?\"#>\"\"!" }{TEXT -1 2 " . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "e21 := `union`(e20,\{ c[20]=2249/2500,seq(a[20,i]=0,i=2..5),seq(a[20,i]=0,i=17..19)\}):\neqs _20 := expand(subs(e21,interp_order_eqns20)):\nnops(eqs_20);\nindets(e qs_20);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#?\"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6 $F'\"\"*&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\"#7&F%6$F'\"#8&F%6$F'\"#9&F%6$F' \"#:&F%6$F'\"#;&F%6$F'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e22 := solve(\{op(eqs_20)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e23 \+ := `union`(e21,e22):\nseq(a[20,i]=subs(e23,a[20,i]),i=1..19):\nevalf[4 0](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "65/&%\"aG6$\"#?\"\"\"$\"IMZ*oy 'GFb`$oI,U++*[ewj!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F 0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"IkhBJ=K8aF6&ec&[.gBv[d!#S/&F%6$F'\"\"( $\"IKLEZq4qD'*Hd3q$4i4R^\"fFD/&F%6$F'\"\")$!I+\\Uco\"\\/;FUGOsl(\\9fRq !#R/&F%6$F'\"\"*$\"I'R#[*QgyRacg'[cQL)>\\\"z8!#Q/&F%6$F'\"#5$!IXQA\"Ho EBt\\FQ'*R*)*zr&))o\"FX/&F%6$F'\"#6$\"I-9_=r(>LDC*e\\!e/\"FX/&F%6 $F'\"#7$!I/@!H?G)*y1i%[)pma77JLG(F+/&F%6$F'\"#8$!IZbw%\\8#p+Xg3B#)*3P- W$\\=F+/&F%6$F'\"#9$!I!HtXxjLw-Sc5xKX&=,6R:FD/&F%6$F'\"#:$!ICB*4nGxtS% oEh\"4dH]a=M\"FD/&F%6$F'\"#;$!I^GgbS!phr*>v?bMXQ55DFFD/&F%6$F'\"#F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#---------------------------------------------- -----------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check w hich of the (adapted) simple order conditions are satisfied at this st age." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_19) do\n tt := con vert(SO7_19[ct],'interpolation_order_condition'):\n if expand(subs(e 23,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop( recd);\nnops(recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\" \"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#= \"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N\" #O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"# `\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 70 "#------------------------------------------------------ ---------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e23" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25745 "e23 := \{a[8,3] = 0, a[11,4] = -99628603013253815961393088965 2/16353068885996164905464325675, c[14] = 1/2-1/14*7^(1/2), a[12,5] = - 4583493974484572912949314673356033540575/45195770365525074715731303427 0335135744, a[11,3] = 0, a[8,4] = -125/2, a[6,3] = 0, a[11,2] = 0, a[2 0,17] = 0, a[20,18] = 0, a[20,19] = 0, a[10,4] = -20462749524591049105 403365239069/454251913499893469596231268750, a[13,5] = 0, a[12,7] = 16 57121559319846802171283690913610698586256573484808662625/1343148041125 5146477259155104956093505361644432088109056, a[12,8] = 345685379554677 052215495825476969226377187500/747711674369300772216672031795513475463 62089, a[17,12] = 4287708272334522522694630075868183565716044496193746 7671/1923535017412709386208278631598776372836079978942871093750-350257 8350717681024540012102744371580394344491444736/48208897679516525970132 2965312976534545383453369140625*7^(1/2), a[17,14] = 287666940234183647 228741967485029371867251084288/768467909630489872386593335682960386276 2451171875+572017064437929229652116249893368639587841024/1229548655408 7837958185493370927366180419921875*7^(1/2), a[17,13] = 575222075052234 8143924684511521705800928/2001596446569898054839108908557891845703125- 1698861030059055060173707215296857930816/20015964465698980548391089085 57891845703125*7^(1/2), a[10,2] = 0, a[14,12] = 1135212809866814665986 1/254668911904014019468056-5215842639928607924801/12733445595200700973 4028*7^(1/2), a[14,13] = 3/392-3/392*7^(1/2), a[14,5] = 0, a[11,10] = \+ -31155237437111730665923206875/392862141594230515010338956291, a[8,6] \+ = 1501408353528689/265697280000000, a[8,7] = 6070139212132283/92502016 000000, a[7,2] = 0, a[7,3] = 0, c[8] = 943/1000, c[9] = 7067558016280/ 7837150160667, c[10] = 909/1000, a[9,2] = 0, a[9,1] = -147251426448621 5803881384708877264246346044433307094207829051978044531801133057155/12 4689480162003200115705962164398602480330155839348790044045363616804606 9686436608, c[4] = 1023/6400, a[6,2] = 0, a[4,2] = 0, a[9,3] = 0, c[5] = 39/100, c[6] = 93/200, c[7] = 31/200, a[5,3] = -3899844/2907025, a[ 14,4] = 0, c[11] = 47/50, c[12] = 1, c[13] = 1, a[2,1] = 1/20, a[3,1] \+ = -7161/1024000, a[5,4] = 3982992/2907025, a[8,1] = -1221101821869329/ 690812928000000, a[8,2] = 0, a[11,1] = -234265984581408683695120714006 5609179073838476242943917/13584809613510567770222314001391587608575321 62795520000, c[2] = 1/20, c[3] = 341/3200, a[7,6] = 5611/283500, a[6,1 ] = 5611/114400, a[7,1] = 21173/343200, a[9,4] = -51722943110856684583 75175655246981230039025336933699114138315270772319372469280000/1246193 8100480914589727863057121529836525707941023625292185093674907648713299 5191, a[4,1] = 1023/25600, a[5,1] = 4202367/11628100, a[14,2] = 0, a[1 4,3] = 0, a[4,3] = 3069/25600, a[3,2] = 116281/1024000, a[5,2] = 0, a[ 6,5] = 923521/5106400, a[9,5] = -1207067925846925480797893644173318794 9484571516120469966534514296406891652614970375/27220311547616572217104 78184531100699497284085048389015085076961673446140398628096, a[16,7] = 125183725987006091358284807772832081560458336415582349576400523766066 98149159914801577172298098152775739801047/1291772597666133989742651196 4385702783984431335430335690360713394329797397731959853551576489184000 0000000000000+17615228617354110108508911790502650111326496849472122723 96211479818027022186347979194359016936089217819331021/3358608753931948 3733308931107402827238359521472118872794937854825257473234103095619234 0988718784000000000000000*7^(1/2), a[16,9] = 6874361487673480006285931 7305274277780650783519032025828979300088523488988111783201634466707776 9765702709805954192296963021552895479234993257819821106998915574193829 101860119823330279/112536814843150050913676542349737304407597710834184 7582170921264810018468142400631425900826929874528478047717120270330442 606124102557418328419689608693222186318096000000000000000000000+296323 2731987952356517772319258878191238255598109988789416736119366244822580 7977952199471123771928706835338605328112324978622704891575788804993019 3669040074737641797319943395161/94568751969033656229980287688854877653 4434544825082001824303583873965099279328261702437669688970191998359426 1514876726408454824391238809482518400073052287279984000000000000000000 00*7^(1/2), a[16,1] = 571262365974722592147508702085674413123074979711 954873812807464116257302576802226313028164589259868449219745217/110772 1982975045860773527525379031925113390736566832412431811518073343206298 7800250708692452000000000000000000000-41805939724167815095839522495249 3598697047777024498277797593332867900350838856267902560305003793843931 777113/521280933164727463893424717825426788288654464266744664673793655 563926214728837658856879644800000000000000000000*7^(1/2), a[12,4] = -1 6957088714171468676387054358954754000/14369041511965468332636822810157 0221, a[10,7] = 318607235173649312405151265849660869927653414425413/67 14716715558965303132938072935465423910912000000, a[12,9] = -3205890962 7170725427914343121527275340081027740232102405713615707572490561670152 30160352087048674542196011/9475695496839658147830151244512736049846577 47127257615372449205973192657306017239103491074738324033259120, a[13,4 ] = 0, a[13,2] = 0, a[13,7] = 3847749490868980348119500000/15517045062 138271618141237517, a[14,6] = 460454863031265521343678403/235579852751 4165229982023734-283438859697640884437065685/4711597055028330459964047 468*7^(1/2), a[14,10] = 188753532739715767256965625/332836662942383840 68886478-168249974410951846618609375/66567332588476768137772956*7^(1/2 ), a[12,3] = 0, a[10,3] = 0, a[14,8] = 671994922390044434145634375/909 47002658165004198514338-463454429627970722580265625/181894005316330008 397028676*7^(1/2), a[10,5] = -180269259803172281163724663224981097/381 00922558256871086579832832000000, a[13,1] = 44901867737754616851973/10 14046409980231013380680, a[11,9] = 30076066976810251783423249756545243 4946672266195876496371874262392684852243925359864884962513/46554433375 0134645558506533660450560376082477961552128575189281031568049236410667 4524398280000, a[12,11] = -6122933601070769591613093993993358877250/10 50517001510235513198246721302027675953, a[11,6] = 20980822345096760292 224086794978105312644533925634933539/377588999200755080387872783911549 4641972212962174156800, a[11,5] = -26053085959256534152588089363841/43 77552804565683061011299942400, a[11,8] = 16102142614312417838907512192 9246710833125/10997207722131034650667041364346422894371443, a[7,4] = 8 602624/76559175, a[7,5] = -26782109/689364000, a[9,6] = 78012515584389 3641323090552530431036567795592568497182701460674803126770111481625/18 3110425412731972197889874507158786859226102980861859505241443073629143 100805376, a[13,9] = 1227476547031319687842881203774063505031923427600 6986398294443554969616342274215316330684448207141/48934514749371551765 0385834143510934888829280686609654482896526796523353052166757299452852 166040, a[13,8] = -13734512432397741476562500000/875132892924995907746 928783, a[9,8] = 10332848184452015604056836767286656859124007796970668 046446015775000000/131270355003603364807383424874072791453797202863895 0165249582733679393783, a[10,1] = -29055573360337415088538618442231036 441314060511/22674759891089577691327962602370597632000000000, a[14,9] \+ = -1520417530253553980380043602085925886494262345387178427997982057187 814442042273423080071783/392192981943480397633664222556992075106628568 847304631390191589139262501961993074284749200+799930289218177898912938 325894667068964649008561582353557148147567546469052859876678259/461403 5081688004678043108500670495001254453751144760369296371636932500023082 27146217352*7^(1/2), a[14,11] = -236307906973303345482653/256372329303 73573654194+173345833072833817590685/51274465860747147308388*7^(1/2), \+ a[14,1] = 307/6800, a[11,7] = 8907229937563791864189296220958338352643 22635782294899/13921242001395112657501941955594013822830119803764736, \+ a[13,3] = 0, a[12,1] = -2866556991825663971778295329101033887534912787 724034363/868226711619262703011213925016143612030669233795338240, a[12 ,2] = 0, a[10,9] = -26984049294008425187211664850871297985622698482295 17793703413951226714583/4695456749139343150770004420808711418846760359 02717550325616728175875000000, a[13,10] = -979836368457773944531250000 0/308722986341456031822630699, a[13,6] = 791638675191615279648100000/2 235604725089973126411512319, a[14,7] = 35514316969207250641724424985/1 47161653169956511475274962258-481702304929173090974271575/294323306339 913022950549924516*7^(1/2), a[10,8] = 21208320243451908228184224553589 4/20022426044775672563822865371173879, a[10,6] = 211276702141728028701 28286992003940810655221489/4679473877997892906145822697976708633673728 000, a[12,6] = 2346305388553404258656258473446184419154740172519949575 /256726716407895402892744978301151486254183185289662464, a[12,10] = 40 279545832706233433100438588458933210937500/889646084279948284691697212 6377338947215101, a[13,11] = 282035543183190840068750/1229540762987304 0425991, a[13,12] = -306814272936976936753/1299331183183744997286, a[9 ,7] = 6641131229599116421347821358391064699281403281605770353571553403 92950009492511875/1517846559858624813633302310729534917527976515008907 8301139943253016877823170816, a[6,4] = 31744/135025, a[8,5] = -1024030 607959889/168929280000000, a[17,2] = 0, a[17,3] = 0, a[16,13] = -56937 7978597854093756310987070088139985275402946482571847134560099124221804 8279830357689/66232602572693238538706352752377192616929954986459591292 2410705613811200000000000000000000+11160589781332902274736436489913088 72297960011076080647100654462877250873303116055359689/6623260257269323 8538706352752377192616929954986459591292241070561381120000000000000000 0000*7^(1/2), a[16,15] = 125667239794778602072676925560041609561972597 583209979258314845487667433090419263759799/190432487583395357730560482 6984240424690363045991229128931126504895049076310016000000000-59028946 7390256817426262854423311264912235967242906167868093396651954027310387 29879/3046919801334325723688967723174784679504580873585966606289802407 832078522096025600000*7^(1/2), a[16,6] = -9974521878866034665001215093 7989082733803619651653658456817485614803504155385560690393554360336092 065584815633/120972156263496171876242158814719485307611948360930108810 6017767355971757142551992626941965147200000000000000000+93320733560040 2394987214451054361465920803702387871787946131778187784567072191817997 5766545310520465203275977/24194431252699234375248431762943897061522389 6721860217621203553471194351428510398525388393029440000000000000000*7^ (1/2), a[16,8] = -5475983898288618345981813338837590800614185855015399 0686846978348863143039759306003458396414251841877395553/30307100180597 4172080490966130104468050126440864014782092582400429042604909625037616 23721182105600000000000+1328436178798707517398824622334536072945626857 9730365891407251773114191650397356739750286947316947113707/35446900796 0203710035661948690180664386112796332181031687230877694786672408918172 65056983838720000000000*7^(1/2), a[16,10] = -1034263338267649013726626 3670703536419445868122740593242876840357971955043876772411590246332491 91823996323/8224244086237771871467831538183993451977137207679607561945 95910492784789119961559213755380633600000000000+2742634254463020699447 6442744991819773159228214144325512485660170991733435364442179829595019 6247775939471/21383034624218206865816361999278382975140556739966979661 05949367281240451711900053955763989647360000000000*7^(1/2), a[16,12] = -17006680248577809473038573186499560848605733156506811085532643136411 97383910289904206751497096329440010629/2704188218517981979215772760946 6664510310133901501817355481149039230552258240217568756800000000000000 000000+181109233759697002284558469350564784363590987534352651033927139 76023265962815946771736872563829234342887/1233489362832763709817720908 5019882057334447042790302653377366228420953661653432575222400000000000 00000000*7^(1/2), a[16,11] = 66239091060403368360262620702166481840242 021269701059559317336313240648392339911854838860941565539670913883/266 1296885681834025392430094576278609597373668886728883839313606883067431 5050279167258323200000000000000000-90093654678717919554006084633237473 3176476066877608935979837245786333301828102316788089290072092611304509 /177419792378788935026162006305085240639824911259115258922620907125537 8287670018611150554880000000000000000*7^(1/2), a[16,14] = 366113642577 8970321788839592892634028961008509259561366501743235532704860828477762 82814967/1799237869153052718979783972379834839141527343363786470802828 1999192834400000000000000000000-10234780275030794031486823421385099351 07943980471305000079396396912342658754866704389719/2878780590644884350 3676543558077357426264437493820583532845251198708535040000000000000000 *7^(1/2), a[16,3] = 0, a[16,4] = 0, a[16,5] = 0, a[15,10] = 5436526402 5507653851977480738208/4466478390356713423359819140625-149629452831113 1416682934929792/235786943683989396406626953125*7^(1/2), a[15,9] = -30 6706645966127099400049304854628613540749389913325084152909271687564134 0142781006771180215525814345889446383/37334071921822778141051165324669 7185431540894078529094301526280820101435128911405410349160893280029296 875000+181664761198407694024184205976438800959946932614625486554558709 4445401874686830164148895260773137/42447565466790579709675174791599368 3544031508184382874685996435033916010685878289836883544921875*7^(1/2), a[15,14] = 172678683744/3814697265625*7^(1/2), a[15,1] = 571966514289 18572875631366996149/1245822888041363915313720703125000+10891702347425 878099023621466959/8357395207277482931896209716796875*7^(1/2), a[15,6] = -245534888302358312351817429197184/87004553202140026192981872558593 75*7^(1/2)+4116869726480787181292612488487584/265985348360828080075687 43896484375, a[15,13] = -15630718068/3814697265625+6167095848/38146972 65625*7^(1/2), a[15,7] = 631476617959696547824260385843267488/27059579 14016858192337687904052734375-33227466731347290455944055132270208/3922 838396059942349838659979248046875*7^(1/2), a[15,11] = -359806679909281 6058284144620546/146286964102678984417724609375+6557086911287295138744 3794088672/5120043743593764454620361328125*7^(1/2), a[15,8] = 59893300 3512786415989929347488/28972691439853083012428515625-14751096540440194 135375671464064/1371942153475395989706173828125*7^(1/2), a[18,4] = 0, \+ a[18,5] = 0, a[17,1] = 14101524042935003010096541750481879321816371268 08924995035277613/3317197909948541665919242431964092415586741982292556 7626953125000-34366992952915414642695091310301270706184846774211372647 3984/118471353926733630925687229713003300556669356510448455810546875*7 ^(1/2), a[17,6] = 1443175047412731084367706398900931302412131919281210 02485536/1059071381820782115429096951498359519812365804784881591796875 -36138650435089373461278335139553986413666973704247679234048/756479558 443415796735069251070256799865975574846343994140625*7^(1/2), a[17,16] \+ = 1413383347739698940972600039030481649401856/264397660355009524692996 65052864808467693125+1861757224737799874854447250922400645120/42303625 656801523950879464084583693548309*7^(1/2), a[17,9] = -1170105558265995 4072431612713805800779875665736054727965423788667287696781753047133223 9610851127136258213798655554999195607667534787/72442848981639410882030 3136848490351295228985814524607088372322520512385396296447478468581426 097365269115793412394100189208984375000+232734140517153928479992141700 4174552247521703277357365728574443247365083894628816391102044670716831 1579553151269834892177196544/25872446064871218172153683458874655403401 0352076615931102990115185897280498677302670881636223606201881827069075 8550357818603515625*7^(1/2), a[17,11] = -54318256075754156430715214780 34676792363131575507490074/5824694411529015933948896260840005226582958 953857421875+43078828428937580969195523720831420116933769072147456/154 092444749444865977484028064550402819654998779296875*7^(1/2), a[17,7] = 241521473964570429990475882210375964811719239806626681198304/14701756 68471334233788212618705488919434060375826495361328125-2073401903747204 401198777073254836154760997520307226750976/362112233613629121622712467 66145047276700994478485107421875*7^(1/2), a[17,8] = 360962960703134304 7012837230915901844578627451417318752/53065735882781308009044598862367 21969496006556752734375-6844570157978013134738308219277865790854278525 4600704/332492079466048295796018789864456263752882616337890625*7^(1/2) , a[17,15] = -180219433009153795201274490416832044056/2212261355670987 711698010267367693359375+88272455565304907970131443124874176/353961816 9073580338716816427788309375*7^(1/2), a[17,10] = 765660571360590504522 410811080563711090654609706246624/187201424910254107980463739378295055 5096740829301171875-18934143677045047338847621384679376883588265306034 176/230543626736766142833083422879673713681864634150390625*7^(1/2), c[ 18] = 13/25, a[16,2] = 0, c[17] = 233/625, a[15,12] = -192755545388391 7676542797699/276054308640062825012207031250, c[20] = 2249/2500, c[19] = 177/250, a[19,18] = 0, a[19,17] = 0, a[18,17] = 0, a[18,10] = 13521 134712758181369289831786617583384500556000/283571389213165986053246847 81564221426317494219-8864019052669836446353435689804066485248/10144541 7221405134333310741139852901561249*7^(1/2), a[18,11] = -62493168743164 1426621118270642785259964248638186/59644870774057123163636697711001653 5202094996875+4693205177799375282949632884666545966117964544/157790663 42343154276094364473809961248732671875*7^(1/2), a[18,7] = 161373973469 595107260482187200001399198950768983264/890804665393281807928479125180 130564201466168548125-133660304370102186532539896833370107734708907929 6/21941001610671965712524116383747058231563206121875*7^(1/2), a[18,8] \+ = 10370962436573077008766029273604620733739288898400/13584828385992014 850315417308766008241909776785287-487371810479052306219873758877980813 83160832/222530646649172192741910614915819094171863921*7^(1/2), a[18,1 2] = 28899294106750011031701193669610820543459670391/11655028744559848 58862294271453933139517246093750-2508789956313772608724700390956700966 59584/32456220396992059561745872220939380103515625*7^(1/2), a[18,16] = 1667634263017147072374514384896000000000/4230362565680152395087946408 4583693548309+1980745822360913756216308531200000000000/423036256568015 23950879464084583693548309*7^(1/2), a[18,6] = 162832569080179434048711 65834200503438819098307881504/1084489094984480886199395278334320148287 86258409971875-145818901909625960229141275905926424005344756683776/286 9018769800213984654484863318307270602811069046875*7^(1/2), a[18,15] = \+ 611266295965159733830102317250/13983676717327724794930632801139+218246 32869622508258865350000/822569218666336752642978400067*7^(1/2), a[18,9 ] = -14710825800192383440364935594012619375541854672955854090580175358 951645433099691448186629741883086125225426868034484205191963/741814773 5719875674319904121328541197263144814740731976584932582610046826458075 6221795182738032370203557457245429155859375000+25355124852747346663348 1686437788192862888412819293403381348833600438230448512838997656884411 0774595925130633317370691456/26493384770428127408285371861887647133082 6600526454713449461877950358815230645557934982795492972750726990918733 675556640625*7^(1/2), a[18,13] = 73226019221752589487279696352758/2277 3719569861951201724972470703125-1209684402178421037903496517328/133963 0562933055953042645439453125*7^(1/2), a[18,1] = 1479372994995191822390 19177721288159839355546541856055637/3396810659787306665901304250331230 633560823789867578125000-462234303531538433172120667936726432477695301 843136/149771193112315108725807065711253555271641260576171875*7^(1/2), a[18,14] = 386910568376965091074907779867460087552/971495233903236579 4121871305424091796875+769360036098797882353443381891543296/1554392374 2451785270594994088678546875*7^(1/2), a[20,2] = 0, a[20,3] = 0, a[20,4 ] = 0, a[20,5] = 0, a[15,4] = 0, a[15,5] = 0, c[15] = 57/125, a[15,2] \+ = 0, a[15,3] = 0, a[19,4] = 0, a[19,5] = 0, c[16] = 1163/10000, a[19,3 ] = 0, a[19,2] = 0, a[20,16] = -42532803574891743576605545724281728090 55232/26439766035500952469299665052864808467693125-1785094777818151142 513049915774791536640/42303625656801523950879464084583693548309*7^(1/2 ), a[20,12] = -2127375361994947297156456662040945063812043101896442491 879/23310057489119697177245885429078662790344921875000000000000+569767 107608351000019229940102305750050165107337869521/817896754004199900955 99597996767237860859375000000000000*7^(1/2), a[20,7] = 198765287186638 5232452907492438015519716639253386233472801741/44540233269664090396423 95625900652821007330842740625000000000+4216023480836739985053256243736 1075865066296050829346660567/76793505637351879993834407343114703810471 2214265625000000000*7^(1/2), a[20,9] = 1736891608571769935079625154730 2241376558283516531108642672846555922149224420740403823313300635899659 1512306005798646123369075131279979/12573131755457416397152379866658544 4021409234148147999603134450552712658075560603765754547013614186785690 60550072738281250000000000000-6398177962709559367921046506897802517559 0723058743214318565031910899102521800732807570082544157640637140135653 0654173520825641881471/74181477357198756743199041213285411972631448147 4073197658493258261004682645807562217951827380323702035574572454291558 59375000000000000*7^(1/2), a[20,10] = -9696392003105334990803611577089 3575920364297479381979101/56714277842633197210649369563128442852634988 43800000000+385003959857900961746835426088855092634524120381038717/488 9161882985620449193911169235210590744395555000000000*7^(1/2), a[20,1] \+ = 8118795743627367733179779787397474929191662340010737513325588701/143 932655075733333300902722471662314981390838553710937500000000000+188959 07275772561552080776506607802183413383927634325825193723987/6793621319 574613331802608500662461267121647579735156250000000000000*7^(1/2), a[2 0,14] = -564882522642913093590541724798848365028989502024553/157382227 89232432586477431514787028710937500000000000-1123251917969593604607379 708431968416738765464119/251811564627718921383638904236592459375000000 00000*7^(1/2), a[20,6] = 246009071900299281262492714773577475558316468 008450336418220241/542244547492240443099697639167160074143931292049859 375000000000+177410828005340205426044251274228964643067060257036358334 08497/387317533923028887928355456547971481531379494321328125000000000* 7^(1/2), a[20,13] = -1354173155655320081817796503704371022429901163/65 588312361202419460967920715625000000000000000+533759995360334526521105 17044084509032058611/65588312361202419460967920715625000000000000000*7 ^(1/2), a[20,8] = -205451776793492176675859548474735272940432496667036 92394259/2716965677198402970063083461753201648381955357057400000000+23 5208174935590378206735630533474656549656002037130641567/11916516128063 17092132931342874211249290331296955000000000*7^(1/2), a[20,11] = 13321 3852436463035837839872058329383492383939986258170076149/11928974154811 424632727339542200330704041899937500000000000-592147783090421103173033 593272783400624371424847208058827/220906928792804159865321102633339457 4822574062500000000000*7^(1/2), a[20,15] = -10282400371704628063167572 16166142820312607/14498276020525385067384080088220915200000000-2773405 513418872586522384915122392541421/115986208164203080539072640705767321 600000*7^(1/2), a[18,2] = 0, a[18,3] = 0, a[17,5] = 0, a[17,4] = 0, a[ 19,16] = -4839265312222765202258896404987936768000/4230362565680152395 0879464084583693548309-1595940816178932602370969394040832000000/423036 25656801523950879464084583693548309*7^(1/2), a[19,1] = 553052300452019 1736960195542348698154104110118263839314563/10661678153758024688955757 2201231344430659880410156250000000+53024609659096171207519665161701297 338237007144082879671/213233563075160493779115144402462688861319760820 31250000000*7^(1/2), a[19,15] = 165094365518528096370316638519506567/2 684865929726923160626681497818688000-459171703963343898329722995722387 /21478927437815385285013451982549504*7^(1/2), a[19,14] = -935233124784 96629327249133347563042523424791/2914485701709709738236561391627227539 062500000-185968295894238188887694028978095988881993/46631771227355355 81178498226603564062500000*7^(1/2), a[19,10] = 61017330127910419916270 8838720280036930719997064817/17749468435935204312221747141053160818695 0241593000+828648390770158536669530237046152912694947771687/1177020453 3113530711022378740751432903643915225000*7^(1/2), a[19,13] = 467817704 897204714306167183851765640617/364379513117791219227599559531250000000 00+26511177545934515439957752429061942951/3643795131177912192275995595 3125000000000*7^(1/2), a[19,7] = 1486234627304027290885603728717958757 029589614659889902501/376364971128661563849782430388605163375119456211 5828125000+2450032695809623249306525774682192560340266795349154999/499 15778664278721995992364773024557476806293927265625000*7^(1/2), a[19,11 ] = -9031882193968573190100226290191444897756098873792197/220906928792 8041598653211026333394574822574062500000-26470101979653635988884791935 78185849983594257475863/1104534643964020799326605513166697287411287031 2500000*7^(1/2), a[19,9] = -181498393767461703325802288869617554427730 9547301192139393251180251003230842991271357378863411390292946236146310 1791056402129653/78582073471608852482202374166615902513380771342592499 7519590315954454112972253773535965918835088667410566284379546142578125 0000-48476338655627429633159249648132448704713025320691295853125059645 919927178833757198048236974746322271158408663982057250777761/628656587 7728708198576189933329272201070461707407399980156722527635632903778030 188287727350680709339284530275036369140625000000*7^(1/2), a[19,12] = 5 4019040833900918183496545826453513602367636600622753/72951846586319052 2769361969910054446586720703125000000+36789486295752866742517550896118 75084920448820393/590703211225255484023774874421096717883984375000000* 7^(1/2), a[19,8] = 146090153650779791143014364411295065496854270804064 7/503141792073778327789459900324666971922584325381000+1168248977101226 31181617536954552563387143135344547/6620286737812872734071840793745618 05161295164975000*7^(1/2), a[19,6] = 317611357960908802165139257838389 05738809726937453499027/1004156569430074894629069702161407544710983874 16640625000+2937256445284077425545167027958391452146185679178908559/71 725469245005349616362121582957681765070276726171875000*7^(1/2)\}: " } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 21" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 253 "The standard (simple) order conditio ns can be adapted to give a method of stage by stage construction for \+ an interpolation scheme that avoids dealing with the weight polynomial s for a given stage (corresponding to an \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_20 := SimpleOrderConditions(7,20,'expanded'):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,21,27,31,32,61,63,64]:\ninterp_order_eqns21 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_20[ct],'interpo lation_order_condition'):\n interp_order_eqns21 := [op(interp_order_ eqns21),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be spec ified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 558 "interp_order_eqns21 := [add(a[21,i],i=1..20)=c[21],seq(op(StageOr derConditions(i,21..21,'expanded')),i=2..7),\n add(a[21,i]*add(a[i, j]*add(a[j,k]*add(a[k,l]*c[l]^2,l=2..k-1),k=2..j-1),\n \+ j=2..i-1),i=2..20)=c[21]^6/360, #21\n add(a[21,i]*add(a[ i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..20)=c[21]^6/120, #27 \n add(a[21,i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..20)=c[21]^6/30, #31 \n add(a[21,i]*c[i]*add(a[i,j]*c[j]^4,j=2..i-1),i=2..20)=c[21]^7/35 , ##61\n add(a[21,i]*add(a[i,j]*c[j]^5,j=2..i-1),i=2..20)=c[21]^7/ 42]: ##63" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We specify " }{XPPEDIT 18 0 "c[21] = 979/10000;" "6#/&%\"cG6# \"#@*&\"$z*\"\"\"\"&++\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[21, 17]=0" "6#/&%\"aG6$\"#@\"#<\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[ 21,18]=0" "6#/&%\"aG6$\"#@\"#=\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[21,19] = 0;" "6#/&%\"aG6$\"#@\"#>\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "a[21,20] = 0;" "6#/&%\"aG6$\"#@\"#?\"\"!" }{TEXT -1 2 " . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "e24 := `union`(e23,\{ c[21]=979/10000,seq(a[21,i]=0,i=2..5),seq(a[21,i]=0,i=17..20)\}):\neqs _21 := expand(subs(e24,interp_order_eqns21)):\nnops(eqs_21);\nindets(e qs_21);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#@\"\"\"&F%6$F'\"\")&F%6$F'\"\"*&F% 6$F'\"#5&F%6$F'\"#6&F%6$F'\"#7&F%6$F'\"#8&F%6$F'\"#9&F%6$F'\"#:&F%6$F' \"#;&F%6$F'\"\"'&F%6$F'\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e25 := solve(\{op(eqs_21)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e26 \+ := `union`(e24,e25):\nseq(a[21,i]=subs(e26,a[21,i]),i=1..20):\nevalf[4 0](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "66/&%\"aG6$\"#@\"\"\"$\"I+3%pn Q+VY\"[xh&\\`rFV&eQ!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\" %F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$!I#3*[_C5$pl'e'>'RJ@BB)>`$F+/&F%6$F'\" \"($!I()>\"H\\2.y3lw?7K<))fH$eYF+/&F%6$F'\"\")$!IaE&*=?\"[T#o'z>%z?=%= 'Qpc!#S/&F%6$F'\"\"*$\"I#H'ynL#G$Q=#)esdv$Rh))R7'FP/&F%6$F'\"#5$!I!\\k 6zE)HS2EawPc+Xu\"zR)FP/&F%6$F'\"#6$\"I\"*4(G$)***o&olia7(=^;Fk!4)FP/&F %6$F'\"#7$!I:ot)*4!e22&)4-5I.ku%eN7F+/&F%6$F'\"#8$!I'*e&ox!3W1%**3EE!Q 'Gl_Ay#!#U/&F%6$F'\"#9$!I?Du')3F0/&F%6$F'\"#?F0" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#---------------------- -----------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check which of the (adapted) simple order conditio ns are satisfied at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops (SO7_20) do\n tt := convert(SO7_20[ct],'interpolation_order_conditio n'):\n if expand(subs(e26,lhs(tt)=rhs(tt))) then recd := [op(recd),c t] end if: \nend do:\nop(recd);\nnops(recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"# 6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G \"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\" #Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"# j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#------------------------ ---------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e26" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29181 "e26 := \{a[21,2] = 0, a[8 ,3] = 0, a[11,4] = -996286030132538159613930889652/1635306888599616490 5464325675, c[14] = 1/2-1/14*7^(1/2), a[12,5] = -458349397448457291294 9314673356033540575/451957703655250747157313034270335135744, a[11,3] = 0, a[8,4] = -125/2, a[6,3] = 0, a[11,2] = 0, a[20,17] = 0, a[20,18] = 0, a[20,19] = 0, a[10,4] = -20462749524591049105403365239069/45425191 3499893469596231268750, a[13,5] = 0, a[21,17] = 0, a[21,18] = 0, a[21, 19] = 0, a[21,20] = 0, a[21,5] = 0, a[12,7] = 165712155931984680217128 3690913610698586256573484808662625/13431480411255146477259155104956093 505361644432088109056, a[12,8] = 3456853795546770522154958254769692263 77187500/74771167436930077221667203179551347546362089, a[17,12] = 4287 7082723345225226946300758681835657160444961937467671/19235350174127093 86208278631598776372836079978942871093750-3502578350717681024540012102 744371580394344491444736/482088976795165259701322965312976534545383453 369140625*7^(1/2), a[17,14] = 2876669402341836472287419674850293718672 51084288/7684679096304898723865933356829603862762451171875+57201706443 7929229652116249893368639587841024/12295486554087837958185493370927366 180419921875*7^(1/2), a[17,13] = 5752220750522348143924684511521705800 928/2001596446569898054839108908557891845703125-1698861030059055060173 707215296857930816/2001596446569898054839108908557891845703125*7^(1/2) , a[21,3] = 0, a[21,4] = 0, a[10,2] = 0, a[14,12] = 113521280986681466 59861/254668911904014019468056-5215842639928607924801/1273344559520070 09734028*7^(1/2), a[14,13] = 3/392-3/392*7^(1/2), a[14,5] = 0, a[11,10 ] = -31155237437111730665923206875/392862141594230515010338956291, a[8 ,6] = 1501408353528689/265697280000000, a[8,7] = 6070139212132283/9250 2016000000, a[7,2] = 0, a[7,3] = 0, c[8] = 943/1000, c[9] = 7067558016 280/7837150160667, c[10] = 909/1000, a[9,2] = 0, a[9,1] = -14725142644 8621580388138470887726424634604443330709420782905197804453180113305715 5/12468948016200320011570596216439860248033015583934879004404536361680 46069686436608, c[4] = 1023/6400, a[6,2] = 0, a[4,2] = 0, a[9,3] = 0, \+ c[5] = 39/100, c[6] = 93/200, c[7] = 31/200, a[5,3] = -3899844/2907025 , a[14,4] = 0, c[11] = 47/50, c[12] = 1, c[13] = 1, a[2,1] = 1/20, a[3 ,1] = -7161/1024000, a[5,4] = 3982992/2907025, a[8,1] = -1221101821869 329/690812928000000, a[8,2] = 0, a[11,1] = -23426598458140868369512071 40065609179073838476242943917/1358480961351056777022231400139158760857 532162795520000, c[2] = 1/20, c[3] = 341/3200, a[7,6] = 5611/283500, a [6,1] = 5611/114400, a[7,1] = 21173/343200, a[9,4] = -5172294311085668 458375175655246981230039025336933699114138315270772319372469280000/124 6193810048091458972786305712152983652570794102362529218509367490764871 32995191, a[4,1] = 1023/25600, a[5,1] = 4202367/11628100, a[14,2] = 0, a[14,3] = 0, a[4,3] = 3069/25600, a[3,2] = 116281/1024000, a[5,2] = 0 , a[6,5] = 923521/5106400, a[9,5] = -120706792584692548079789364417331 87949484571516120469966534514296406891652614970375/2722031154761657221 710478184531100699497284085048389015085076961673446140398628096, a[16, 7] = 12518372598700609135828480777283208156045833641558234957640052376 606698149159914801577172298098152775739801047/129177259766613398974265 1196438570278398443133543033569036071339432979739773195985355157648918 40000000000000000+1761522861735411010850891179050265011132649684947212 272396211479818027022186347979194359016936089217819331021/335860875393 1948373330893110740282723835952147211887279493785482525747323410309561 92340988718784000000000000000*7^(1/2), a[16,9] = 687436148767348000628 5931730527427778065078351903202582897930008852348898811178320163446670 7776976570270980595419229696302155289547923499325781982110699891557419 3829101860119823330279/11253681484315005091367654234973730440759771083 4184758217092126481001846814240063142590082692987452847804771712027033 0442606124102557418328419689608693222186318096000000000000000000000+29 6323273198795235651777231925887819123825559810998878941673611936624482 2580797795219947112377192870683533860532811232497862270489157578880499 30193669040074737641797319943395161/9456875196903365622998028768885487 7653443454482508200182430358387396509927932826170243766968897019199835 9426151487672640845482439123880948251840007305228727998400000000000000 000000*7^(1/2), a[16,1] = 57126236597472259214750870208567441312307497 9711954873812807464116257302576802226313028164589259868449219745217/11 0772198297504586077352752537903192511339073656683241243181151807334320 62987800250708692452000000000000000000000-4180593972416781509583952249 5249359869704777702449827779759333286790035083885626790256030500379384 3931777113/52128093316472746389342471782542678828865446426674466467379 3655563926214728837658856879644800000000000000000000*7^(1/2), a[12,4] \+ = -16957088714171468676387054358954754000/1436904151196546833263682281 01570221, a[10,7] = 31860723517364931240515126584966086992765341442541 3/6714716715558965303132938072935465423910912000000, a[12,9] = -320589 0962717072542791434312152727534008102774023210240571361570757249056167 015230160352087048674542196011/947569549683965814783015124451273604984 657747127257615372449205973192657306017239103491074738324033259120, a[ 13,4] = 0, a[13,2] = 0, a[13,7] = 3847749490868980348119500000/1551704 5062138271618141237517, a[14,6] = 460454863031265521343678403/23557985 27514165229982023734-283438859697640884437065685/471159705502833045996 4047468*7^(1/2), a[14,10] = 188753532739715767256965625/33283666294238 384068886478-168249974410951846618609375/66567332588476768137772956*7^ (1/2), a[12,3] = 0, a[10,3] = 0, a[14,8] = 671994922390044434145634375 /90947002658165004198514338-463454429627970722580265625/18189400531633 0008397028676*7^(1/2), a[10,5] = -180269259803172281163724663224981097 /38100922558256871086579832832000000, a[13,1] = 4490186773775461685197 3/1014046409980231013380680, a[11,9] = 3007606697681025178342324975654 52434946672266195876496371874262392684852243925359864884962513/4655443 3375013464555850653366045056037608247796155212857518928103156804923641 06674524398280000, a[12,11] = -612293360107076959161309399399335887725 0/1050517001510235513198246721302027675953, a[11,6] = 2098082234509676 0292224086794978105312644533925634933539/37758899920075508038787278391 15494641972212962174156800, a[11,5] = -2605308595925653415258808936384 1/4377552804565683061011299942400, a[11,8] = 1610214261431241783890751 21929246710833125/10997207722131034650667041364346422894371443, a[7,4] = 8602624/76559175, a[7,5] = -26782109/689364000, a[9,6] = 7801251558 4389364132309055253043103656779559256849718270146067480312677011148162 5/18311042541273197219788987450715878685922610298086185950524144307362 9143100805376, a[13,9] = 122747654703131968784288120377406350503192342 76006986398294443554969616342274215316330684448207141/4893451474937155 1765038583414351093488882928068660965448289652679652335305216675729945 2852166040, a[13,8] = -13734512432397741476562500000/87513289292499590 7746928783, a[9,8] = 1033284818445201560405683676728665685912400779697 0668046446015775000000/13127035500360336480738342487407279145379720286 38950165249582733679393783, a[10,1] = -2905557336033741508853861844223 1036441314060511/22674759891089577691327962602370597632000000000, a[14 ,9] = -152041753025355398038004360208592588649426234538717842799798205 7187814442042273423080071783/39219298194348039763366422255699207510662 8568847304631390191589139262501961993074284749200+79993028921817789891 2938325894667068964649008561582353557148147567546469052859876678259/46 1403508168800467804310850067049500125445375114476036929637163693250002 308227146217352*7^(1/2), a[14,11] = -236307906973303345482653/25637232 930373573654194+173345833072833817590685/51274465860747147308388*7^(1/ 2), a[14,1] = 307/6800, a[11,7] = 890722993756379186418929622095833835 264322635782294899/139212420013951126575019419555940138228301198037647 36, a[13,3] = 0, a[12,1] = -286655699182566397177829532910103388753491 2787724034363/868226711619262703011213925016143612030669233795338240, \+ a[12,2] = 0, a[10,9] = -2698404929400842518721166485087129798562269848 229517793703413951226714583/469545674913934315077000442080871141884676 035902717550325616728175875000000, a[13,10] = -97983636845777394453125 00000/308722986341456031822630699, a[13,6] = 7916386751916152796481000 00/2235604725089973126411512319, a[14,7] = 355143169692072506417244249 85/147161653169956511475274962258-481702304929173090974271575/29432330 6339913022950549924516*7^(1/2), a[10,8] = 2120832024345190822818422455 35894/20022426044775672563822865371173879, a[10,6] = 21127670214172802 870128286992003940810655221489/467947387799789290614582269797670863367 3728000, a[12,6] = 234630538855340425865625847344618441915474017251994 9575/256726716407895402892744978301151486254183185289662464, a[12,10] \+ = 40279545832706233433100438588458933210937500/88964608427994828469169 72126377338947215101, a[13,11] = 282035543183190840068750/122954076298 73040425991, a[13,12] = -306814272936976936753/1299331183183744997286, a[9,7] = 664113122959911642134782135839106469928140328160577035357155 340392950009492511875/151784655985862481363330231072953491752797651500 89078301139943253016877823170816, a[6,4] = 31744/135025, a[8,5] = -102 4030607959889/168929280000000, a[17,2] = 0, a[17,3] = 0, a[16,13] = -5 6937797859785409375631098707008813998527540294648257184713456009912422 18048279830357689/6623260257269323853870635275237719261692995498645959 12922410705613811200000000000000000000+1116058978133290227473643648991 308872297960011076080647100654462877250873303116055359689/662326025726 9323853870635275237719261692995498645959129224107056138112000000000000 00000000*7^(1/2), a[16,15] = 12566723979477860207267692556004160956197 2597583209979258314845487667433090419263759799/19043248758339535773056 04826984240424690363045991229128931126504895049076310016000000000-5902 8946739025681742626285442331126491223596724290616786809339665195402731 038729879/304691980133432572368896772317478467950458087358596660628980 2407832078522096025600000*7^(1/2), a[16,6] = -997452187886603466500121 5093798908273380361965165365845681748561480350415538556069039355436033 6092065584815633/12097215626349617187624215881471948530761194836093010 88106017767355971757142551992626941965147200000000000000000+9332073356 0040239498721445105436146592080370238787178794613177818778456707219181 79975766545310520465203275977/2419443125269923437524843176294389706152 2389672186021762120355347119435142851039852538839302944000000000000000 0*7^(1/2), a[16,8] = -547598389828861834598181333883759080061418585501 53990686846978348863143039759306003458396414251841877395553/3030710018 0597417208049096613010446805012644086401478209258240042904260490962503 761623721182105600000000000+132843617879870751739882462233453607294562 68579730365891407251773114191650397356739750286947316947113707/3544690 0796020371003566194869018066438611279633218103168723087769478667240891 817265056983838720000000000*7^(1/2), a[16,10] = -103426333826764901372 6626367070353641944586812274059324287684035797195504387677241159024633 249191823996323/822424408623777187146783153818399345197713720767960756 194595910492784789119961559213755380633600000000000+274263425446302069 9447644274499181977315922821414432551248566017099173343536444217982959 50196247775939471/2138303462421820686581636199927838297514055673996697 966105949367281240451711900053955763989647360000000000*7^(1/2), a[16,1 2] = -1700668024857780947303857318649956084860573315650681108553264313 641197383910289904206751497096329440010629/270418821851798197921577276 0946666451031013390150181735548114903923055225824021756875680000000000 0000000000+18110923375969700228455846935056478436359098753435265103392 713976023265962815946771736872563829234342887/123348936283276370981772 0908501988205733444704279030265337736622842095366165343257522240000000 000000000000*7^(1/2), a[16,11] = 6623909106040336836026262070216648184 0242021269701059559317336313240648392339911854838860941565539670913883 /266129688568183402539243009457627860959737366888672888383931360688306 74315050279167258323200000000000000000-9009365467871791955400608463323 7473317647606687760893597983724578633330182810231678808929007209261130 4509/17741979237878893502616200630508524063982491125911525892262090712 55378287670018611150554880000000000000000*7^(1/2), a[16,14] = 36611364 2577897032178883959289263402896100850925956136650174323553270486082847 776282814967/179923786915305271897978397237983483914152734336378647080 28281999192834400000000000000000000-1023478027503079403148682342138509 935107943980471305000079396396912342658754866704389719/287878059064488 4350367654355807735742626443749382058353284525119870853504000000000000 0000*7^(1/2), c[21] = 979/10000, a[16,3] = 0, a[16,4] = 0, a[16,5] = 0 , a[15,10] = 54365264025507653851977480738208/446647839035671342335981 9140625-1496294528311131416682934929792/235786943683989396406626953125 *7^(1/2), a[15,9] = -3067066459661270994000493048546286135407493899133 250841529092716875641340142781006771180215525814345889446383/373340719 2182277814105116532466971854315408940785290943015262808201014351289114 05410349160893280029296875000+1816647611984076940241842059764388009599 469326146254865545587094445401874686830164148895260773137/424475654667 9057970967517479159936835440315081843828746859964350339160106858782898 36883544921875*7^(1/2), a[15,14] = 172678683744/3814697265625*7^(1/2), a[15,1] = 57196651428918572875631366996149/12458228880413639153137207 03125000+10891702347425878099023621466959/8357395207277482931896209716 796875*7^(1/2), a[15,6] = -245534888302358312351817429197184/870045532 0214002619298187255859375*7^(1/2)+4116869726480787181292612488487584/2 6598534836082808007568743896484375, a[15,13] = -15630718068/3814697265 625+6167095848/3814697265625*7^(1/2), a[15,7] = 6314766179596965478242 60385843267488/2705957914016858192337687904052734375-33227466731347290 455944055132270208/3922838396059942349838659979248046875*7^(1/2), a[15 ,11] = -3598066799092816058284144620546/146286964102678984417724609375 +65570869112872951387443794088672/5120043743593764454620361328125*7^(1 /2), a[15,8] = 598933003512786415989929347488/289726914398530830124285 15625-14751096540440194135375671464064/1371942153475395989706173828125 *7^(1/2), a[18,4] = 0, a[18,5] = 0, a[17,1] = 141015240429350030100965 4175048187932181637126808924995035277613/33171979099485416659192424319 640924155867419822925567626953125000-343669929529154146426950913103012 707061848467742113726473984/118471353926733630925687229713003300556669 356510448455810546875*7^(1/2), a[17,6] = 14431750474127310843677063989 0093130241213191928121002485536/10590713818207821154290969514983595198 12365804784881591796875-3613865043508937346127833513955398641366697370 4247679234048/75647955844341579673506925107025679986597557484634399414 0625*7^(1/2), a[17,16] = 1413383347739698940972600039030481649401856/2 6439766035500952469299665052864808467693125+18617572247377998748544472 50922400645120/42303625656801523950879464084583693548309*7^(1/2), a[17 ,9] = -117010555826599540724316127138058007798756657360547279654237886 672876967817530471332239610851127136258213798655554999195607667534787/ 7244284898163941088203031368484903512952289858145246070883723225205123 85396296447478468581426097365269115793412394100189208984375000+2327341 4051715392847999214170041745522475217032773573657285744432473650838946 288163911020446707168311579553151269834892177196544/258724460648712181 7215368345887465540340103520766159311029901151858972804986773026708816 362236062018818270690758550357818603515625*7^(1/2), a[17,11] = -543182 5607575415643071521478034676792363131575507490074/58246944115290159339 48896260840005226582958953857421875+4307882842893758096919552372083142 0116933769072147456/15409244474944486597748402806455040281965499877929 6875*7^(1/2), a[17,7] = 2415214739645704299904758822103759648117192398 06626681198304/1470175668471334233788212618705488919434060375826495361 328125-2073401903747204401198777073254836154760997520307226750976/3621 1223361362912162271246766145047276700994478485107421875*7^(1/2), a[17, 8] = 3609629607031343047012837230915901844578627451417318752/530657358 8278130800904459886236721969496006556752734375-68445701579780131347383 082192778657908542785254600704/332492079466048295796018789864456263752 882616337890625*7^(1/2), a[17,15] = -180219433009153795201274490416832 044056/2212261355670987711698010267367693359375+8827245556530490797013 1443124874176/3539618169073580338716816427788309375*7^(1/2), a[17,10] \+ = 765660571360590504522410811080563711090654609706246624/1872014249102 541079804637393782950555096740829301171875-189341436770450473388476213 84679376883588265306034176/2305436267367661428330834228796737136818646 34150390625*7^(1/2), c[18] = 13/25, a[16,2] = 0, c[17] = 233/625, a[15 ,12] = -1927555453883917676542797699/276054308640062825012207031250, c [20] = 2249/2500, c[19] = 177/250, a[19,18] = 0, a[19,17] = 0, a[18,17 ] = 0, a[18,10] = 13521134712758181369289831786617583384500556000/2835 7138921316598605324684781564221426317494219-88640190526698364463534356 89804066485248/101445417221405134333310741139852901561249*7^(1/2), a[1 8,11] = -624931687431641426621118270642785259964248638186/596448707740 571231636366977110016535202094996875+469320517779937528294963288466654 5966117964544/15779066342343154276094364473809961248732671875*7^(1/2), a[18,7] = 161373973469595107260482187200001399198950768983264/8908046 65393281807928479125180130564201466168548125-1336603043701021865325398 968333701077347089079296/219410016106719657125241163837470582315632061 21875*7^(1/2), a[18,8] = 103709624365730770087660292736046207337392888 98400/13584828385992014850315417308766008241909776785287-4873718104790 5230621987375887798081383160832/22253064664917219274191061491581909417 1863921*7^(1/2), a[18,12] = 288992941067500110317011936696108205434596 70391/1165502874455984858862294271453933139517246093750-25087899563137 7260872470039095670096659584/32456220396992059561745872220939380103515 625*7^(1/2), a[18,16] = 1667634263017147072374514384896000000000/42303 625656801523950879464084583693548309+198074582236091375621630853120000 0000000/42303625656801523950879464084583693548309*7^(1/2), a[18,6] = 1 6283256908017943404871165834200503438819098307881504/10844890949844808 8619939527833432014828786258409971875-14581890190962596022914127590592 6424005344756683776/28690187698002139846544848633183072706028110690468 75*7^(1/2), a[18,15] = 611266295965159733830102317250/1398367671732772 4794930632801139+21824632869622508258865350000/82256921866633675264297 8400067*7^(1/2), a[18,9] = -147108258001923834403649355940126193755418 5467295585409058017535895164543309969144818662974188308612522542686803 4484205191963/74181477357198756743199041213285411972631448147407319765 849325826100468264580756221795182738032370203557457245429155859375000+ 2535512485274734666334816864377881928628884128192934033813488336004382 304485128389976568844110774595925130633317370691456/264933847704281274 0828537186188764713308266005264547134494618779503588152306455579349827 95492972750726990918733675556640625*7^(1/2), a[18,13] = 73226019221752 589487279696352758/22773719569861951201724972470703125-120968440217842 1037903496517328/1339630562933055953042645439453125*7^(1/2), a[18,1] = 147937299499519182239019177721288159839355546541856055637/33968106597 87306665901304250331230633560823789867578125000-4622343035315384331721 20667936726432477695301843136/1497711931123151087258070657112535552716 41260576171875*7^(1/2), a[18,14] = 38691056837696509107490777986746008 7552/9714952339032365794121871305424091796875+769360036098797882353443 381891543296/15543923742451785270594994088678546875*7^(1/2), a[20,2] = 0, a[20,3] = 0, a[20,4] = 0, a[20,5] = 0, a[15,4] = 0, a[15,5] = 0, c [15] = 57/125, a[15,2] = 0, a[15,3] = 0, a[19,4] = 0, a[19,5] = 0, c[1 6] = 1163/10000, a[19,3] = 0, a[19,2] = 0, a[20,16] = -425328035748917 4357660554572428172809055232/26439766035500952469299665052864808467693 125-1785094777818151142513049915774791536640/4230362565680152395087946 4084583693548309*7^(1/2), a[20,12] = -21273753619949472971564566620409 45063812043101896442491879/2331005748911969717724588542907866279034492 1875000000000000+56976710760835100001922994010230575005016510733786952 1/81789675400419990095599597996767237860859375000000000000*7^(1/2), a[ 20,7] = 1987652871866385232452907492438015519716639253386233472801741/ 4454023326966409039642395625900652821007330842740625000000000+42160234 808367399850532562437361075865066296050829346660567/767935056373518799 938344073431147038104712214265625000000000*7^(1/2), a[20,9] = 17368916 0857176993507962515473022413765582835165311086426728465559221492244207 404038233133006358996591512306005798646123369075131279979/125731317554 5741639715237986665854440214092341481479996031344505527126580755606037 6575454701361418678569060550072738281250000000000000-63981779627095593 6792104650689780251755907230587432143185650319108991025218007328075700 825441576406371401356530654173520825641881471/741814773571987567431990 4121328541197263144814740731976584932582610046826458075622179518273803 2370203557457245429155859375000000000000*7^(1/2), a[20,10] = -96963920 031053349908036115770893575920364297479381979101/567142778426331972106 4936956312844285263498843800000000+38500395985790096174683542608885509 2634524120381038717/48891618829856204491939111692352105907443955550000 00000*7^(1/2), a[20,1] = 811879574362736773317977978739747492919166234 0010737513325588701/14393265507573333330090272247166231498139083855371 0937500000000000+18895907275772561552080776506607802183413383927634325 825193723987/679362131957461333180260850066246126712164757973515625000 0000000000*7^(1/2), a[20,14] = -56488252264291309359054172479884836502 8989502024553/15738222789232432586477431514787028710937500000000000-11 23251917969593604607379708431968416738765464119/2518115646277189213836 3890423659245937500000000000*7^(1/2), a[20,6] = 2460090719002992812624 92714773577475558316468008450336418220241/5422445474922404430996976391 67160074143931292049859375000000000+1774108280053402054260442512742289 6464306706025703635833408497/38731753392302888792835545654797148153137 9494321328125000000000*7^(1/2), a[20,13] = -13541731556553200818177965 03704371022429901163/65588312361202419460967920715625000000000000000+5 3375999536033452652110517044084509032058611/65588312361202419460967920 715625000000000000000*7^(1/2), a[20,8] = -2054517767934921766758595484 7473527294043249666703692394259/27169656771984029700630834617532016483 81955357057400000000+2352081749355903782067356305334746565496560020371 30641567/1191651612806317092132931342874211249290331296955000000000*7^ (1/2), a[20,11] = 1332138524364630358378398720583293834923839399862581 70076149/11928974154811424632727339542200330704041899937500000000000-5 92147783090421103173033593272783400624371424847208058827/2209069287928 041598653211026333394574822574062500000000000*7^(1/2), a[20,15] = -102 8240037170462806316757216166142820312607/14498276020525385067384080088 220915200000000-2773405513418872586522384915122392541421/1159862081642 03080539072640705767321600000*7^(1/2), a[21,10] = -3367866957586616893 410427922346783366803414426599864670257531/392589842653389223059445491 9646702322522240223650611200000000+13148535922868036966154017210526778 15605731754623429227/1925468052163530824271949687212080796476900761600 00000000*7^(1/2), a[21,6] = -10178794478318584493456761019943726701607 1190464018877029806010097/22210336665282168549363615300286876636935425 72236224000000000000000+2428640427040191383422157563483694492277686588 77917014777973/6114204412643952383530057232128921217684242528000000000 0000000*7^(1/2), a[21,14] = -24770990232711883946737230511891929328893 07401930799/795848895613531405854463697340341600000000000000000000-492 5636955223785191465921698950765057056567478577/12733582329816502493671 41915744546560000000000000000*7^(1/2), a[21,9] = 169706398248230084613 8313538268592369211816913471928666858105103135476066677562700267829370 63810984608721258380881299166411075429341429273/2762248465955328251092 2115709965186130899855601070943796447167143973410728338434316770278954 0891443957973949888434384000000000000000000000-29514527805659251443993 8398160315143268345175914323543636443629467758230778382804554710223942 5710343537548781708565063174753684260829/39460692370790403587031593871 3788373298569365729584911377816673485334438976263347382432556486987777 0828199284120491200000000000000000000*7^(1/2), a[21,8] = -136265141478 7912165741809695851592598033579174576934113855593/22257382827609317130 75677971868222790354497828501422080000000+1326117821821340476216817208 523201301529093076210962351207/774762699373757906250235996890915758268 76142735360000000000*7^(1/2), a[21,13] = -8862359037124771899000982418 6596875337790442861/29849969714609456679124955916800000000000000000000 +2106559645860581354011260769689534502174173117/2984996971460945667912 4955916800000000000000000000*7^(1/2), a[21,11] = 425367433252938473435 88895318047408491040870064772965070443067/4886107813810759529565118276 4852554563755622144000000000000000-96926074196368670424240730776622393 9081981738348816070613/41697455314991974138633881861113291145038080000 000000000000*7^(1/2), a[21,12] = -204691877744661751352838010376042086 10497860027843551773783243/1466889203213490296256532345023504388307611 200000000000000000000+493538563854902573481849668931831700041829917115 11242739/81698089847590659774799907826427423464640000000000000000000*7 ^(1/2), a[21,16] = 51081506620150022938242528276699656513851617/423036 256568015239508794640845836935483090000-154800222479534476770344934051 340614195/42303625656801523950879464084583693548309*7^(1/2), a[21,15] \+ = 181042800053064361629137655490956268617036587/3665736949387159088642 295805021782016000000000-12161820932752853244110559996271097503243/586 5179119019454541827673288034851225600000*7^(1/2), a[21,7] = -182460995 578637394052817159406681129508991835141311094293412398453/308318184348 5995531057417669743453498368978585285286400000000000000+37621711341339 0755239825969009852998391976488205415554634927/79022309567876122754351 165138516772306376736704000000000000000*7^(1/2), a[21,1] = 32998328281 0951157549731408864906325882596395205074486910908910866533/86958352890 55505064707338880847950421915708902061000000000000000000000+1231775471 2126023040069358494577291492268178954465488679493923/51068704015653917 614227324965935031208544649600000000000000000000*7^(1/2), a[18,2] = 0, a[18,3] = 0, a[17,5] = 0, a[17,4] = 0, a[19,16] = -483926531222276520 2258896404987936768000/42303625656801523950879464084583693548309-15959 40816178932602370969394040832000000/4230362565680152395087946408458369 3548309*7^(1/2), a[19,1] = 5530523004520191736960195542348698154104110 118263839314563/106616781537580246889557572201231344430659880410156250 000000+53024609659096171207519665161701297338237007144082879671/213233 56307516049377911514440246268886131976082031250000000*7^(1/2), a[19,15 ] = 165094365518528096370316638519506567/26848659297269231606266814978 18688000-459171703963343898329722995722387/214789274378153852850134519 82549504*7^(1/2), a[19,14] = -9352331247849662932724913334756304252342 4791/2914485701709709738236561391627227539062500000-185968295894238188 887694028978095988881993/4663177122735535581178498226603564062500000*7 ^(1/2), a[19,10] = 610173301279104199162708838720280036930719997064817 /177494684359352043122217471410531608186950241593000+82864839077015853 6669530237046152912694947771687/11770204533113530711022378740751432903 643915225000*7^(1/2), a[19,13] = 4678177048972047143061671838517656406 17/36437951311779121922759955953125000000000+2651117754593451543995775 2429061942951/36437951311779121922759955953125000000000*7^(1/2), a[19, 7] = 1486234627304027290885603728717958757029589614659889902501/376364 9711286615638497824303886051633751194562115828125000+24500326958096232 49306525774682192560340266795349154999/4991577866427872199599236477302 4557476806293927265625000*7^(1/2), a[19,11] = -90318821939685731901002 26290191444897756098873792197/2209069287928041598653211026333394574822 574062500000-2647010197965363598888479193578185849983594257475863/1104 5346439640207993266055131666972874112870312500000*7^(1/2), a[19,9] = - 1814983937674617033258022888696175544277309547301192139393251180251003 2308429912713573788634113902929462361463101791056402129653/78582073471 6088524822023741666159025133807713425924997519590315954454112972253773 5359659188350886674105662843795461425781250000-48476338655627429633159 2496481324487047130253206912958531250596459199271788337571980482369747 46322271158408663982057250777761/6286565877728708198576189933329272201 0704617074073999801567225276356329037780301882877273506807093392845302 75036369140625000000*7^(1/2), a[19,12] = 54019040833900918183496545826 453513602367636600622753/729518465863190522769361969910054446586720703 125000000+3678948629575286674251755089611875084920448820393/5907032112 25255484023774874421096717883984375000000*7^(1/2), a[19,8] = 146090153 6507797911430143644112950654968542708040647/50314179207377832778945990 0324666971922584325381000+11682489771012263118161753695455256338714313 5344547/662028673781287273407184079374561805161295164975000*7^(1/2), a [19,6] = 31761135796090880216513925783838905738809726937453499027/1004 15656943007489462906970216140754471098387416640625000+2937256445284077 425545167027958391452146185679178908559/717254692450053496163621215829 57681765070276726171875000*7^(1/2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "calculation of the inte rpolation coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order condition gives rise t o a group \{list) of equations to be satisfied by the \"d\" coefficie nts of the weight polynomials for a given stage (corresponding to an \+ \"approximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO8_21 := SimpleOrderCon ditions(8,21,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "whch := [1,2,4,8,16,21,27,31,32,38 ,48,54,63,64,102,117,121,123,125,127,128]:\nordeqns := []:\nfor ct in \+ whch do\n eqn_group := convert(SO8_21[ct],'polynom_order_conditions' ,8):\n ordeqns := [op(ordeqns),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for al l known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "eq ns := []:\nfor ct to nops(ordeqns) do\n eqns := [op(eqns),expand(sub s(e26,ordeqns[ct]))];\nend do:\nnops(eqns);\nnops(indets(eqns));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$o\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$o\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Solve the system of equations. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dd := solve(\{op(eqns)\}):\ninfolevel[solve]:=0:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(map(rhs,dd));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "dd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9853 "dd := \{d[ 10,5] = -16561116707927434368430749746093750000000/2666381583079411771 4882047199796111, d[3,2] = 0, d[2,2] = 0, d[16,2] = 0, d[17,5] = -3150 754324678763694915771484375/551563366125069359782367136, d[15,5] = 0, \+ d[13,5] = -21044490071134970568941875/33644155975075275442848, d[15,8] = 0, d[15,3] = 0, d[15,6] = 0, d[5,5] = 0, d[11,2] = -221951084323087 927500849639942387500/59114154491986076909894021245311, d[4,2] = 0, d[ 20,2] = 527103525659203968750000000000/21042001853631209820331300117, \+ d[17,8] = 257978485189378261566162109375/241308972679717844904785622, \+ d[17,6] = 26875995663206206989288330078125/386094356287548551847656995 2, d[6,1] = 0, d[3,7] = 0, d[6,5] = 2234495771304940698617759771092752 50000000/32245181391358395728085235636683762597, d[17,3] = -2927878473 26545784576416015625/551563366125069359782367136, d[14,2] = 0, d[14,5] = 0, d[7,2] = -161256051718398551812329473144499000000/39729525385802 61729585271390877529877, d[18,4] = 594890026372866297347046875/8184288 84658927261404048, d[19,8] = 198756832107543945312500000000/2252338425 23147675227082001, d[19,1] = 0, d[4,1] = 0, d[21,7] = -420211648303600 0000000000000000000000/1518201707178031814054255492746701, d[1,6] = -9 936300382598080556798045420067912379280078125/146315493677943720634525 39353156282844292934, d[13,1] = 0, d[1,5] = 30175606080542971688972320 33215502362626867125/4877183122598124021150846451052094281430978, d[12 ,5] = -43301017050975443033339586768516250/937043324765785837103930455 5009, d[1,3] = 603974687560059968000436359347917013177431311/665070425 8088350937932972433252855838314970, d[2,6] = 0, d[2,4] = 0, d[2,5] = 0 , d[13,8] = 1753295541014862060546875/14719318239095433006246, d[1,8] \+ = -225341210088867395112620553906856689453125000/243859156129906201057 5423225526047140715489, d[2,3] = 0, d[13,2] = 98133428109818708317997/ 19625757652127244008328, d[19,4] = 156013402657853530279062500000/7507 7947507715891742360667, d[19,7] = -265409368837976074218750000000/7507 7947507715891742360667, d[5,2] = 0, d[5,1] = 0, d[13,4] = 211121913355 5967775344525/7597067478242804132256, d[2,8] = 0, d[11,6] = -281436616 469811842014401190966796875000/532027390427874692189046191207799, d[17 ,4] = 9583240825437812534149169921875/3860943562875485518476569952, d[ 11,1] = 0, d[4,3] = 0, d[6,2] = -6229890756596799780029333366840122000 00/10748393797119465242695078545561254199, d[14,1] = 0, d[1,2] = -1312 119830254443264105643722266214288615323/961969057711661542633303047544 79177148540, d[13,7] = -13892101481308973388671875/2943863647819086601 2492, d[9,2] = -669783505006608915182461274952081868611397162955866057 00229793827351418521835661466480054765502528714398961/1631291802612930 5665674278345857134198144609002961128127431311134969911629245737637313 197585417592242220, d[4,8] = 0, d[18,6] = 3196454529204356505126953125 /2455286653976781784212144, d[11,8] = -1377126675699174023773193359375 0000000/177342463475958230729682063735933, d[16,7] = 0, d[18,1] = 0, d [8,2] = 10808531054681950895375225742640625000000/42074848260985307232 19920448250447943, d[16,1] = 0, d[18,3] = -65847429978680223862703125/ 350755236282397397744592, d[21,3] = -272466115969755668480000000000000 00/59150715864079161586529434782339, d[8,4] = 175943547272529256076280 9439327929687500000/12622454478295592169659761344751343829, d[4,6] = 0 , d[20,7] = -1621653662302978515625000000000000/8206380722916171829929 20704563, d[8,1] = 0, d[1,1] = 1, d[8,8] = 670630489863170970535278320 312500000000000/12622454478295592169659761344751343829, d[12,2] = 6494 1094147564984157121130927/1680192441753246973469482617, d[11,7] = 3964 25193373467031179446533203125000000/1241397244331707615107774446151531 , d[14,8] = 0, d[20,4] = 1096405697556323584937500000000000/8206380722 91617182992920704563, d[17,1] = 0, d[13,3] = -622306056594546202685803 1/100932467925225826328544, d[20,5] = -2378788557978784219375000000000 000/820638072291617182992920704563, d[3,5] = 0, d[7,3] = 6800193460955 3442543624002187154000000/136998363399319369985699013478535513, d[1,4] = -6187322114614576292806575955718201415777240485/1950873249039249608 4603385804208377125723912, d[7,5] = 3258221295582757564912514269612571 250000000/671428979020064232299910865058302549213, d[3,6] = 0, d[14,7] = 0, d[11,4] = -36129665445955873813993752722181406250/17734246347595 8230729682063735933, d[3,3] = 0, d[7,1] = 0, d[19,2] = 834754671403732 6577031250000/225233842523147675227082001, d[5,8] = 0, d[20,1] = 0, d[ 21,6] = 21197732329279595200000000000000000000/45546051215340954421627 66478240103, d[5,6] = 0, d[19,5] = -352787822182508792334375000000/750 77947507715891742360667, d[14,4] = 0, d[15,2] = 0, d[21,5] = -87640586 3727817963520000000000000000/216885958168290259150607927535243, d[15,4 ] = 0, d[20,6] = 2749343243595084863281250000000000/820638072291617182 992920704563, d[16,5] = 0, d[7,8] = -563635179326510793181567382812500 000000000/671428979020064232299910865058302549213, d[13,6] = 534871782 205496121951171875/706527275476580784299808, d[3,4] = 0, d[12,3] = -78 624525982843456768820693241782/166339051733571450373478779083, d[8,7] \+ = -19305037533397184826216735839843750000000000/8835718134806914518761 8329413259406803, d[20,3] = -249926433236077324250000000000000/8206380 72291617182992920704563, d[16,6] = 0, d[18,5] = -157299012164159942045 703125/116918412094132465914864, d[10,6] = 158238485470895929312200927 7343750000000/2161931013307631166071517340524009, d[9,7] = 12939928298 3871024397413264713092169242862404380925602998973328839396007278531178 6311114312011242835750976562500/37054793296352717819579123262614480331 08547935022620254146022324308415426583169304315692831527606077820273, \+ d[18,8] = 6294729580402374267578125/51151805291182953837753, d[17,2] = 6754649117969844085693359375/160872648453145229936523748, d[14,6] = 0 , d[21,1] = 0, d[10,2] = 273214539477776345146478171875000000/52591352 723459798254205221301373, d[7,7] = 16225027726164670296841871308593750 000000000/4700002853140449626099376055408117844491, d[18,7] = -6497725 5150147247314453125/102303610582365907675506, d[4,4] = 0, d[4,7] = 0, \+ d[10,4] = 7516180149572117780156544882226562500000/2666381583079411771 4882047199796111, d[3,8] = 0, d[16,4] = 0, d[9,6] = -61243494754871645 9990855585770122915922702837260634498416363585032594767838499580295803 874630658918414266015625/105870837989579193770226066464612800945958512 4292177215470292092659547264738048372661626523293601736520078, d[21,4] = 93273272950221879296000000000000000/4897424861864618755013727395957 1, d[5,3] = 0, d[8,3] = -1189631425917076722284551432177718750000000/3 7867363434886776508979284034254031487, d[2,7] = 0, d[19,6] = 427573615 261502517089843750000/75077947507715891742360667, d[15,7] = 0, d[7,4] \+ = -1478727470898732037593655621252169587500000/67142897902006423229991 0865058302549213, d[1,7] = 6688503327997100188538318464353772999023437 500/17070140929093434074027962578682329985008423, d[19,3] = -344246198 39398271445625000000/75077947507715891742360667, d[9,1] = 0, d[21,2] = 22532261717673989632000000000000000/506067235726010604684751830915567 , d[9,3] = 26579841878480723233465723226892486128923990882412778092935 2716974648494983235975645648394492594931731722116463/52935418994789596 8851130332323064004729792562146088607735146046329773632369024186330813 2616468008682600390, d[10,7] = -82469713805707522095031738281250000000 000/186646710815558824004174330398572777, d[16,3] = 0, d[8,5] = -38767 32013339226718183330236113281250000000/1262245447829559216965976134475 1343829, d[9,4] = -943462298825075939411875259627611964050712024636755 375934340977821350197438435564564605490029827000754238117865/423483351 9583167750809042658584512037838340497168708861881168370638189058952193 490646506093174406946080312, d[16,8] = 0, d[20,8] = 384062674865722656 250000000000000/820638072291617182992920704563, d[9,5] = 5197051204680 4390706449113866207890311807111676450502771807983206640014671762639217 5882246018287455378945360375/10587083798957919377022606646461280094595 85124292177215470292092659547264738048372661626523293601736520078, d[5 ,4] = 0, d[6,7] = 1112716189430464554983499308593750000000000/22571626 9739508770096596649456786338179, d[12,1] = 0, d[3,1] = 0, d[4,5] = 0, \+ d[17,7] = -2080526078308522701263427734375/482617945359435689809571244 , d[21,8] = 1011308203750000000000000000000000000/15182017071780318140 54255492746701, d[12,4] = 39303900884779814446171071292885075/18740866 495315716742078609110018, d[10,3] = -103693506972082795713937878125000 0000/16321454293487523596132654886633, d[2,1] = 0, d[6,6] = -213502043 90091817170873761213281250000000/2614474166866896950925829916487872643 , d[15,1] = 0, d[18,2] = 172622909057200527921875/10492678008447785402 616, d[5,7] = 0, d[8,6] = 13705346014937882713356111389160156250000000 /37867363434886776508979284034254031487, d[14,3] = 0, d[10,1] = 0, d[1 1,5] = 79607938362535275613893136931734375000/177342463475958230729682 063735933, d[12,7] = -215626913732250803030921855468750000/65593032733 605008597275131885063, d[7,6] = -3839578464159148196797269575230468750 000000/671428979020064232299910865058302549213, d[12,6] = 153081363230 771842853174751144531250/28111299742973575113117913665027, d[6,8] = -3 8654232187090589826567382812500000000000/32245181391358395728085235636 683762597, d[12,8] = 7490582835375413494946289062500000/93704332476578 58371039304555009, d[11,3] = 24428850099176219502725741550176825000/53 2027390427874692189046191207799, d[9,8] = -449515337047602424747148878 3352283343622766458498652245469156575306255984719561273070319011794607 2998046875000/52935418994789596885113033232306400472979256214608860773 5146046329773632369024186330813261646800868260039, d[10,8] = 286488459 1533068466186523437500000000000/26663815830794117714882047199796111, d [6,4] = -101411475184796600869849435111443827500000/322451813913583957 28085235636683762597, d[6,3] = 685687424737287683122335386085656332000 00/96735544174075187184255706910051287791\}: " }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "subs(dd,matrix([seq([seq(d[j,i],j=1..11)],i=1..8)])) :\nevalf[8](%);\nsubs(dd,matrix([seq([seq(d[j,i],j=12..21)],i=1..8)])) :\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7-$\" \"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+7-$!)Q*RO\"!\"'F+F+F+F+$!)K6'z&F/$!)n% )eSF/$\")=))oD!\"%$!)t%e5%F6$\")o/&>&F6$!)$=Yv$F67-$\")WO\"3*F/F+F+F+F +$\")wE)3(!\"&$\")Uqj\\FB$!)XdTJ!\"$$\")H=@]FG$!)F?`jFG$\")Fl\"f%FG7-$ !)acrJFBF+F+F+F+$!)=,XJF6$!)(eB?#F6$\")L*QR\"!\"#$!);'yA#FW$\")*o)=GFW $!)BGP?FW7-$\")s4(='FBF+F+F+F+$\")XqHpF6$\")rm_[F6$!)\")HrIFW$\")+')3 \\FW$!)G36iFW$\")$Q*)[%FW7-$!)/,\"z'FBF+F+F+F+$!)i:m\")F6$!)w^=dF6$\") HI>OFW$!)pt%y&FW$\")CJ>tFW$!)'*))*G&FW7-$\")tC=RFBF+F+F+F+$\")>rH\\F6$ \"):8_MF6$!)h)[=#FW$\")l5#\\$FW$!)F\\=WFW$\")!zL>$FW7-$!)&H1C*F/F+F+F+ F+$!)+w)>\"F6$!)9c%R)FB$\")f*HJ&FG$!))o<\\)FG$\")mWu5FW$!)@NlxFGQ(ppri nt06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7,$\"\"!F)F(F( F(F(F(F(F(F(F(7,$\")')4lQ!\"'$\")kB+]!\"(F(F(F($\")bv)>%F-$\")\\$! )VIx=F>$!)R=&e%F>$!)N^XIF>$!)HI1YF>7,$\")(Hs4#!\"%$\")A**yFF>F(F(F($\" )\")4#[#FN$\")KoosF>$\")!>!y?FN$\")0/O8FN$\")r`/>FN7,$!)h-@YFN$!)5-biF >F(F(F($!)oS7dFN$!)UPX8FN$!)S&*)p%FN$!)hq)*GFN$!).'3/%FN7,$\")caXaFN$ \")OVqvF>F(F(F($\");*4'pFN$\")h'=I\"FN$\")F1&p&FN$\")2D]LFN$\")=8aYFN7 ,$!)[M(G$FN$!)I+>ZF>F(F(F($!)u\"4J%FN$!)PT^jF>$!)v6NNFN$!)()3w>FN$!)\\ #yw#FN7,$\")!\\Q*zF>$\")F:\">\"F>F(F(F($\")&z!p5FN$\")xfI7F>$\")dYC))F >$\")%\\+o%F>$\")VAhmF>Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of ord er conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "nm := NULL:\nfor ct to nops(SO8_21) do\n eqn_group := convert(S O8_21[ct],'polynom_order_conditions',8):\n tt := expand(subs(\{op(e2 6),op(dd)\},eqn_group));\n tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),t t);\n if add(op(i,tt),i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\s\"\"\"\"\"# \"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"# =\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\"#K\"#L\"#M\"#N \"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^\"#_ \"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k\"#l\"#m\"#n\"#o\"#p\" #q\"#r\"#s\"#t\"#u\"#v\"#w\"#x\"#y\"#z\"#!)\"#\")\"##)\"#$)\"#%)\"#&) \"#')\"#()\"#))\"#*)\"#!*\"#\"*\"##*\"#$*\"#%*\"#&*\"#'*\"#(*\"#)*\"#* *\"$+\"\"$,\"\"$-\"\"$.\"\"$/\"\"$0\"\"$1\"\"$2\"\"$3\"\"$4\"\"$5\"\"$ 6\"\"$7\"\"$8\"\"$9\"\"$:\"\"$;\"\"$<\"\"$=\"\"$>\"\"$?\"\"$@\"\"$A\" \"$B\"\"$C\"\"$D\"\"$E\"\"$F\"\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"$G\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "principle error graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "The interpolation scheme amount s to having a Runge-Kutta method for each value of the parameter " } {TEXT 270 1 "u" }{TEXT -1 8 " where " }{XPPEDIT 18 0 "0<=u" "6#1\"\"! %\"uG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "The nodes and linking coefficients are ..." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "e_u := map(_U->lhs(_U)=rhs(_ U)/u,e26):" }}}{PARA 0 "" 0 "" {TEXT -1 43 " ... and the weight polyno mials (of degree " }{XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 33 " and re-using the weight symbol " }{XPPEDIT 18 0 "b[j]" "6#&%\"bG 6#%\"jG" }{TEXT -1 10 ") are ... " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "pols := [seq(b[j]=add(simplify(subs(dd,d[j,i]))*u^(i- 1),i=1..8),j=1..21)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "The whole interpolation scheme (Runge-Kutta scheme \+ with a parameter), including the weights, is given by the set of equat ions:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eu := `union`(e_u, \{op(pols)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can calculate the principal error norm, that is, the \+ root mean square of the residues of the principal error terms. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "errterms8_21 := PrincipalErr orTerms(8,21,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "sm := 0:\nfor ct to nops(errterms8_ 21) do\n sm := sm+expand(subs(eu,errterms8_21[ct]))^2;\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Because t he step has width " }{XPPEDIT 18 0 "u*h" "6#*&%\"uG\"\"\"%\"hGF%" } {TEXT -1 17 " we multiply by " }{XPPEDIT 18 0 "u^9;" "6#*$%\"uG\"\"* " }{TEXT -1 45 " in order to provide appropriate weighting. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ssm := sqrt(sm)*u^9:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(ssm,u=0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 459 282 282 {PLOTDATA 2 "6%-%'CURVESG6$7eq7$$\"3`*****\\n5;\"o!#@$\"3))*R0TU`@V*!# G7$$\"3#******\\8ABO\"!#?$\"3ItZp+NePP!#F7$$\"33+++-K[V?F1$\"3m#**eXG. 3L)F47$$\"3#)******pUkCFF1$\"3!*Rj\"GwQrY\"!#E7$$\"3s*****\\Smp3%F1$\" 3'pmg(>AFRKF?7$$\"3k******R&)G\\aF1$\"3%RQQUsl0l&F?7$$\"3Y******4G$R<) F1$\"35?Lh3%[QA\"!#D7$$\"3%******zqd)*3\"!#>$\"3=F)pc'yM;FS$\"3i9lOc*=+O%FO7$$\"3')*****fT:(z@FS$\"3sa<3.\"**e;(FO7$$ \"3#*******zZ*z7$FS$\"3'p#4MDS*RG\"!#C7$$\"33+++XTFwSFS$\"3'=sBz#G)3*= F_o7$$\"3=+++qMrU^FS$\"39%*Hq9*=Mb#F_o7$$\"3&******4z_\"4iFS$\"3?xbE& \\tP9$F_o7$$\"3y*****\\;hEG(FS$\"3P)*))f:#=>j$F_o7$$\"3o******R&phN)FS $\"3u)**)*Qc#G(*RF_o7$$\"3!)*****\\rvXU*FS$\"3h9l.s#[iB%F_o7$$\"3++++* =)H\\5!#=$\"37,;LmC[eVF_o7$$\"3++++A>1u5Faq$\"3')e6'oQe;P%F_o7$$\"3#** ****RlD))4\"Faq$\"3y>A&*p\"3'zVF_o7$$\"3++++'Q*eB6Faq$\"3#\\\"Faq$\"3em2% f*[WkVF_o7$$\"3%******z/3uC\"Faq$\"3&>B([G'=FL%F_o7$$\"31+++!pt*\\8Faq $\"3MBAI9[BHUF_o7$$\"35+++J$RDX\"Faq$\"3a)[C=5cS4%F_o7$$\"37+++)R'ok;F aq$\"3%HvhjeQ2\"QF_o7$$\"31+++_(>/x\"Faq$\"3kGz&*[$Q%3PF_o7$$\"3-+++1J :w=Faq$\"3I;+&3.P*[OF_o7$$\"3,+++#)H`I>Faq$\"3,g()otU=OOF_o7$$\"3#**** **p&G\"\\)>Faq$\"3\"4Ic.fK^j$F_o7$$\"36+++KFHR?Faq$\"3zBaE#p&*[k$F_o7$ $\"33+++3En$4#Faq$\"3)y//KCfUm$F_o7$$\"31+++c#o%*=#Faq$\"3V$fLwGRtr$F_ o7$$\"3-+++/RE&G#Faq$\"3?c=u9ul(y$F_o7$$\"3\")*****\\K]4]#Faq$\"3%3)R/ *p&QuRF_o7$$\"3$******\\PAvr#Faq$\"3_`JthuNsTF_o7$$\"3)******\\nHi#HFa q$\"3qv?M&4UkQ%F_o7$$\"3*)*****p*ev:JFaq$\"3&)=ifwo')QYF_o7$$\"3$)**** *z!47TLFaq$\"3cYj=z#o#e]F_o7$$\"3?+++LY.KNFaq$\"3Lnv,x9R%e%GoF_o7$$ \"3?+++\"=lj;%Faq$\"33avG;;'>`(F_o7$$\"3++++V&R(QZ%\\\") F_o7$$\"3/+++Xh-'e%Faq$\"3Uu+(>R+;p)F_o7$$\"3!*******Q\"3Gy%Faq$\"3'\\ K)>@r3o!*F_o7$$\"3')*****47O*))[Faq$\"3as^Y2aFaq$\"3CO\\y9h<;%*F_o7$$\"37+++]K56bFaq$\"3%[7S#p/gr$*F_o7$$\" 3\\+++yXu9cFaq$\"3Ukb>\"\\>7J*F_o7$$\"3d******[y))GeFaq$\"3![k&oGd[n\" *F_o7$$\"3J+++bljLfFaq$\"3b]'y(f7\"Q5*F_o7$$\"3.+++i_QQgFaq$\"3KD()eed Ha!*F_o7$$\"30+++?]tRhFaq$\"3+3q0^SHD!*F_o7$$\"3A+++!y%3TiFaq$\"3EX-& \\Bv$=!*F_o7$$\"3K+++5kh`jFaq$\"3oW\\^bI(z.*F_o7$$\"35+++O![hY'Faq$\"3 MV$3*y@O%3*F_o7$$\"3I+++#Qx$omFaq$\"3_Dq^`F8:#*F_o7$$\"3s*****RP+V)oFa q$\"3kQnt'>+YO*F_o7$$\"35+++N&H@)pFaq$\"3_]%ekpPIT*F_o7$$\"3Y*****ppe* zqFaq$\"3&ouX%*>)QQ%*F_o7$$\"3?+++b_VLrFaq$\"3MAGs^p/S%*F_o7$$\"3e**** **4=\"p=(Faq$\"3h-$oFu6>V*F_o7$$\"30+++l$)QSsFaq$\"3GfTAO=F8%*F_o7$$\" 3u*****R#\\'QH(Faq$\"3[BGGhmh$Q*F_o7$$\"3'*******p%*\\%R(Faq$\"3!oMeyM CrH*F_o7$$\"3#******H,M^\\(Faq$\"3cH=_9]Kr\"*F_o7$$\"3S+++0#=bq(Faq$\" 3PfW5W_(Q!))F_o7$$\"3Y*****p?27\"zFaq$\"3#*4:!\\K\"z(Q)F_o7$$\"3d***** *pe()=!)Faq$\"37#)4R.yh%>)F_o7$$\"3a+++IXaE\")Faq$\"3gpMN\\8L[!)F_o7$$ \"3a+++!R$Ry\")Faq$\"3(\\Ws7MC2+)F_o7$$\"3a+++]ACI#)Faq$\"31[Od'=!QqzF _o7$$\"3)*******zm;c#)Faq$\"39GshQa0izF_o7$$\"3a+++564#G)Faq$\"3#zvLE2 =%ezF_o7$$\"3)*******Rb,3$)Faq$\"3ei;fkv]fzF_o7$$\"37+++l*RRL)Faq$\"37 q@Jg$>`'zF_o7$$\"3#)******4H'pQ)Faq$\"3Y*yxnVa;*zF_o7$$\"3&*******fe)* R%)Faq$\"3aE2o**HZO!)F_o7$$\"3m*****\\!)3I\\)Faq$\"3y\"f$QJw*z4)F_o7$$ \"3i*****HvJga)Faq$\"3e%Q)\\/mvt\")F_o7$$\"3s*****HJnjv)Faq$\"3M%Ga*y! Q*[&)F_o7$$\"3g******zb+`))Faq$\"3W%fX&f0s:()F_o7$$\"3k******[Qk\\*)Fa q$\"3wiJPk2!*)F_o7$$\"3M+++5ASg! *Faq$\"3KuF65k(4v$*)F_o7$$\"3Y*****\\;C-F*Faq$ \"39qh(=Xp2())F_o7$$\"3[*****\\w(Gp$*Faq$\"3!HJ%y[P*=v)F_o7$$\"3;+++?- \"\\Z*Faq$\"3iD()pBoC'f)F_o7$$\"37+++!oK0e*Faq$\"3?/SBA*H-X)F_o7$$\"3= +++l(z5j*Faq$\"3s\"[\")*\\:Y&R)F_o7$$\"3C+++]oi\"o*Faq$\"3#4h#R7b!\\N) F_o7$$\"3G+++NR " 0 "" {MPLTEXT 1 0 33 "read \"C:\\\\Maple /procdrs/roots.m\";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "findmax(ssm,u=0.64..0.75);\nfindmax(ssm,u =0.48..0.58);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+c&zi6(!#5$\"+/E aS%*!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+xD;p_!#5$\"+%=.-W*!#; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "abreviated calculation for stages 15 to 21" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6681 "e5 := \{a[ 10,2] = 0, a[11,2] = 0, a[11,3] = 0, a[12,7] = 16571215593198468021712 83690913610698586256573484808662625/1343148041125514647725915510495609 3505361644432088109056, a[12,8] = 345685379554677052215495825476969226 377187500/74771167436930077221667203179551347546362089, a[14,4] = 0, a [14,5] = 0, a[11,10] = -31155237437111730665923206875/3928621415942305 15010338956291, a[14,2] = 0, a[14,3] = 0, a[9,1] = -147251426448621580 3881384708877264246346044433307094207829051978044531801133057155/12468 9480162003200115705962164398602480330155839348790044045363616804606968 6436608, c[4] = 1023/6400, c[11] = 47/50, c[12] = 1, c[13] = 1, a[11,1 ] = -2342659845814086836951207140065609179073838476242943917/135848096 1351056777022231400139158760857532162795520000, c[2] = 1/20, c[3] = 34 1/3200, a[7,2] = 0, a[7,3] = 0, c[8] = 943/1000, c[9] = 7067558016280/ 7837150160667, c[10] = 909/1000, a[9,2] = 0, a[9,3] = 0, c[5] = 39/100 , c[6] = 93/200, c[7] = 31/200, a[5,3] = -3899844/2907025, a[5,4] = 39 82992/2907025, a[8,1] = -1221101821869329/690812928000000, a[8,2] = 0, a[8,6] = 1501408353528689/265697280000000, a[8,7] = 6070139212132283/ 92502016000000, a[2,1] = 1/20, a[3,1] = -7161/1024000, a[7,6] = 5611/2 83500, a[6,1] = 5611/114400, a[6,2] = 0, a[4,2] = 0, a[4,3] = 3069/256 00, a[3,2] = 116281/1024000, a[4,1] = 1023/25600, a[5,1] = 4202367/116 28100, a[5,2] = 0, a[6,5] = 923521/5106400, a[7,1] = 21173/343200, a[9 ,4] = -517229431108566845837517565524698123003902533693369911413831527 0772319372469280000/12461938100480914589727863057121529836525707941023 6252921850936749076487132995191, a[9,5] = -120706792584692548079789364 41733187949484571516120469966534514296406891652614970375/2722031154761 657221710478184531100699497284085048389015085076961673446140398628096, a[8,3] = 0, a[8,4] = -125/2, a[6,3] = 0, a[6,4] = 31744/135025, a[8,5 ] = -1024030607959889/168929280000000, a[9,6] = 7801251558438936413230 90552530431036567795592568497182701460674803126770111481625/1831104254 1273197219788987450715878685922610298086185950524144307362914310080537 6, a[7,4] = 8602624/76559175, a[7,5] = -26782109/689364000, a[10,8] = \+ 212083202434519082281842245535894/20022426044775672563822865371173879, a[10,6] = 21127670214172802870128286992003940810655221489/46794738779 97892906145822697976708633673728000, a[10,9] = -2698404929400842518721 166485087129798562269848229517793703413951226714583/469545674913934315 077000442080871141884676035902717550325616728175875000000, a[13,11] = \+ 282035543183190840068750/12295407629873040425991, a[13,12] = -30681427 2936976936753/1299331183183744997286, a[11,4] = -996286030132538159613 930889652/16353068885996164905464325675, a[13,1] = 4490186773775461685 1973/1014046409980231013380680, a[10,5] = -180269259803172281163724663 224981097/38100922558256871086579832832000000, a[10,4] = -204627495245 91049105403365239069/454251913499893469596231268750, a[13,5] = 0, a[13 ,6] = 791638675191615279648100000/2235604725089973126411512319, a[12,5 ] = -4583493974484572912949314673356033540575/451957703655250747157313 034270335135744, a[12,6] = 2346305388553404258656258473446184419154740 172519949575/256726716407895402892744978301151486254183185289662464, a [12,10] = 40279545832706233433100438588458933210937500/889646084279948 2846916972126377338947215101, a[12,4] = -16957088714171468676387054358 954754000/143690415119654683326368228101570221, a[10,7] = 318607235173 649312405151265849660869927653414425413/671471671555896530313293807293 5465423910912000000, a[12,9] = -32058909627170725427914343121527275340 08102774023210240571361570757249056167015230160352087048674542196011/9 4756954968396581478301512445127360498465774712725761537244920597319265 7306017239103491074738324033259120, a[12,1] = -28665569918256639717782 95329101033887534912787724034363/8682267116192627030112139250161436120 30669233795338240, a[12,2] = 0, a[12,3] = 0, a[13,3] = 0, a[13,4] = 0, a[13,2] = 0, a[13,7] = 3847749490868980348119500000/15517045062138271 618141237517, a[13,8] = -13734512432397741476562500000/875132892924995 907746928783, a[13,9] = 1227476547031319687842881203774063505031923427 6006986398294443554969616342274215316330684448207141/48934514749371551 7650385834143510934888829280686609654482896526796523353052166757299452 852166040, a[13,10] = -9798363684577739445312500000/308722986341456031 822630699, a[12,11] = -6122933601070769591613093993993358877250/105051 7001510235513198246721302027675953, a[11,9] = 300760669768102517834232 497565452434946672266195876496371874262392684852243925359864884962513/ 4655443337501346455585065336604505603760824779615521285751892810315680 492364106674524398280000, a[14,1] = 307/6800, a[11,7] = 89072299375637 9186418929622095833835264322635782294899/13921242001395112657501941955 594013822830119803764736, a[11,8] = 1610214261431241783890751219292467 10833125/10997207722131034650667041364346422894371443, a[11,6] = 20980 822345096760292224086794978105312644533925634933539/377588999200755080 3878727839115494641972212962174156800, a[11,5] = -26053085959256534152 588089363841/4377552804565683061011299942400, a[10,3] = 0, a[9,7] = 66 4113122959911642134782135839106469928140328160577035357155340392950009 492511875/151784655985862481363330231072953491752797651500890783011399 43253016877823170816, a[9,8] = 103328481844520156040568367672866568591 24007796970668046446015775000000/1312703550036033648073834248740727914 537972028638950165249582733679393783, a[10,1] = -290555733603374150885 38618442231036441314060511/2267475989108957769132796260237059763200000 0000, a[14,7] = 35514316969207250641724424985/147161653169956511475274 962258-481702304929173090974271575/294323306339913022950549924516*7^(1 /2), a[14,8] = 671994922390044434145634375/90947002658165004198514338- 463454429627970722580265625/181894005316330008397028676*7^(1/2), a[14, 6] = 460454863031265521343678403/2355798527514165229982023734-28343885 9697640884437065685/4711597055028330459964047468*7^(1/2), a[14,10] = 1 88753532739715767256965625/33283666294238384068886478-1682499744109518 46618609375/66567332588476768137772956*7^(1/2), a[14,9] = -15204175302 5355398038004360208592588649426234538717842799798205718781444204227342 3080071783/39219298194348039763366422255699207510662856884730463139019 1589139262501961993074284749200+79993028921817789891293832589466706896 4649008561582353557148147567546469052859876678259/46140350816880046780 4310850067049500125445375114476036929637163693250002308227146217352*7^ (1/2), a[14,11] = -236307906973303345482653/25637232930373573654194+17 3345833072833817590685/51274465860747147308388*7^(1/2), c[14] = 1/2-1/ 14*7^(1/2), a[14,12] = 11352128098668146659861/25466891190401401946805 6-5215842639928607924801/127334455952007009734028*7^(1/2), a[14,13] = \+ 3/392-3/392*7^(1/2)\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "set up order relations etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "SO7 _14 := SimpleOrderConditions(7,14,'expanded'):\nSO7_15 := SimpleOrderC onditions(7,15,'expanded'):\nSO7_16 := SimpleOrderConditions(7,16,'exp anded'):\nSO7_17 := SimpleOrderConditions(7,17,'expanded'):\nSO7_18 := SimpleOrderConditions(7,18,'expanded'):\nSO7_19 := SimpleOrderConditi ons(7,19,'expanded'):\nSO7_20 := SimpleOrderConditions(7,20,'expanded' ):\nSO8_21 := SimpleOrderConditions(8,21,'expanded'):\nerrterms8_21 := PrincipalErrorTerms(8,21,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1799 "whch := [1,2,4,8,16,2 1,27,31,32,64]:\ninterp_order_eqns15 := []:\nfor ct in whch do\n tem p_eqn := convert(SO7_14[ct],'interpolation_order_condition'):\n inte rp_order_eqns15 := [op(interp_order_eqns15),temp_eqn];\nend do:\nwhch \+ := [1,2,4,8,16,21,27,31,32,63,64]:\ninterp_order_eqns16 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_15[ct],'interpolation_order_co ndition'):\n interp_order_eqns16 := [op(interp_order_eqns16),temp_eq n];\nend do:\nwhch := [1,2,4,8,16,17,25,27,32,61,63,64]:\ninterp_order _eqns17 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_16[ct],' interpolation_order_condition'):\n interp_order_eqns17 := [op(interp _order_eqns17),temp_eqn];\nend do:\nwhch := [1,2,4,8,16,17,25,27,32,61 ,63,64]:\ninterp_order_eqns18 := []:\nfor ct in whch do\n temp_eqn : = convert(SO7_17[ct],'interpolation_order_condition'):\n interp_orde r_eqns18 := [op(interp_order_eqns18),temp_eqn];\nend do:\nwhch := [1,2 ,4,8,16,17,25,27,32,61,63,64]:\ninterp_order_eqns19 := []:\nfor ct in \+ whch do\n temp_eqn := convert(SO7_18[ct],'interpolation_order_condit ion'):\n interp_order_eqns19 := [op(interp_order_eqns19),temp_eqn]; \nend do:\nwhch := [1,2,4,8,16,17,25,27,32,61,63,64]:\ninterp_order_eq ns20 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_19[ct],'int erpolation_order_condition'):\n interp_order_eqns20 := [op(interp_or der_eqns20),temp_eqn];\nend do:\nwhch := [1,2,4,8,16,21,27,31,32,61,63 ,64]:\ninterp_order_eqns21 := []:\nfor ct in whch do\n temp_eqn := c onvert(SO7_20[ct],'interpolation_order_condition'):\n interp_order_e qns21 := [op(interp_order_eqns21),temp_eqn];\nend do:\nwhch := [1,2,4, 8,16,21,27,31,32,38,48,54,63,64,102,117,121,123,125,127,128]:\nordeqns := []:\nfor ct in whch do\n eqn_group := convert(SO8_21[ct],'polyno m_order_conditions',8):\n ordeqns := [op(ordeqns),op(eqn_group)];\ne nd do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1960 "calc_coeffs := proc()\n local eqns,pols ,e_u,eu,ct,eqs_15,eqs_16,eqs_17,eqs_18,eqs_19,eqs_20,eqs_21;\n globa l dd,sm,e5,e6,e7,e8,e9,e10,e11,e12,e13,e14,e15,e16,e17,e18,e19,e20,e21 ,\n e22,e23,e24,e25,e26;\n\n e6 := `union`(e5,\{c[15]=c_15, seq(a[15,i]=0,i=2..5)\}):\n eqs_15 := expand(subs(e6,interp_order_eq ns15)):\n e7 := solve(\{op(eqs_15)\}):\n e8 := `union`(e6,e7):\n \+ e9 := `union`(e8,\{c[16]=c_16,seq(a[16,i]=0,i=2..5)\}):\n eqs_16 := expand(subs(e9,interp_order_eqns16)):\n e10 := solve(\{op(eqs_16)\} ):\n e11 := `union`(e9,e10):\n e12 := `union`(e11,\{c[17]=c_17,seq (a[17,i]=0,i=2..5)\}):\n eqs_17 := expand(subs(e12,interp_order_eqns 17)):\n e13 := solve(\{op(eqs_17)\}):\n e14 := `union`(e12,e13):\n e15 := `union`(e14,\{c[18]=c_18,seq(a[18,i]=0,i=2..5),a[18,17]=0\}) :\n eqs_18 := expand(subs(e15,interp_order_eqns18)):\n e16 := solv e(\{op(eqs_18)\}):\n e17 := `union`(e15,e16):\n e18 := `union`(e17 ,\{c[19]=c_19,seq(a[19,i]=0,i=2..5),a[19,17]=0,a[19,18]=0\}):\n eqs_ 19 := expand(subs(e18,interp_order_eqns19)):\n e19 := solve(\{op(eqs _19)\}):\n e20 := `union`(e18,e19):\n e21 := `union`(e20,\{c[20]=c _20,seq(a[20,i]=0,i=2..5),seq(a[20,i]=0,i=17..19)\}):\n eqs_20 := ex pand(subs(e21,interp_order_eqns20)):\n e22 := solve(\{op(eqs_20)\}): \n e23 := `union`(e21,e22):\n e24 := `union`(e23,\{c[21]=c_21,seq( a[21,i]=0,i=2..5),seq(a[21,i]=0,i=17..20)\}):\n eqs_21 := expand(sub s(e24,interp_order_eqns21)):\n e25 := solve(\{op(eqs_21)\}):\n e26 := `union`(e24,e25):\n eqns := []:\n for ct to nops(ordeqns) do\n eqns := [op(eqns),expand(subs(e26,ordeqns[ct]))];\n end do:\n \+ dd := solve(\{op(eqns)\}):\n e_u := map(_U->lhs(_U)=rhs(_U)/u,e26) :\n pols := [seq(b[j]=add(simplify(subs(dd,d[j,i]))*u^(i-1),i=1..8), j=1..21)]:\n eu := `union`(e_u,\{op(pols)\}):\n sm := 0:\n for c t to 286 do\n sm := sm+expand(subs(eu,errterms8_21[ct]))^2;\n e nd do:\n return(c[15]=c_15,c[16]=c_16,c[17]=c_17,c[18]=c_18,c[19]=c_ 19,c[20]=c_20,c[21]=c_21);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 17 "Sample comparison" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "c_15 := 456/1000:\nc_16 := 112/1000:\nc_17 := 373/1000:\nc_18 : = 450/1000:\nc_19 := 700/1000:\nc_20 := 900/1000:\nc_21 := 97/1000:\nc alc_coeffs();\nssmA := sqrt(sm)*u^9:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6)/&%\"cG6#\"#:#\"#d\"$D\"/&F%6#\"#;#\"#9F*/&F%6#\"#<#\"$t$\"%+5/&F%6# \"#=#\"\"*\"#?/&F%6#\"#>#\"\"(\"#5/&F%6#F>#F=FE/&F%6#\"#@#\"#(*F7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "c_15 := 4560/10000:\nc_16 := 1156/10000:\nc_17 := 3737/10000:\nc_ 18 := 4500/10000:\nc_19 := 7082/10000:\nc_20 := 9000/10000:\nc_21 := 9 70/10000:\ncalc_coeffs();\nssmB := sqrt(sm)*u^9:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"#:#\"#d\"$D\"/&F%6#\"#;#\"$*G\"%+D/&F%6#\"#< #\"%PP\"&++\"/&F%6#\"#=#\"\"*\"#?/&F%6#\"#>#\"%TN\"%+]/&F%6#F?#F>\"#5/ &F%6#\"#@#\"#(*\"%+5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "c_15 := 4560/10000:\nc_16 := 1163/10000: \nc_17 := 3728/10000:\nc_18 := 5200/10000:\nc_19 := 7080/10000:\nc_20 \+ := 8996/10000:\nc_21 := 979/10000:\ncalc_coeffs();\nssmC := sqrt(sm)*u ^9:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"#:#\"#d\"$D\"/&F%6# \"#;#\"%j6\"&++\"/&F%6#\"#<#\"$L#\"$D'/&F%6#\"#=#\"#8\"#D/&F%6#\"#>#\" $x\"\"$]#/&F%6#\"#?#\"%\\A\"%+D/&F%6#\"#@#\"$z*F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "plot([ssmA,s smB,ssmC],u=0..1,color=[COLOR(RGB,.4,0,1),COLOR(RGB,0,.65,0),brown]); " }}{PARA 13 "" 1 "" {GLPLOT2D 505 278 278 {PLOTDATA 2 "6'-%'CURVESG6$ 7er7$$\"3`*****\\n5;\"o!#@$\"3Kr,sM*fA)p!#G7$$\"3#******\\8ABO\"!#?$\" 3Ea=:Re@lF!#F7$$\"33+++-K[V?F1$\"3'ecT)=J&*fhF47$$\"3#)******pUkCFF1$ \"3Y$\"3eW))=USjO:!#D7$$\"3*)*****>c'yM;FR$\"3#\\aW2^(f% =$FU7$$\"3')*****fT:(z@FR$\"3oo%))pX)**3_FU7$$\"3#*******zZ*z7$FR$\"3_ 7'>2#G%\\D*FU7$$\"33+++XTFwSFR$\"35R5RDBQ^8!#C7$$\"3=+++qMrU^FR$\"3%ep iW'eq2=Fdo7$$\"3&******4z_\"4iFR$\"3%oVg!*HBc?#Fdo7$$\"3y*****\\;hEG(F 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C" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 34 "nodes and linking coefficients: ee" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29180 "ee := \{c [2] = 1/20, c[3] = 341/3200, c[4] = 1023/6400, c[5] = 39/100, c[6] = 9 3/200, c[7] = 31/200, c[8] = 943/1000, c[9] = 7067558016280/7837150160 667, c[10] = 909/1000, c[11] = 47/50, c[12] = 1, c[13] = 1, c[14] = 1/ 2-1/14*7^(1/2), c[15] = 57/125, c[16] = 1163/10000, c[17] = 233/625, c [18] = 13/25, c[19] = 177/250, c[20] = 2249/2500, c[21] = 979/10000,\n a[2,1] = 1/20, a[3,1] = -7161/1024000, a[3,2] = 116281/1024000, a[4,1] = 1023/25600, a[4,2] = 0, a[4,3] = 3069/25600, a[5,1] = 4202367/11628 100, a[5,2] = 0, a[5,3] = -3899844/2907025, a[5,4] = 3982992/2907025, \+ a[6,1] = 5611/114400, a[6,2] = 0, a[6,3] = 0, a[6,4] = 31744/135025, a [6,5] = 923521/5106400, a[7,1] = 21173/343200, a[7,2] = 0, a[7,3] = 0, a[7,4] = 8602624/76559175, a[7,5] = -26782109/689364000, a[7,6] = 561 1/283500, a[8,1] = -1221101821869329/690812928000000, a[8,2] = 0, a[8, 3] = 0, a[8,4] = -125/2, a[8,5] = -1024030607959889/168929280000000, a [8,6] = 1501408353528689/265697280000000, a[8,7] = 6070139212132283/92 502016000000, a[9,1] = -1472514264486215803881384708877264246346044433 307094207829051978044531801133057155/124689480162003200115705962164398 6024803301558393487900440453636168046069686436608, a[9,2] = 0, a[9,3] \+ = 0, a[9,4] = -5172294311085668458375175655246981230039025336933699114 138315270772319372469280000/124619381004809145897278630571215298365257 079410236252921850936749076487132995191, a[9,5] = -1207067925846925480 7978936441733187949484571516120469966534514296406891652614970375/27220 3115476165722171047818453110069949728408504838901508507696167344614039 8628096, a[9,6] = 7801251558438936413230905525304310365677955925684971 82701460674803126770111481625/1831104254127319721978898745071587868592 26102980861859505241443073629143100805376, a[9,7] = 664113122959911642 134782135839106469928140328160577035357155340392950009492511875/151784 6559858624813633302310729534917527976515008907830113994325301687782317 0816, a[9,8] = 1033284818445201560405683676728665685912400779697066804 6446015775000000/13127035500360336480738342487407279145379720286389501 65249582733679393783, a[10,1] = -2905557336033741508853861844223103644 1314060511/22674759891089577691327962602370597632000000000, a[10,2] = \+ 0, a[10,3] = 0, a[10,4] = -20462749524591049105403365239069/4542519134 99893469596231268750, a[10,5] = -180269259803172281163724663224981097/ 38100922558256871086579832832000000, a[10,6] = 21127670214172802870128 286992003940810655221489/467947387799789290614582269797670863367372800 0, a[10,7] = 318607235173649312405151265849660869927653414425413/67147 16715558965303132938072935465423910912000000, a[10,8] = 21208320243451 9082281842245535894/20022426044775672563822865371173879, a[10,9] = -26 9840492940084251872116648508712979856226984822951779370341395122671458 3/46954567491393431507700044208087114188467603590271755032561672817587 5000000, a[11,1] = -23426598458140868369512071400656091790738384762429 43917/1358480961351056777022231400139158760857532162795520000, a[11,2] = 0, a[11,3] = 0, a[11,4] = -996286030132538159613930889652/163530688 85996164905464325675, a[11,5] = -26053085959256534152588089363841/4377 552804565683061011299942400, a[11,6] = 2098082234509676029222408679497 8105312644533925634933539/37758899920075508038787278391154946419722129 62174156800, a[11,7] = 89072299375637918641892962209583383526432263578 2294899/13921242001395112657501941955594013822830119803764736, a[11,8] = 161021426143124178389075121929246710833125/109972077221310346506670 41364346422894371443, a[11,9] = 30076066976810251783423249756545243494 6672266195876496371874262392684852243925359864884962513/46554433375013 4645558506533660450560376082477961552128575189281031568049236410667452 4398280000, a[11,10] = -31155237437111730665923206875/3928621415942305 15010338956291, a[12,1] = -2866556991825663971778295329101033887534912 787724034363/868226711619262703011213925016143612030669233795338240, a [12,2] = 0, a[12,3] = 0, a[12,4] = -1695708871417146867638705435895475 4000/143690415119654683326368228101570221, a[12,5] = -4583493974484572 912949314673356033540575/451957703655250747157313034270335135744, a[12 ,6] = 2346305388553404258656258473446184419154740172519949575/25672671 6407895402892744978301151486254183185289662464, a[12,7] = 165712155931 9846802171283690913610698586256573484808662625/13431480411255146477259 155104956093505361644432088109056, a[12,8] = 3456853795546770522154958 25476969226377187500/74771167436930077221667203179551347546362089, a[1 2,9] = -32058909627170725427914343121527275340081027740232102405713615 70757249056167015230160352087048674542196011/9475695496839658147830151 2445127360498465774712725761537244920597319265730601723910349107473832 4033259120, a[12,10] = 40279545832706233433100438588458933210937500/88 96460842799482846916972126377338947215101, a[12,11] = -612293360107076 9591613093993993358877250/1050517001510235513198246721302027675953, a[ 13,1] = 44901867737754616851973/1014046409980231013380680, a[13,2] = 0 , a[13,3] = 0, a[13,4] = 0, a[13,5] = 0, a[13,6] = 7916386751916152796 48100000/2235604725089973126411512319, a[13,7] = 384774949086898034811 9500000/15517045062138271618141237517, a[13,8] = -13734512432397741476 562500000/875132892924995907746928783, a[13,9] = 122747654703131968784 2881203774063505031923427600698639829444355496961634227421531633068444 8207141/48934514749371551765038583414351093488882928068660965448289652 6796523353052166757299452852166040, a[13,10] = -9798363684577739445312 500000/308722986341456031822630699, a[13,11] = 28203554318319084006875 0/12295407629873040425991, a[13,12] = -306814272936976936753/129933118 3183744997286, a[14,1] = 307/6800, a[14,2] = 0, a[14,3] = 0, a[14,4] = 0, a[14,5] = 0, a[14,6] = 460454863031265521343678403/235579852751416 5229982023734-283438859697640884437065685/4711597055028330459964047468 *7^(1/2), a[14,7] = 35514316969207250641724424985/14716165316995651147 5274962258-481702304929173090974271575/294323306339913022950549924516* 7^(1/2), a[14,8] = 671994922390044434145634375/90947002658165004198514 338-463454429627970722580265625/181894005316330008397028676*7^(1/2), a [14,9] = -152041753025355398038004360208592588649426234538717842799798 2057187814442042273423080071783/39219298194348039763366422255699207510 6628568847304631390191589139262501961993074284749200+79993028921817789 8912938325894667068964649008561582353557148147567546469052859876678259 /461403508168800467804310850067049500125445375114476036929637163693250 002308227146217352*7^(1/2), a[14,10] = 188753532739715767256965625/332 83666294238384068886478-168249974410951846618609375/665673325884767681 37772956*7^(1/2), a[14,11] = -236307906973303345482653/256372329303735 73654194+173345833072833817590685/51274465860747147308388*7^(1/2), a[1 4,12] = 11352128098668146659861/254668911904014019468056-5215842639928 607924801/127334455952007009734028*7^(1/2), a[14,13] = 3/392-3/392*7^( 1/2), a[15,1] = 57196651428918572875631366996149/124582288804136391531 3720703125000+10891702347425878099023621466959/83573952072774829318962 09716796875*7^(1/2), a[15,2] = 0, a[15,3] = 0, a[15,4] = 0, a[15,5] = \+ 0, a[15,6] = -245534888302358312351817429197184/8700455320214002619298 187255859375*7^(1/2)+4116869726480787181292612488487584/26598534836082 808007568743896484375, a[15,7] = 631476617959696547824260385843267488/ 2705957914016858192337687904052734375-33227466731347290455944055132270 208/3922838396059942349838659979248046875*7^(1/2), a[15,8] = 598933003 512786415989929347488/28972691439853083012428515625-147510965404401941 35375671464064/1371942153475395989706173828125*7^(1/2), a[15,9] = -306 7066459661270994000493048546286135407493899133250841529092716875641340 142781006771180215525814345889446383/373340719218227781410511653246697 1854315408940785290943015262808201014351289114054103491608932800292968 75000+1816647611984076940241842059764388009599469326146254865545587094 445401874686830164148895260773137/424475654667905797096751747915993683 544031508184382874685996435033916010685878289836883544921875*7^(1/2), \+ a[15,10] = 54365264025507653851977480738208/44664783903567134233598191 40625-1496294528311131416682934929792/235786943683989396406626953125*7 ^(1/2), a[15,11] = -3598066799092816058284144620546/146286964102678984 417724609375+65570869112872951387443794088672/512004374359376445462036 1328125*7^(1/2), a[15,12] = -1927555453883917676542797699/276054308640 062825012207031250, a[15,13] = -15630718068/3814697265625+6167095848/3 814697265625*7^(1/2), a[15,14] = 172678683744/3814697265625*7^(1/2), a [16,1] = 5712623659747225921475087020856744131230749797119548738128074 64116257302576802226313028164589259868449219745217/1107721982975045860 7735275253790319251133907365668324124318115180733432062987800250708692 452000000000000000000000-418059397241678150958395224952493598697047777 024498277797593332867900350838856267902560305003793843931777113/521280 9331647274638934247178254267882886544642667446646737936555639262147288 37658856879644800000000000000000000*7^(1/2), a[16,2] = 0, a[16,3] = 0, a[16,4] = 0, a[16,5] = 0, a[16,6] = -99745218788660346650012150937989 0827338036196516536584568174856148035041553855606903935543603360920655 84815633/1209721562634961718762421588147194853076119483609301088106017 767355971757142551992626941965147200000000000000000+933207335600402394 9872144510543614659208037023878717879461317781877845670721918179975766 545310520465203275977/241944312526992343752484317629438970615223896721 860217621203553471194351428510398525388393029440000000000000000*7^(1/2 ), a[16,7] = 125183725987006091358284807772832081560458336415582349576 40052376606698149159914801577172298098152775739801047/1291772597666133 9897426511964385702783984431335430335690360713394329797397731959853551 5764891840000000000000000+17615228617354110108508911790502650111326496 84947212272396211479818027022186347979194359016936089217819331021/3358 6087539319483733308931107402827238359521472118872794937854825257473234 1030956192340988718784000000000000000*7^(1/2), a[16,8] = -547598389828 8618345981813338837590800614185855015399068684697834886314303975930600 3458396414251841877395553/30307100180597417208049096613010446805012644 086401478209258240042904260490962503761623721182105600000000000+132843 6178798707517398824622334536072945626857973036589140725177311419165039 7356739750286947316947113707/35446900796020371003566194869018066438611 279633218103168723087769478667240891817265056983838720000000000*7^(1/2 ), a[16,9] = 687436148767348000628593173052742777806507835190320258289 7930008852348898811178320163446670777697657027098059541922969630215528 95479234993257819821106998915574193829101860119823330279/1125368148431 5005091367654234973730440759771083418475821709212648100184681424006314 2590082692987452847804771712027033044260612410255741832841968960869322 2186318096000000000000000000000+29632327319879523565177723192588781912 3825559810998878941673611936624482258079779521994711237719287068353386 053281123249786227048915757888049930193669040074737641797319943395161/ 9456875196903365622998028768885487765344345448250820018243035838739650 9927932826170243766968897019199835942615148767264084548243912388094825 1840007305228727998400000000000000000000*7^(1/2), a[16,10] = -10342633 3826764901372662636707035364194458681227405932428768403579719550438767 7241159024633249191823996323/82242440862377718714678315381839934519771 3720767960756194595910492784789119961559213755380633600000000000+27426 3425446302069944764427449918197731592282141443255124856601709917334353 644421798295950196247775939471/213830346242182068658163619992783829751 4055673996697966105949367281240451711900053955763989647360000000000*7^ (1/2), a[16,11] = 6623909106040336836026262070216648184024202126970105 9559317336313240648392339911854838860941565539670913883/26612968856818 3402539243009457627860959737366888672888383931360688306743150502791672 58323200000000000000000-9009365467871791955400608463323747331764760668 77608935979837245786333301828102316788089290072092611304509/1774197923 7878893502616200630508524063982491125911525892262090712553782876700186 11150554880000000000000000*7^(1/2), a[16,12] = -1700668024857780947303 8573186499560848605733156506811085532643136411973839102899042067514970 96329440010629/2704188218517981979215772760946666451031013390150181735 5481149039230552258240217568756800000000000000000000+18110923375969700 2284558469350564784363590987534352651033927139760232659628159467717368 72563829234342887/1233489362832763709817720908501988205733444704279030 265337736622842095366165343257522240000000000000000000*7^(1/2), a[16,1 3] = -5693779785978540937563109870700881399852754029464825718471345600 991242218048279830357689/662326025726932385387063527523771926169299549 864595912922410705613811200000000000000000000+111605897813329022747364 3648991308872297960011076080647100654462877250873303116055359689/66232 6025726932385387063527523771926169299549864595912922410705613811200000 000000000000000*7^(1/2), a[16,14] = 3661136425778970321788839592892634 02896100850925956136650174323553270486082847776282814967/1799237869153 0527189797839723798348391415273433637864708028281999192834400000000000 000000000-102347802750307940314868234213850993510794398047130500007939 6396912342658754866704389719/28787805906448843503676543558077357426264 437493820583532845251198708535040000000000000000*7^(1/2), a[16,15] = 1 2566723979477860207267692556004160956197259758320997925831484548766743 3090419263759799/19043248758339535773056048269842404246903630459912291 28931126504895049076310016000000000-5902894673902568174262628544233112 6491223596724290616786809339665195402731038729879/30469198013343257236 88967723174784679504580873585966606289802407832078522096025600000*7^(1 /2), a[17,1] = 1410152404293500301009654175048187932181637126808924995 035277613/331719790994854166591924243196409241558674198229255676269531 25000-343669929529154146426950913103012707061848467742113726473984/118 471353926733630925687229713003300556669356510448455810546875*7^(1/2), \+ a[17,2] = 0, a[17,3] = 0, a[17,4] = 0, a[17,5] = 0, a[17,6] = 14431750 4741273108436770639890093130241213191928121002485536/10590713818207821 15429096951498359519812365804784881591796875-3613865043508937346127833 5139553986413666973704247679234048/75647955844341579673506925107025679 9865975574846343994140625*7^(1/2), a[17,7] = 2415214739645704299904758 82210375964811719239806626681198304/1470175668471334233788212618705488 919434060375826495361328125-207340190374720440119877707325483615476099 7520307226750976/36211223361362912162271246766145047276700994478485107 421875*7^(1/2), a[17,8] = 36096296070313430470128372309159018445786274 51417318752/5306573588278130800904459886236721969496006556752734375-68 445701579780131347383082192778657908542785254600704/332492079466048295 796018789864456263752882616337890625*7^(1/2), a[17,9] = -1170105558265 9954072431612713805800779875665736054727965423788667287696781753047133 2239610851127136258213798655554999195607667534787/72442848981639410882 0303136848490351295228985814524607088372322520512385396296447478468581 426097365269115793412394100189208984375000+232734140517153928479992141 7004174552247521703277357365728574443247365083894628816391102044670716 8311579553151269834892177196544/25872446064871218172153683458874655403 4010352076615931102990115185897280498677302670881636223606201881827069 0758550357818603515625*7^(1/2), a[17,10] = 765660571360590504522410811 080563711090654609706246624/187201424910254107980463739378295055509674 0829301171875-18934143677045047338847621384679376883588265306034176/23 0543626736766142833083422879673713681864634150390625*7^(1/2), a[17,11] = -5431825607575415643071521478034676792363131575507490074/5824694411 529015933948896260840005226582958953857421875+430788284289375809691955 23720831420116933769072147456/1540924447494448659774840280645504028196 54998779296875*7^(1/2), a[17,12] = 42877082723345225226946300758681835 657160444961937467671/192353501741270938620827863159877637283607997894 2871093750-3502578350717681024540012102744371580394344491444736/482088 976795165259701322965312976534545383453369140625*7^(1/2), a[17,13] = 5 752220750522348143924684511521705800928/200159644656989805483910890855 7891845703125-1698861030059055060173707215296857930816/200159644656989 8054839108908557891845703125*7^(1/2), a[17,14] = 287666940234183647228 741967485029371867251084288/768467909630489872386593335682960386276245 1171875+572017064437929229652116249893368639587841024/1229548655408783 7958185493370927366180419921875*7^(1/2), a[17,15] = -18021943300915379 5201274490416832044056/2212261355670987711698010267367693359375+882724 55565304907970131443124874176/3539618169073580338716816427788309375*7^ (1/2), a[17,16] = 1413383347739698940972600039030481649401856/26439766 035500952469299665052864808467693125+186175722473779987485444725092240 0645120/42303625656801523950879464084583693548309*7^(1/2), a[18,1] = 1 47937299499519182239019177721288159839355546541856055637/3396810659787 306665901304250331230633560823789867578125000-462234303531538433172120 667936726432477695301843136/149771193112315108725807065711253555271641 260576171875*7^(1/2), a[18,2] = 0, a[18,3] = 0, a[18,4] = 0, a[18,5] = 0, a[18,6] = 16283256908017943404871165834200503438819098307881504/10 8448909498448088619939527833432014828786258409971875-14581890190962596 0229141275905926424005344756683776/28690187698002139846544848633183072 70602811069046875*7^(1/2), a[18,7] = 161373973469595107260482187200001 399198950768983264/890804665393281807928479125180130564201466168548125 -1336603043701021865325398968333701077347089079296/2194100161067196571 2524116383747058231563206121875*7^(1/2), a[18,8] = 1037096243657307700 8766029273604620733739288898400/13584828385992014850315417308766008241 909776785287-48737181047905230621987375887798081383160832/222530646649 172192741910614915819094171863921*7^(1/2), a[18,9] = -1471082580019238 3440364935594012619375541854672955854090580175358951645433099691448186 629741883086125225426868034484205191963/741814773571987567431990412132 8541197263144814740731976584932582610046826458075622179518273803237020 3557457245429155859375000+25355124852747346663348168643778819286288841 2819293403381348833600438230448512838997656884411077459592513063331737 0691456/26493384770428127408285371861887647133082660052645471344946187 7950358815230645557934982795492972750726990918733675556640625*7^(1/2), a[18,10] = 13521134712758181369289831786617583384500556000/2835713892 1316598605324684781564221426317494219-88640190526698364463534356898040 66485248/101445417221405134333310741139852901561249*7^(1/2), a[18,11] \+ = -624931687431641426621118270642785259964248638186/596448707740571231 636366977110016535202094996875+469320517779937528294963288466654596611 7964544/15779066342343154276094364473809961248732671875*7^(1/2), a[18, 12] = 28899294106750011031701193669610820543459670391/1165502874455984 858862294271453933139517246093750-250878995631377260872470039095670096 659584/32456220396992059561745872220939380103515625*7^(1/2), a[18,13] \+ = 73226019221752589487279696352758/22773719569861951201724972470703125 -1209684402178421037903496517328/1339630562933055953042645439453125*7^ (1/2), a[18,14] = 386910568376965091074907779867460087552/971495233903 2365794121871305424091796875+769360036098797882353443381891543296/1554 3923742451785270594994088678546875*7^(1/2), a[18,15] = 611266295965159 733830102317250/13983676717327724794930632801139+218246328696225082588 65350000/822569218666336752642978400067*7^(1/2), a[18,16] = 1667634263 017147072374514384896000000000/423036256568015239508794640845836935483 09+1980745822360913756216308531200000000000/42303625656801523950879464 084583693548309*7^(1/2), a[18,17] = 0, a[19,1] = 553052300452019173696 0195542348698154104110118263839314563/10661678153758024688955757220123 1344430659880410156250000000+53024609659096171207519665161701297338237 007144082879671/213233563075160493779115144402462688861319760820312500 00000*7^(1/2), a[19,2] = 0, a[19,3] = 0, a[19,4] = 0, a[19,5] = 0, a[1 9,6] = 31761135796090880216513925783838905738809726937453499027/100415 656943007489462906970216140754471098387416640625000+293725644528407742 5545167027958391452146185679178908559/71725469245005349616362121582957 681765070276726171875000*7^(1/2), a[19,7] = 14862346273040272908856037 28717958757029589614659889902501/3763649711286615638497824303886051633 751194562115828125000+245003269580962324930652577468219256034026679534 9154999/49915778664278721995992364773024557476806293927265625000*7^(1/ 2), a[19,8] = 1460901536507797911430143644112950654968542708040647/503 141792073778327789459900324666971922584325381000+116824897710122631181 617536954552563387143135344547/662028673781287273407184079374561805161 295164975000*7^(1/2), a[19,9] = -1814983937674617033258022888696175544 2773095473011921393932511802510032308429912713573788634113902929462361 463101791056402129653/785820734716088524822023741666159025133807713425 9249975195903159544541129722537735359659188350886674105662843795461425 781250000-484763386556274296331592496481324487047130253206912958531250 59645919927178833757198048236974746322271158408663982057250777761/6286 5658777287081985761899333292722010704617074073999801567225276356329037 78030188287727350680709339284530275036369140625000000*7^(1/2), a[19,10 ] = 610173301279104199162708838720280036930719997064817/17749468435935 2043122217471410531608186950241593000+82864839077015853666953023704615 2912694947771687/11770204533113530711022378740751432903643915225000*7^ (1/2), a[19,11] = -903188219396857319010022629019144489775609887379219 7/2209069287928041598653211026333394574822574062500000-264701019796536 3598888479193578185849983594257475863/11045346439640207993266055131666 972874112870312500000*7^(1/2), a[19,12] = 5401904083390091818349654582 6453513602367636600622753/72951846586319052276936196991005444658672070 3125000000+3678948629575286674251755089611875084920448820393/590703211 225255484023774874421096717883984375000000*7^(1/2), a[19,13] = 4678177 04897204714306167183851765640617/3643795131177912192275995595312500000 0000+26511177545934515439957752429061942951/36437951311779121922759955 953125000000000*7^(1/2), a[19,14] = -935233124784966293272491333475630 42523424791/2914485701709709738236561391627227539062500000-18596829589 4238188887694028978095988881993/46631771227355355811784982266035640625 00000*7^(1/2), a[19,15] = 165094365518528096370316638519506567/2684865 929726923160626681497818688000-459171703963343898329722995722387/21478 927437815385285013451982549504*7^(1/2), a[19,16] = -483926531222276520 2258896404987936768000/42303625656801523950879464084583693548309-15959 40816178932602370969394040832000000/4230362565680152395087946408458369 3548309*7^(1/2), a[19,17] = 0, a[19,18] = 0, a[20,1] = 811879574362736 7733179779787397474929191662340010737513325588701/14393265507573333330 0902722471662314981390838553710937500000000000+18895907275772561552080 776506607802183413383927634325825193723987/679362131957461333180260850 0662461267121647579735156250000000000000*7^(1/2), a[20,2] = 0, a[20,3] = 0, a[20,4] = 0, a[20,5] = 0, a[20,6] = 2460090719002992812624927147 73577475558316468008450336418220241/5422445474922404430996976391671600 74143931292049859375000000000+1774108280053402054260442512742289646430 6706025703635833408497/38731753392302888792835545654797148153137949432 1328125000000000*7^(1/2), a[20,7] = 1987652871866385232452907492438015 519716639253386233472801741/445402332696640903964239562590065282100733 0842740625000000000+42160234808367399850532562437361075865066296050829 346660567/767935056373518799938344073431147038104712214265625000000000 *7^(1/2), a[20,8] = -2054517767934921766758595484747352729404324966670 3692394259/2716965677198402970063083461753201648381955357057400000000+ 235208174935590378206735630533474656549656002037130641567/119165161280 6317092132931342874211249290331296955000000000*7^(1/2), a[20,9] = 1736 8916085717699350796251547302241376558283516531108642672846555922149224 4207404038233133006358996591512306005798646123369075131279979/12573131 7554574163971523798666585444021409234148147999603134450552712658075560 60376575454701361418678569060550072738281250000000000000-6398177962709 5593679210465068978025175590723058743214318565031910899102521800732807 5700825441576406371401356530654173520825641881471/74181477357198756743 1990412132854119726314481474073197658493258261004682645807562217951827 38032370203557457245429155859375000000000000*7^(1/2), a[20,10] = -9696 3920031053349908036115770893575920364297479381979101/56714277842633197 21064936956312844285263498843800000000+3850039598579009617468354260888 55092634524120381038717/4889161882985620449193911169235210590744395555 000000000*7^(1/2), a[20,11] = 1332138524364630358378398720583293834923 83939986258170076149/1192897415481142463272733954220033070404189993750 0000000000-592147783090421103173033593272783400624371424847208058827/2 209069287928041598653211026333394574822574062500000000000*7^(1/2), a[2 0,12] = -2127375361994947297156456662040945063812043101896442491879/23 310057489119697177245885429078662790344921875000000000000+569767107608 351000019229940102305750050165107337869521/817896754004199900955995979 96767237860859375000000000000*7^(1/2), a[20,13] = -1354173155655320081 817796503704371022429901163/655883123612024194609679207156250000000000 00000+53375999536033452652110517044084509032058611/6558831236120241946 0967920715625000000000000000*7^(1/2), a[20,14] = -56488252264291309359 0541724798848365028989502024553/15738222789232432586477431514787028710 937500000000000-1123251917969593604607379708431968416738765464119/2518 1156462771892138363890423659245937500000000000*7^(1/2), a[20,15] = -10 28240037170462806316757216166142820312607/1449827602052538506738408008 8220915200000000-2773405513418872586522384915122392541421/115986208164 203080539072640705767321600000*7^(1/2), a[20,16] = -425328035748917435 7660554572428172809055232/26439766035500952469299665052864808467693125 -1785094777818151142513049915774791536640/4230362565680152395087946408 4583693548309*7^(1/2), a[20,17] = 0, a[20,18] = 0, a[20,19] = 0, a[21, 1] = 32998328281095115754973140886490632588259639520507448691090891086 6533/86958352890555050647073388808479504219157089020610000000000000000 00000+12317754712126023040069358494577291492268178954465488679493923/5 1068704015653917614227324965935031208544649600000000000000000000*7^(1/ 2), a[21,2] = 0, a[21,3] = 0, a[21,4] = 0, a[21,5] = 0, a[21,6] = -101 787944783185844934567610199437267016071190464018877029806010097/222103 3666528216854936361530028687663693542572236224000000000000000+24286404 2704019138342215756348369449227768658877917014777973/61142044126439523 835300572321289212176842425280000000000000000*7^(1/2), a[21,7] = -1824 60995578637394052817159406681129508991835141311094293412398453/3083181 843485995531057417669743453498368978585285286400000000000000+376217113 413390755239825969009852998391976488205415554634927/790223095678761227 54351165138516772306376736704000000000000000*7^(1/2), a[21,8] = -13626 51414787912165741809695851592598033579174576934113855593/2225738282760 931713075677971868222790354497828501422080000000+132611782182134047621 6817208523201301529093076210962351207/77476269937375790625023599689091 575826876142735360000000000*7^(1/2), a[21,9] = 16970639824823008461383 1353826859236921181691347192866685810510313547606667756270026782937063 810984608721258380881299166411075429341429273/276224846595532825109221 1570996518613089985560107094379644716714397341072833843431677027895408 91443957973949888434384000000000000000000000-2951452780565925144399383 9816031514326834517591432354363644362946775823077838280455471022394257 10343537548781708565063174753684260829/3946069237079040358703159387137 8837329856936572958491137781667348533443897626334738243255648698777708 28199284120491200000000000000000000*7^(1/2), a[21,10] = -3367866957586 616893410427922346783366803414426599864670257531/392589842653389223059 4454919646702322522240223650611200000000+13148535922868036966154017210 52677815605731754623429227/1925468052163530824271949687212080796476900 76160000000000*7^(1/2), a[21,11] = 42536743325293847343588895318047408 491040870064772965070443067/488610781381075952956511827648525545637556 22144000000000000000-9692607419636867042424073077662239390819817383488 16070613/41697455314991974138633881861113291145038080000000000000000*7 ^(1/2), a[21,12] = -20469187774466175135283801037604208610497860027843 551773783243/146688920321349029625653234502350438830761120000000000000 0000000+49353856385490257348184966893183170004182991711511242739/81698 089847590659774799907826427423464640000000000000000000*7^(1/2), a[21,1 3] = -88623590371247718990009824186596875337790442861/2984996971460945 6679124955916800000000000000000000+21065596458605813540112607696895345 02174173117/29849969714609456679124955916800000000000000000000*7^(1/2) , a[21,14] = -2477099023271188394673723051189192932889307401930799/795 848895613531405854463697340341600000000000000000000-492563695522378519 1465921698950765057056567478577/12733582329816502493671419157445465600 00000000000000*7^(1/2), a[21,15] = 18104280005306436162913765549095626 8617036587/3665736949387159088642295805021782016000000000-121618209327 52853244110559996271097503243/5865179119019454541827673288034851225600 000*7^(1/2), a[21,16] = 51081506620150022938242528276699656513851617/4 23036256568015239508794640845836935483090000-1548002224795344767703449 34051340614195/42303625656801523950879464084583693548309*7^(1/2), a[21 ,17] = 0, a[21,18] = 0, a[21,19] = 0, a[21,20] = 0\}: " }{TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "interpolat ion coefficients: dd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9853 "dd := \{d[1,1] = 1, d[2,1] = 0, d[3,1] \+ = 0, d[4,1] = 0, d[5,1] = 0, d[6,1] = 0, d[7,1] = 0, d[8,1] = 0, d[9,1 ] = 0, d[10,1] = 0, d[11,1] = 0, d[12,1] = 0, d[13,1] = 0, d[14,1] = 0 , d[15,1] = 0, d[16,1] = 0, d[17,1] = 0, d[18,1] = 0, d[19,1] = 0, d[2 0,1] = 0, d[21,1] = 0, d[1,2] = -1312119830254443264105643722266214288 615323/96196905771166154263330304754479177148540, d[2,2] = 0, d[3,2] = 0, d[4,2] = 0, d[5,2] = 0, d[6,2] = -62298907565967997800293333668401 2200000/10748393797119465242695078545561254199, d[7,2] = -161256051718 398551812329473144499000000/3972952538580261729585271390877529877, d[8 ,2] = 10808531054681950895375225742640625000000/4207484826098530723219 920448250447943, d[9,2] = -6697835050066089151824612749520818686113971 6295586605700229793827351418521835661466480054765502528714398961/16312 9180261293056656742783458571341981446090029611281274313111349699116292 45737637313197585417592242220, d[10,2] = 27321453947777634514647817187 5000000/52591352723459798254205221301373, d[11,2] = -22195108432308792 7500849639942387500/59114154491986076909894021245311, d[12,2] = 649410 94147564984157121130927/1680192441753246973469482617, d[13,2] = 981334 28109818708317997/19625757652127244008328, d[14,2] = 0, d[15,2] = 0, d [16,2] = 0, d[17,2] = 6754649117969844085693359375/1608726484531452299 36523748, d[18,2] = 172622909057200527921875/10492678008447785402616, \+ d[19,2] = 8347546714037326577031250000/225233842523147675227082001, d[ 20,2] = 527103525659203968750000000000/21042001853631209820331300117, \+ d[21,2] = 22532261717673989632000000000000000/506067235726010604684751 830915567, d[1,3] = 603974687560059968000436359347917013177431311/6650 704258088350937932972433252855838314970, d[2,3] = 0, d[3,3] = 0, d[4,3 ] = 0, d[5,3] = 0, d[6,3] = 68568742473728768312233538608565633200000/ 96735544174075187184255706910051287791, d[7,3] = 680019346095534425436 24002187154000000/136998363399319369985699013478535513, d[8,3] = -1189 631425917076722284551432177718750000000/378673634348867765089792840342 54031487, d[9,3] = 265798418784807232334657232268924861289239908824127 780929352716974648494983235975645648394492594931731722116463/529354189 9478959688511303323230640047297925621460886077351460463297736323690241 863308132616468008682600390, d[10,3] = -103693506972082795713937878125 0000000/16321454293487523596132654886633, d[11,3] = 244288500991762195 02725741550176825000/532027390427874692189046191207799, d[12,3] = -786 24525982843456768820693241782/166339051733571450373478779083, d[13,3] \+ = -6223060565945462026858031/100932467925225826328544, d[14,3] = 0, d[ 15,3] = 0, d[16,3] = 0, d[17,3] = -292787847326545784576416015625/5515 63366125069359782367136, d[18,3] = -65847429978680223862703125/3507552 36282397397744592, d[19,3] = -34424619839398271445625000000/7507794750 7715891742360667, d[20,3] = -249926433236077324250000000000000/8206380 72291617182992920704563, d[21,3] = -2724661159697556684800000000000000 0/59150715864079161586529434782339, d[1,4] = -618732211461457629280657 5955718201415777240485/19508732490392496084603385804208377125723912, d [2,4] = 0, d[3,4] = 0, d[4,4] = 0, d[5,4] = 0, d[6,4] = -1014114751847 96600869849435111443827500000/32245181391358395728085235636683762597, \+ d[7,4] = -1478727470898732037593655621252169587500000/6714289790200642 32299910865058302549213, d[8,4] = 175943547272529256076280943932792968 7500000/12622454478295592169659761344751343829, d[9,4] = -943462298825 0759394118752596276119640507120246367553759343409778213501974384355645 64605490029827000754238117865/4234833519583167750809042658584512037838 340497168708861881168370638189058952193490646506093174406946080312, d[ 10,4] = 7516180149572117780156544882226562500000/266638158307941177148 82047199796111, d[11,4] = -36129665445955873813993752722181406250/1773 42463475958230729682063735933, d[12,4] = 39303900884779814446171071292 885075/18740866495315716742078609110018, d[13,4] = 2111219133555967775 344525/7597067478242804132256, d[14,4] = 0, d[15,4] = 0, d[16,4] = 0, \+ d[17,4] = 9583240825437812534149169921875/3860943562875485518476569952 , d[18,4] = 594890026372866297347046875/818428884658927261404048, d[19 ,4] = 156013402657853530279062500000/75077947507715891742360667, d[20, 4] = 1096405697556323584937500000000000/820638072291617182992920704563 , d[21,4] = 93273272950221879296000000000000000/4897424861864618755013 7273959571, d[1,5] = 3017560608054297168897232033215502362626867125/48 77183122598124021150846451052094281430978, d[2,5] = 0, d[3,5] = 0, d[4 ,5] = 0, d[5,5] = 0, d[6,5] = 2234495771304940698617759771092752500000 00/32245181391358395728085235636683762597, d[7,5] = 325822129558275756 4912514269612571250000000/671428979020064232299910865058302549213, d[8 ,5] = -3876732013339226718183330236113281250000000/1262245447829559216 9659761344751343829, d[9,5] = 5197051204680439070644911386620789031180 7111676450502771807983206640014671762639217588224601828745537894536037 5/10587083798957919377022606646461280094595851242921772154702920926595 47264738048372661626523293601736520078, d[10,5] = -1656111670792743436 8430749746093750000000/26663815830794117714882047199796111, d[11,5] = \+ 79607938362535275613893136931734375000/1773424634759582307296820637359 33, d[12,5] = -43301017050975443033339586768516250/9370433247657858371 039304555009, d[13,5] = -21044490071134970568941875/336441559750752754 42848, d[14,5] = 0, d[15,5] = 0, d[16,5] = 0, d[17,5] = -3150754324678 763694915771484375/551563366125069359782367136, d[18,5] = -15729901216 4159942045703125/116918412094132465914864, d[19,5] = -3527878221825087 92334375000000/75077947507715891742360667, d[20,5] = -2378788557978784 219375000000000000/820638072291617182992920704563, d[21,5] = -87640586 3727817963520000000000000000/216885958168290259150607927535243, d[1,6] = -9936300382598080556798045420067912379280078125/1463154936779437206 3452539353156282844292934, d[2,6] = 0, d[3,6] = 0, d[4,6] = 0, d[5,6] \+ = 0, d[6,6] = -21350204390091817170873761213281250000000/2614474166866 896950925829916487872643, d[7,6] = -3839578464159148196797269575230468 750000000/671428979020064232299910865058302549213, d[8,6] = 1370534601 4937882713356111389160156250000000/37867363434886776508979284034254031 487, d[9,6] = -6124349475487164599908555857701229159227028372606344984 16363585032594767838499580295803874630658918414266015625/1058708379895 7919377022606646461280094595851242921772154702920926595472647380483726 61626523293601736520078, d[10,6] = 15823848547089592931220092773437500 00000/2161931013307631166071517340524009, d[11,6] = -28143661646981184 2014401190966796875000/532027390427874692189046191207799, d[12,6] = 15 3081363230771842853174751144531250/28111299742973575113117913665027, d [13,6] = 534871782205496121951171875/706527275476580784299808, d[14,6] = 0, d[15,6] = 0, d[16,6] = 0, d[17,6] = 2687599566320620698928833007 8125/3860943562875485518476569952, d[18,6] = 3196454529204356505126953 125/2455286653976781784212144, d[19,6] = 42757361526150251708984375000 0/75077947507715891742360667, d[20,6] = 274934324359508486328125000000 0000/820638072291617182992920704563, d[21,6] = 21197732329279595200000 000000000000000/4554605121534095442162766478240103, d[1,7] = 668850332 7997100188538318464353772999023437500/17070140929093434074027962578682 329985008423, d[2,7] = 0, d[3,7] = 0, d[4,7] = 0, d[5,7] = 0, d[6,7] = 1112716189430464554983499308593750000000000/2257162697395087700965966 49456786338179, d[7,7] = 16225027726164670296841871308593750000000000/ 4700002853140449626099376055408117844491, d[8,7] = -193050375333971848 26216735839843750000000000/88357181348069145187618329413259406803, d[9 ,7] = 1293992829838710243974132647130921692428624043809256029989733288 393960072785311786311114312011242835750976562500/370547932963527178195 7912326261448033108547935022620254146022324308415426583169304315692831 527606077820273, d[10,7] = -82469713805707522095031738281250000000000/ 186646710815558824004174330398572777, d[11,7] = 3964251933734670311794 46533203125000000/1241397244331707615107774446151531, d[12,7] = -21562 6913732250803030921855468750000/65593032733605008597275131885063, d[13 ,7] = -13892101481308973388671875/29438636478190866012492, d[14,7] = 0 , d[15,7] = 0, d[16,7] = 0, d[17,7] = -2080526078308522701263427734375 /482617945359435689809571244, d[18,7] = -64977255150147247314453125/10 2303610582365907675506, d[19,7] = -265409368837976074218750000000/7507 7947507715891742360667, d[20,7] = -1621653662302978515625000000000000/ 820638072291617182992920704563, d[21,7] = -420211648303600000000000000 0000000000/1518201707178031814054255492746701, d[1,8] = -2253412100888 67395112620553906856689453125000/2438591561299062010575423225526047140 715489, d[2,8] = 0, d[3,8] = 0, d[4,8] = 0, d[5,8] = 0, d[6,8] = -3865 4232187090589826567382812500000000000/32245181391358395728085235636683 762597, d[7,8] = -563635179326510793181567382812500000000000/671428979 020064232299910865058302549213, d[8,8] = 67063048986317097053527832031 2500000000000/12622454478295592169659761344751343829, d[9,8] = -449515 3370476024247471488783352283343622766458498652245469156575306255984719 5612730703190117946072998046875000/52935418994789596885113033232306400 4729792562146088607735146046329773632369024186330813261646800868260039 , d[10,8] = 2864884591533068466186523437500000000000/26663815830794117 714882047199796111, d[11,8] = -13771266756991740237731933593750000000/ 177342463475958230729682063735933, d[12,8] = 7490582835375413494946289 062500000/9370433247657858371039304555009, d[13,8] = 17532955410148620 60546875/14719318239095433006246, d[14,8] = 0, d[15,8] = 0, d[16,8] = \+ 0, d[17,8] = 257978485189378261566162109375/24130897267971784490478562 2, d[18,8] = 6294729580402374267578125/51151805291182953837753, d[19,8 ] = 198756832107543945312500000000/225233842523147675227082001, d[20,8 ] = 384062674865722656250000000000000/820638072291617182992920704563, \+ d[21,8] = 1011308203750000000000000000000000000/1518201707178031814054 255492746701\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Suppose that we are given the initial value problem: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=f(x,y)" "6# /*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$%\"xG%\"yG" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "y(x[k])=y[k]" "6#/-%\"yG6#&%\"xG6#%\"kG&F%6#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "When a Runge-Kutta step of wi dth " }{TEXT 273 1 "h" }{TEXT -1 69 " has been made using the basic sc heme, we wish to obtain the result " }{XPPEDIT 18 0 "y[k](u)" "6#-&% \"yG6#%\"kG6#%\"uG" }{TEXT -1 21 " of a step of width " }{XPPEDIT 18 0 "h*u" "6#*&%\"hG\"\"\"%\"uGF%" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 107 "ee2 := map(_U->lhs(_U)=rhs(_U)/u,ee):\nsubs(ee2,m atrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..8)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*,$*&\"\"\"F**&\"#?F*%\"uGF*! \"\"F*F(%!GF/F/F/F/F/7*,$*(\"$T$F*\"%+KF.F-F.F*,$*(\"%hrF*\"(+S-\"F.F- F.F.,$*(\"'\"G;\"F*F8F.F-F.F*F/F/F/F/F/7*,$*(\"%B5F*\"%+kF.F-F.F*,$*(F ?F*\"&+c#F.F-F.F*\"\"!,$*(\"%pIF*FCF.F-F.F*F/F/F/F/7*,$*(\"#RF*\"$+\"F .F-F.F*,$*(\"(nB?%F*\")+\"G;\"F.F-F.F*FD,$*(\"(W)**QF*\"(Dq!HF.F-F.F., $*(\"(#*H)RF*FTF.F-F.F*F/F/F/7*,$*(\"#$*F*\"$+#F.F-F.F*,$*(\"%6cF*\"'+ W6F.F-F.F*FDFD,$*(\"&W<$F*\"'D]8F.F-F.F*,$*(\"'@N#*F*\"(+k5&F.F-F.F*F/ F/7*,$*(\"#JF*FfnF.F-F.F*,$*(\"&t6#F*\"'+KMF.F-F.F*FDFD,$*(\"(CEg)F*\" )v\"fl(F.F-F.F*,$*(\")4@yEF*\"*+SO*oF.F-F.F.,$*(FinF*\"'+NGF.F-F.F*F/7 *,$*(\"$V*F*\"%+5F.F-F.F*,$*(\"1H$p=#=5@7F*\"0+++GH\"3pF.F-F.F.FDFD,$* (\"$D\"F*\"\"#F.F-F.F.,$*(\"1*))fzgIS-\"F*\"0+++!GH*o\"F.F-F.F.,$*(\"1 *oGNN39]\"F*\"0+++!G(pl#F.F-F.F*,$*(\"1$GK@@R,2'F*\"/+++;?]#*F.F-F.F*Q (pprint46\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The new polynom ials (of degree " }{XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 55 " ) are obtained as follows (re-using the weight symbol " } {XPPEDIT 18 0 "b[j]" "6#&%\"bG6#%\"jG" }{TEXT -1 3 "). " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "[seq(b[j]=add(d[j,i]*u^(i-1),i=1..8 ),j=1..21)]:\npols := eval(subs(dd,%)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The first few non-zero polynomials with rough approximations for the coefficients are . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct to 12 do\n if rhs(pols[ct] )<>0 then print(evalf[6](pols[ct])) end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",2$F'\"\"!F'*&$\"'*RO\"!\"%F'%\"uGF'! \"\"*&$\"'O\"3*F.F')F/\"\"#F'F'*&$\"'drJ!\"$F')F/\"\"$F'F0*&$\"'5(='F9 F')F/\"\"%F'F'*&$\"',\"z'F9F')F/\"\"&F'F0*&$\"'D=RF9F')F/\"\"'F'F'*&$ \"'jS#*F.F')F/\"\"(F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" \"',0*&$\"'6'z&!\"%\"\"\"%\"uGF-!\"\"*&$\"'F)3(!\"$F-)F.\"\"#F-F-*&$\" ',XJ!\"#F-)F.\"\"$F-F/*&$\"'qHpF9F-)F.\"\"%F-F-*&$\"';m\")F9F-)F.\"\"& F-F/*&$\"'rH\\F9F-)F.F'F-F-*&$\"'w)>\"F9F-)F.\"\"(F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(,0*&$\"'&)eS!\"%\"\"\"%\"uGF-!\"\"*& $\"'qj\\!\"$F-)F.\"\"#F-F-*&$\"'O-A!\"#F-)F.\"\"$F-F/*&$\"'n_[F9F-)F. \"\"%F-F-*&$\"'_=dF9F-)F.\"\"&F-F/*&$\"'8_MF9F-)F.\"\"'F-F-*&$\"'c%R)F 3F-)F.F'F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"),0*&$\"' ))oD!\"#\"\"\"%\"uGF-F-*&$\"'dTJ!\"\"F-)F.\"\"#F-F2*&$\"'*QR\"\"\"!F-) F.\"\"$F-F-*&$\"'IrIF8F-)F.\"\"%F-F2*&$\"'I>OF8F-)F.\"\"&F-F-*&$\"'*[= #F8F-)F.\"\"'F-F2*&$\"'+8`F2F-)F.\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*,0*&$\"'&e5%!\"#\"\"\"%\"uGF-!\"\"*&$\"'= @]F/F-)F.\"\"#F-F-*&$\"''yA#\"\"!F-)F.\"\"$F-F/*&$\"'')3\\F8F-)F.\"\"% F-F-*&$\"'u%y&F8F-)F.\"\"&F-F/*&$\"'6#\\$F8F-)F.\"\"'F-F-*&$\"'x\"\\)F /F-)F.\"\"(F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5,0*&$ \"'0&>&!\"#\"\"\"%\"uGF-F-*&$\"'?`j!\"\"F-)F.\"\"#F-F2*&$\"'()=G\"\"!F -)F.\"\"$F-F-*&$\"'36iF8F-)F.\"\"%F-F2*&$\"'J>tF8F-)F.\"\"&F-F-*&$\"' \\=WF8F-)F.\"\"'F-F2*&$\"'Xu5F8F-)F.\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6,0*&$\"'iaP!\"#\"\"\"%\"uGF-!\"\"*&$\"'l\" f%F/F-)F.\"\"#F-F-*&$\"'GP?\"\"!F-)F.\"\"$F-F/*&$\"'%*)[%F8F-)F.\"\"%F -F-*&$\"'*)*G&F8F-)F.\"\"&F-F/*&$\"'Q$>$F8F-)F.\"\"'F-F-*&$\"'NlxF/F-) F.\"\"(F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7,0*&$\"'5l Q!\"%\"\"\"%\"uGF-F-*&$\"'wEZ!\"$F-)F.\"\"#F-!\"\"*&$\"'B(4#!\"#F-)F. \"\"$F-F-*&$\"'.@YF9F-)F.\"\"%F-F5*&$\"'bXaF9F-)F.\"\"&F-F-*&$\"'M(G$F 9F-)F.\"\"'F-F5*&$\"'&Q*zF2F-)F.\"\"(F-F-" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 74 "The whole scheme, including the weights, is given by th e set of equations:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ee3 : = `union`(ee2,\{op(pols)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 134 "We can now check that this scheme satisf ies the order conditions (and row sum conditions) for a 21 stage, orde r 8 Runge-Kutta scheme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 " RK8_21eqs := [op(RowSumConditions(21,'expanded')),op(OrderConditions(8 ,21,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "simplify(subs(ee3,RK8_21eqs)):\nmap(u->lhs( u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7hx\"\"!F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$?#" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 43 "#--------------------------------------- ---" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }