{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 "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 61 "Interpolation for Verner's \"Mapl e\" order 8 Runge-Kutta scheme" }}{PARA 0 "" 0 "" {TEXT -1 47 "by Pete r 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 "load proc edures for constructing Runge-Kutta schemes " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 9 "butcher.m" }{TEXT -1 32 " is r equired by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follow s, 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 purpose of this worksheet " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 122 "In this worksheet the interpolation scheme for Maple's \"built in\" dverk78 Runge Kutta scheme as supplied by J. Verner. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The \+ Maple scheme (scheme A) was constructed to provide a lower order inter polation scheme as well as a higher order scheme." }}{PARA 0 "" 0 "" {TEXT -1 136 "An alternative interpolation scheme (scheme B) is also o btained where the construction of a high order scheme is the only cons ideration." }}{PARA 0 "" 0 "" {TEXT -1 135 "Although this scheme appea rs to have a principal error curve which is marginally improved, it st ill has a maximum that is considerably " }}{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 th ird interpolation scheme (scheme C) has an improved principal error cu rve 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/pol ynom_order_conditions,convert/interpolation_order_condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Procedures for \+ converting a standard order condition into: " }}{PARA 0 "" 0 "" {TEXT -1 147 "(i) a list of order conditions for coefficients of an (approxi mate) weight polynomial associated with the construction of an interpo lation scheme, " }}{PARA 0 "" 0 "" {TEXT -1 98 "(ii) a single related \+ order condition associated with the construction of an interpolation s cheme." }}{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(ordco n);\n if not type(LS,polynom) then\n error \"the left side of t he 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 th e 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 \"th e second argument must be greater than or equal to the order,%1, of th e order condition\", dg;\n end if;\n st := max(op(map(_u->op(1,_u) ,select(has,indets(LS),b))));\n step := [];\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(step),trm];\n else\n step := [op(step),lhs(trm)=0];\n end if;\n end do:\nend proc:\n\n`c onvert/interpolation_order_condition` := proc(ord::`=`)\n local LS,R S,st,deg;\n LS := lhs(ord);\n if not type(LS,polynom) then\n \+ error \"the left side of the argument must be a polynomial\"\n end i f;\n RS := rhs(ord);\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 deg := degree(LS,indets(LS));\n st := max( op(map(_u->op(1,_u),select(has,indets(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;\ne nd 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 "Stag e by stage construction of the interpolation scheme A .. [7 stage sche me] .. (longer method)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Start with linking coefficients using the weights o f the 12 stage scheme as the linking coefficients for the first new st age." }}{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 3978 "e1 := \{c[2]=1/16,c[3]=112/1065,c [4]=56/355,c[5]=39/100,c[6]=7/15,c[7]=39/250,c[8]=24/25,\n c[9]=14435 868/16178861,c[10]=11/12,c[11]=19/20,c[12]=1,c[13]=1,\n a[2,1]=1/16,a [3,1]=18928/1134225,a[3,2]=100352/1134225,a[4,1]=14/355,a[4,2]=0,\n a [4,3]=42/355,\n a[5,1]=94495479/250880000,a[5,2]=0,a[5,3]=-352806597/ 250880000,a[5,4]=178077159/125440000,\n a[6,1]=12089/252720,a[6,2]=0, a[6,3]=0,a[6,4]=2505377/10685520,a[6,5]=960400/5209191,\n a[7,1]=2140 0899/350000000,a[7,2]=0,a[7,3]=0,a[7,4]=3064329829899/27126050000000, \n a[7,5]=-21643947/592609375,a[7,6]=124391943/6756250000,\n a[8,1]= -15365458811/13609565775,a[8,2]=0,a[8,3]=0,a[8,4]=-7/5,\n a[8,5]=-833 9128164608/939060038475,a[8,6]=341936800488/47951126225,\n a[8,7]=199 3321838240/380523459069,\n a[9,1]=-1840911252282376584438157336464708 426954728061551/\n 299192361517115192159625381348 3118262195533733898,a[9,2]=0,a[9,3]=0,\n a[9,4]=-14764960804048657303 638372252908780219281424435/\n 29816921025650219756117 11269209606363661854518,\n a[9,5]=-8753250485021304411186134217852667 42862694404520560000/\n 17021203042889441839557167757596 1339495435011888324169,\n a[9,6]=763205196415429092566184979837064563 7589377834346780/\n 173408725741881158304980034758186526 0479233950396659,\n a[9,7]=751983479197113751704853217965234772989930 3513750000/\n 104567730350231759659789070781234983263733 9039997351,\n a[9,8]=1366042683489166351293315549358278750/\n \+ 144631418224267718165055326464180836641,\n a[10,1]=-630777367 05254280154824845013881/78369357853786633855112190394368,\n a[10,2]=0 ,a[10,3]=0,a[10,4]=-31948346510820970247215/6956009216960026632192,\n \+ a[10,5]=-3378604805394255292453489375/517042670569824692230499952,\n \+ a[10,6]=1001587844183325981198091450220795/18423268420772250370166995 3872896,\n a[10,7]=187023075231349900768014890274453125/2522469884980 8178010752575653374848,\n a[10,8]=1908158550070998850625/117087067039 189929394176,\n a[10,9]=-52956818288156668227044990077324877908565/\n 2912779959477433986349822224412353951940608,\n a[1 1,1]=-10116106591826909534781157993685116703/9562819945036894030442231 411871744000,\n a[11,2]=0,a[11,3]=0,a[11,4]=-9623541317323077848129/3 864449564977792573440,\n a[11,5]=-4823348333146829406881375/576413233 634141239944816,\n a[11,6]=6566119246514996884067001154977284529/9703 05487021846325473990863582315520,\n a[11,7]=2226455130519213549256016 892506730559375/364880443159675255577435648380047355776,\n a[11,8]=39 747262782380466933662225/1756032802431424164410720256,\n a[11,9]=4817 5771419260955335244683805171548038966866545122229/\n 198978 6420513815146528880165952064118903852843612160000,\n a[11,10]=-237829 2068163246/47768728487211875,\n a[12,1]=-3218022174758599831659045535 578571/1453396753634469525663775847094384,\n a[12,2]=0,a[12,3]=0,a[12 ,4]=26290092604284231996745/5760876126062860430544,\n a[12,5]=-697069 297560926452045586710000/41107967755245430594036502319,\n a[12,6]=182 7357820434213461438077550902273440/13938101391424531770956768083964169 7,\n a[12,7]=643504802814241550941949227194107500000/2421246091188365 50860494007545333945331,\n a[12,8]=162259938151380266113750/590910828 35244183497007,\n a[12,9]=-230282516328735238185454148568570156166785 75554130463402/\n 200131691831914445034439052404056033499 78424504151629055,\n a[12,10]=7958341351371843889152/3284467988443203 581305,\n a[12,11]=-507974327957860843878400/121555654819179042718967 ,\n a[13,1]=4631674879841/103782082379976,a[13,2]=0,a[13,3]=0,a[13,4] =0,a[13,5]=0,\n a[13,6]=14327219974204125/40489566827933216,\n a[13, 7]=2720762324010009765625000/10917367480696813922225349,\n a[13,8]=-4 98533005859375/95352091037424,\n a[13,9]=4059320304637772479267050305 96175437402459637909765779/\n 78803919436321841083201886 041201537229769115088303952,\n a[13,10]=-10290327637248/1082076946951 ,\n a[13,11]=863264105888000/85814662253313,\n a[13,12]=-29746300739 /247142463456\}:" }}}{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),``$(13-i)],i=2..13)])):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7/$\"++++]i!#6F(%!GF+F+F+F+F+F+F+F+F+F+7/ $\"+#>V;0\"!#5$\"+!p/)o;F*$\"+MsiZ))F*F+F+F+F+F+F+F+F+F+F+7/$\"+*ykud \"F/$\"+s>mVRF*$\"\"!F:$\"+#f)4$=\"F/F+F+F+F+F+F+F+F+F+7/$\"+++++RF/$ \"+m3cmPF/F9$!+&HwiS\"!\"*$\"+4-i>9FDF+F+F+F+F+F+F+F+7/$\"+nmmmYF/$\"+ A\\b$y%F*F9F9$\"+unkWBF/$\"++WmV=F/F+F+F+F+F+F+F+7/$\"++++g:F/$\"+rDa9 hF*F9F9$\"+@JmH6F/$!+AEJ_OF*$\"+U)Q6%=F*F+F+F+F+F+F+7/$\"+++++'*F/$!+M !>!H6FDF9F9$!+++++9FD$!+qGH!)))FD$\"+(4W48(FD$\"+3yOQ_FDF+F+F+F+F+7/$ \"+;EnA*)F/$!+K`$H:'F/F9F9$!+jI(=&\\FD$!+#)obU^FD$\"+wD>,WFD$\"+rJN\"> (FD$\"+gE*\\W*!#7F+F+F+F+7/$\"+nmmm\"*F/$!+&\\v([!)F/F9F9$!+%>8Hf%FD$! +V'zW`'FD$\"+;s`OaFD$\"+oPG9uFD$\"+))>pH;F*$!+&4&3==F*F+F+F+7/$\"+++++ &*F/$!+1\"ey0\"FDF9F9$!+(yu-\\#FD$!+k`'yO)FD$\"+9R1nnFD$\"+ed(=5'FD$\" +m#pME#F*$\"+!*G:@CF*$!+/Owy\\F*F+F+7/$\"\"\"F:$!+@)QT@#FDF9F9$\"+Z!eN c%FD$!+@Oq&p\"!\")$\"+z@068F_s$\"+ZAudEFD$\"+T$Hfu#FD$!+A\\l]6FD$\"+hH -BCFD$!+TW%*yTFDF+7/Fgr$\"+(*[)GY%F*F9F9F9F9$\"+Gn\\QNF/$\"+y89#\\#F/$ !+r%Q$G_FD$\"+#\\l6:&FD$!+sGz4&*FD$\"+9M'f+\"F_s$!+i%4O?\"F/Q(pprint06 \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 14" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order condition gives rise to a \+ group \{list) of equations to be satisfied by the \"d\" coefficients o f the weight polynomials for a given stage (corresponding to an \"appr oximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_13 := SimpleOrderConditi ons(7,13,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "whch := [1,2,3,6,7,8,12,15,16,27,3 1,32,64]:\nordeqns1 := []:\nfor ct in whch do\n eqn_group := convert (SO7_13[ct],'polynom_order_conditions',7):\n ordeqns1 := [op(ordeqns 1),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known coefficients." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns1 := []:\nfor ct to nop s(ordeqns1) do\n eqns1 := [op(eqns1),expand(subs(e1,ordeqns1[ct]))]; \nend do:\nnops(eqns1);\nnops(indets(eqns1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d1 := solve(\{op(eqns1)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs(d1,ma trix([seq([seq(d[j,i],j=1..13)],i=1..7)])):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7/$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+ F+F+F+F+7/$!'%4V'!\"&F+F+F+F+$!'x4))F/$\"'.05!\"%$!'F-F!\"#$\"'Rn6F7$! 'thFF7$\"'I#H%F7$\"']FrF/$\"'KhBF/7/$\"'K=>F4F+F+F+F+$\"'?&H'F4$!'lAVF 4$\"'@F@!\"\"$!'EM\"*F7$\"'9m@FK$!'WwLFK$!'$QZ&F4$!'b2>F47/$!'W#4$F4F+ F+F+F+$!'eA:!\"$$\"'%e8)F4$!'?ggFK$\"'-$e#FK$!'\\[hFK$\"'0<'*FK$\"'0w8 Fen$\"')>/&F47/$\"';#z#F4F+F+F+F+$\"'+Y#)FK$\"'A6]F4$\"'GN?F47/$\"'pIEF/ F+F+F+F+$\"'ud@F4$!'tW#)F/$\"'Cy7FK$!'7b`F7$\"'#=H\"FK$!':O?FKF+F+Q(pp rint16\"" }}}{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 nops(SO7_13) do\n eqn_group := convert(SO7_13[ct],'polynom_order_co nditions',7):\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. .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 "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 at the node " }{XPPEDIT 18 0 "c[14]=1/2-sqr t(7)/14" "6#/&%\"cG6#\"#9,&*&\"\"\"F*\"\"#!\"\"F**&-%%sqrtG6#\"\"(F*F' F,F," }{TEXT -1 83 " to obtain the linking coefficients in the next s tage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "eqs14 := \{seq(a[14,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#R!)y/&%\"cG6#\"\"##\"\"\"\"#;/&F66#\"\"'#\"\"(\"#:/ &F66#F)#\"#R\"$+\"/&F66#\"\"%#\"#c\"$b$/&F66#\"\"$#\"$7\"\"%l5/&F&6$FL F:#\"#9FO/&F&6$FSF8#\"'_.5\"(DU8\"/&F&6$FSF:#\"&G*=F[o/&F&6$F8F:F9/&F6 6#F0F:/&F66#\"#7F:/&F66#\"#6#\"#>\"#?/&F66#\"#5#F^pFjo/&F66#F(#\")oeV9 \")h)yh\"/&F66#\"\")#\"#C\"#D/&F66#FA#FG\"$]#/&F&6$F?F8\"\"!/&F&6$F?F: #\"&*37\"'?FD/&F&6$F)FL#\"*fr2y\"\"*++WD\"/&F&6$F)FS#!*(f1GN\"*++)3D/& F&6$F)F8F\\r/&F&6$F)F:#\")za\\%*F^s/&F&6$FLFS#\"#UFO/&F&6$FLF8F\\r/&F& 6$FAF)#!)ZRk@\"*v$4Ef/&F&6$FAFL#\".**)H)HV1$\"/+++]g7F/&F&6$FAFSF\\r/& F&6$FAF8F\\r/&F&6$FAF:#\")**3S@\"*+++]$/&F&6$F?F)#\"'+/'*\"(\">4_/&F&6 $F?FL#\"(x`]#\")?bo5/&F&6$F?FSF\\r/&F&6$F`qFA#\".S#Q=K$*>\"-p!fM_!Q/&F &6$F`qF?#\"-)[+o$>M\",Di7^z%/&F&6$F`qF)#!.3Y;G\"R$)\"-v%Q+1R*/&F&6$F`q FL#!\"(F)/&F&6$F^pF(#\"VHA7Xl'o'*Q![:<0QoW_Lb4E>9xv\"[\"X++;7O%G&Q!*=T 1_f;!))Gl9:Q^?ky*)>/&F&6$F`qFSF\\r/&F&6$F`qF8F\\r/&F&6$F`qF:#!,6)eaO: \",vdc4O\"/&F&6$FAF?#\"*V>RC\"\"+++Dcn/&F&6$FepF:#!A\")Q,X[#[:!Ga_qOx2 j\"AoVR!>7^&Qj'y`yNp$y/&F&6$F(F`q#\"F](y#e$\\bJ$H^j;*[$oUgO\"\"HTm$3=k kKb];=xEC#=9jW\"/&F&6$F(FA#\"U++v8NI**)HxM_'z@`[q^P6(>zM)>v\"U^t**R!Rt jK)\\B\"yq!*yf'f-y3HDs$QOId'[S!3'\\w9\"O=X &=mjjg4#p7r6c(>-lD5#p\")H/&F&6$F(FSF\\r/&F&6$F(F8F\\r/&F&6$F(F:#!R^:1G Z&pU3ZYOt:QWewBG_7\"4%=\"R)*QtLb>i#=J[8QD'f@>:r^hB>*H/&F&6$FepFA#\"EDJ Xu-*[,o2!*\\8BvI-(=\"D[[P`cd_2,y\"3)\\))pC_#/&F&6$FepF?#\"C&z?-X\"4)> \")fK$=Wye,5\"B'*G(Q&*p;q.Ds2UoKU=/&F&6$FepF)#!=v$*[`CHbUR0[gyL\"<_** \\IApC)p0nUq^/&F&6$FepFL#!8:sCq4#3^Y$[>$\"7#>Km-gp@4g&p/&F&6$FepFSF\\r /&F&6$FepF8F\\r/&F&6$F^pFA#\"Iv$f0t1D*o,c#\\N@>08bkA#\"HwdNZ+Q[cVxbDv' fJW!)[O/&F&6$F^pF?#\"FHXGx\\:,q1%)o*\\^Y#>hc'\"E?bJ#ej3*RZDj%=-([0.(*/ &F&6$F^pF)#!:v8)oSHo9L$[L#[\"9;[%*R79MOB8kd/&F&6$F^pFL#!7H\"[y2BtJTNi* \"7SMd#zx\\c\\W'Q/&F&6$F^pFSF\\r/&F&6$F^pF8F\\r/&F&6$F^pF:#!G.n6&o$*z: \"yM&4p#=f1h65\"F+Sur=TJAWIS*o.X*>Gc*/&F&6$FepF(#!Jl&3z([Kx+*\\/F#om:) G=o&H&\"L31%>&RN7WAA)\\j)RVx%f*zF\"H/&F&6$FepF`q#\"7D1&))*42]&e\"3>\"9 wTRH**=Rq1(3<\"/&F&6$FjoF(#!Y-MYITbv&ymh:q&o&[TX&=Q_tGj^#GI#\"Yb!H;:/X Uy*\\Lg0/C0RW.XW\">$=pJ,?/&F&6$F(F?#\"U!yYV$yP*ePckq$)z\\=mD4HaT'>0Kw \"UfmR]RBz/El=eZ.!)\\Ie6)=uD(3MF#\\>%4bTU\"G![]V'\"HJ`%RLXv+%\\g3bO)= \"4Y7U#/&F&6$FjoF?#\"FSMF-4bx!Q9Y8UV?yNF=\"E(pT'R3on&4xJXU\"R,\"QR\"/& F&6$FjoF)#!?++r'eX?XE4c(Hpqp\">>B]OSfIaCbx'z5T/&F&6$FjoFL#\"8Xn*>B%G/E 4!HE\"7W0VgG1Eh(3w&/&F&6$FjoFSF\\r/&F&6$FjoF8F\\r/&F&6$FjoF:#!Cr&yb`X! 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FcelFAFdelFcel/&F&6$FenFSF\\r/&F&6$FenF8F\\r/&F&6$FenFLF\\r/&F66#Fen,& FdelF:*&FenFcelFAFdelFcel/&F&6$FenFA,&#\"<+]i:gmd<5yZ$)H\"\"<,@/*=#)Q9 al+^\\`F:*(\";+vV['*Qu'H:TmF)F:\">\\H1jsA]I:@$*f7i#FcelFAFdelFcel" }}} {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 40 "[seq(a[14,i]=subs(e2,a[14,i]),i=1..13)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7//&%\"aG6$\"#9\"\"\",&#\"0*)HVBr]B*\"2'Hvk9)GT .#F)*(\"1Z+m<*>'GPF)\"4KgJ=M&\\ytz!\"\"\"\"(#F)\"\"#F)/&F&6$F(F4\"\"!/ &F&6$F(\"\"$F8/&F&6$F(\"\"%F8/&F&6$F(\"\"&F8/&F&6$F(\"\"',&#\"4v8%)Rs` 7I`\"\"4O.\"\\F)4bf$zF)*(\"5vyh/`Tc%\\5\"F)\"6KKINx!***3V%>F1F2F3F1/&F &6$F(F2,&#\"<+]i:gmd<5yZ$)H\"\"<,@/*=#)Q9al+^\\`F)*(\";+vV['*Qu'H:TmF) F)\">\\H1jsA]I:@$*f7i#F1F2F3F1/&F&6$F(\"\"),&#\"2Dc^`ZchI\"\"2/^LV)4!* o=F)*(\"5v$f$oh&oGZ5$F)\"5)oid00:KTc#F1F2F3F)/&F&6$F(\"\"*,&#\"XLb\"[w ivUo966E5'e6B))\\ocE)G%*z6K\"Y#fuvIdluMqH,b2k'pvI_33>&4#obW:F1*(\"YHho xLz.0zLGQm<\")>0W3bsY&eg#*4c$F)\"Y?$[#y1+Fa-wNfv?&oGD$o0X%y4&)R*fjF1F2 F3F1/&F&6$F(\"#5,&#\"/C]oPCIP\"/*f+/x@I&F)*(\"2ORWCJ9Ud\"F)\"2bn9[PL!* H\"F1F2F3F)/&F&6$F(\"#6,&#\"1+?4Amnr]\"1PBT]%=\\?%F1*(\"3+?LY!)**y.QF) \"38X?qS+Tg?F1F2F3F1/&F&6$F(\"#7,&#\".Vt7815\"\"/wt$G#*R%[F)*(\",R2IY( HF)\".kGh.ZU\"F1F2F3F1/&F&6$F(\"#8,&#F<\"$#RF)*(F " 0 "" {MPLTEXT 1 0 48 "[seq(a[14,i]=expand(subs(ee,a[14,i])),i=1..13)];" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7//&%\"aG6$\"#9\"\"\",&#\"0*)HVBr]B*\"2'Hvk9)GT. #F)*(\"1Z+m<*>'GPF)\"4KgJ=M&\\ytz!\"\"\"\"(#F)\"\"#F)/&F&6$F(F4\"\"!/& F&6$F(\"\"$F8/&F&6$F(\"\"%F8/&F&6$F(\"\"&F8/&F&6$F(\"\"',&#\"4v8%)Rs`7 I`\"\"4O.\"\\F)4bf$zF)*(\"5vyh/`Tc%\\5\"F)\"6KKINx!***3V%>F1F2F3F1/&F& 6$F(F2,&#\"<+]i:gmd<5yZ$)H\"\"<,@/*=#)Q9al+^\\`F)*(\";+vV['*Qu'H:TmF)F )\">\\H1jsA]I:@$*f7i#F1F2F3F1/&F&6$F(\"\"),&#\"2Dc^`ZchI\"\"2/^LV)4!*o =F)*(\"5v$f$oh&oGZ5$F)\"5)oid00:KTc#F1F2F3F)/&F&6$F(\"\"*,&#\"XLb\"[wi vUo966E5'e6B))\\ocE)G%*z6K\"Y#fuvIdluMqH,b2k'pvI_33>&4#obW:F1*(\"YHhox Lz.0zLGQm<\")>0W3bsY&eg#*4c$F)\"Y?$[#y1+Fa-wNfv?&oGD$o0X%y4&)R*fjF1F2F 3F1/&F&6$F(\"#5,&#\"/C]oPCIP\"/*f+/x@I&F)*(\"2ORWCJ9Ud\"F)\"2bn9[PL!*H \"F1F2F3F)/&F&6$F(\"#6,&#\"1+?4Amnr]\"1PBT]%=\\?%F1*(\"3+?LY!)**y.QF) \"38X?qS+Tg?F1F2F3F1/&F&6$F(\"#7,&#\".Vt7815\"\"/wt$G#*R%[F)*(\",R2IY( HF)\".kGh.ZU\"F1F2F3F1/&F&6$F(\"#8,&#F<\"$#RF)*(F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calculation for stage 1 5" }}{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_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]:\norde qns2 := []:\nfor ct in whch do\n eqn_group := convert(SO7_14[ct],'po lynom_order_conditions',7):\n ordeqns2 := [op(ordeqns2),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 "eqns2 := []:\nfor ct to nops(ordeqns2) do \n eqns2 := [op(eqns2),expand(subs(e2,ordeqns2[ct]))];\nend do:\nnop s(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(eqns 2)\}):\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$!'UI s!\"&F+F+F+F+$\"'>PVF/$\"'53:!\"%$!'/*p'!\"$$\"'=!p#F7$!'O4mF7$\"'wh5! \"#F+$\"'3dA!\"'$!'oM7F470$\"'J^CF4F+F+F+F+$!'XpCF4$!'VwwF4$\"'1BxF>$! '4XJF>$\"'$!'vA7!\"\"$!'m@sF/$!'1Q[F/$\"'AJ#)F470$!'m\"H%F4F+F+F+ F+$\"'k%\\%F4$\"'=o:F7$!'n6IFS$\"'YN7FS$!'I(*HFS$\"'CrZFS$\"'?pIF4$\"' cQ=F4$!'-_=F770$\"'V6RF4F+F+F+F+$!&yX*F4$!'$\"'#=H\"FS$!':O? FSF+F+F+Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions a re satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "nm := NULL :\nfor ct to nops(SO7_14) do\n eqn_group := convert(SO7_14[ct],'poly nom_order_conditions',7):\n tt := expand(subs(\{op(e2),op(d2)\},eqn_ group));\n tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(o p(i,tt),i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops( [%]);\nop(\{seq(i,i=1..64)\} minus \{nm\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6J\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#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 47 "Evaluate the \"weight\" polynomials at the node " } {XPPEDIT 18 0 "c[15] = 9/20;" "6#/&%\"cG6#\"#:*&\"\"*\"\"\"\"#?!\"\"" }{TEXT -1 83 " to obtain the linking coefficients in the next stage i n the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "eqs15 := \{seq(a[15,j]=add(expand(subs(\{op(d2),c[15]=9/20\},d[j, i]*c[15]^i)),i=1..7),j=1..14)\}:\ne3 := `union`(eqs15,\{c[15]=9/20\},e 2):" }}}{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#R!)y/& %\"cG6#\"\"##\"\"\"\"#;/&F66#\"\"'#\"\"(\"#:/&F66#F)#\"#R\"$+\"/&F66# \"\"%#\"#c\"$b$/&F66#\"\"$#\"$7\"\"%l5/&F&6$FLF:#\"#9FO/&F&6$FSF8#\"'_ .5\"(DU8\"/&F&6$FSF:#\"&G*=F[o/&F&6$F8F:F9/&F66#F0F:/&F66#\"#7F:/&F66# \"#6#\"#>\"#?/&F66#\"#5#F^pFjo/&F66#F(#\")oeV9\")h)yh\"/&F66#\"\")#\"# C\"#D/&F66#FA#FG\"$]#/&F&6$F?F8\"\"!/&F&6$F?F:#\"&*37\"'?FD/&F&6$F)FL# \"*fr2y\"\"*++WD\"/&F&6$F)FS#!*(f1GN\"*++)3D/&F&6$F)F8F\\r/&F&6$F)F:# \")za\\%*F^s/&F&6$FLFS#\"#UFO/&F&6$FLF8F\\r/&F&6$FAF)#!)ZRk@\"*v$4Ef/& F&6$FAFL#\".**)H)HV1$\"/+++]g7F/&F&6$FAFSF\\r/&F&6$FAF8F\\r/&F&6$FAF:# \")**3S@\"*+++]$/&F&6$F?F)#\"'+/'*\"(\">4_/&F&6$F?FL#\"(x`]#\")?bo5/&F &6$F?FSF\\r/&F&6$F`qFA#\".S#Q=K$*>\"-p!fM_!Q/&F&6$F`qF?#\"-)[+o$>M\",D i7^z%/&F&6$F`qF)#!.3Y;G\"R$)\"-v%Q+1R*/&F&6$F`qFL#!\"(F)/&F&6$F^pF(#\" VHA7Xl'o'*Q![:<0QoW_Lb4E>9xv\"[\"X++;7O%G&Q!*=T1_f;!))Gl9:Q^?ky*)>/&F& 6$F`qFSF\\r/&F&6$F`qF8F\\r/&F&6$F`qF:#!,6)eaO:\",vdc4O\"/&F&6$FAF?#\"* V>RC\"\"+++Dcn/&F&6$FepF:#!A\")Q,X[#[:!Ga_qOx2j\"AoVR!>7^&Qj'y`yNp$y/& F&6$F(F`q#\"F](y#e$\\bJ$H^j;*[$oUgO\"\"HTm$3=kkKb];=xEC#=9jW\"/&F&6$F( FA#\"U++v8NI**)HxM_'z@`[q^P6(>zM)>v\"U^t**R!RtjK)\\B\"yq!*yf'f-y3HDs$QOId'[S!3'\\w9\"O=X&=mjjg4#p7r6c(>-lD5#p\") H/&F&6$F(FSF\\r/&F&6$F(F8F\\r/&F&6$F(F:#!R^:1GZ&pU3ZYOt:QWewBG_7\"4%= \"R)*QtLb>i#=J[8QD'f@>:r^hB>*H/&F&6$FepFA#\"EDJXu-*[,o2!*\\8BvI-(=\"D[ [P`cd_2,y\"3)\\))pC_#/&F&6$FepF?#\"C&z?-X\"4)>\")fK$=Wye,5\"B'*G(Q&*p; 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'ceYF7$\"'Q\"G\"FO$!'CK?FO$\"'KyNF2$\"'9z8F2$\"'*yL\"F7$!'&)\\6F771$!' 2z`F2F+F+F+F+$!'AJ@F7$\"'n%G#F7$!'nOXFO$\"'d))=FO$!'$42&FO$\"'B')yFO$! '+t8F7$!'yQaF2$!')G'QF7$\"'q\"\\%F771$\"'2!z&F2F+F+F+F+$\"'(QO%F7$!'*> u#F7$\"'<-$)FO$!')Gf$FO$\"'4h%*FO$!'h[9!\"\"$\"'&y\\#F7$\"'l:5F7$\"'a- _F7$!'#*fxF771$!'%y>$F2F+F+F+F+$!'\\-RF7$\"'I?;F7$!'WSrFO$\"'j.KFO$!'c $G)FO$\"'4]7F^q$!'@N@F7$!'(4%*)F2$!'sLLF7$\"'efhF771$\"'O\"3(F/F+F+F+F +$\"',s7F7$!'*pv$F2$\"'$GJ#FO$!'\\r5FO$\"'2EFFO$!'8hSFO$\"'uvoF2$\"'`y HF2$\"' " 0 "" {MPLTEXT 1 0 318 "nm := NULL:\nfor ct to nops(SO7_15) do\n eqn_group := convert(S O7_15[ct],'polynom_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(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 "6R\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\" \"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"# E\"#F\"#G\"#H\"#I\"#J\"#K\"#O\"#R\"#V\"#Y\"#Z\"#\\\"#^\"#`\"#a\"#c\"#d \"#f\"#g\"#i\"#j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"#L\"#M\"#N\"#P\"#Q\"#S\"#T\"#U\"#W\"#X\"#[ \"#]\"#_\"#b\"#e\"#h" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate the \"weight\" polynomials at the node " } {XPPEDIT 18 0 "c[16] = 1/10;" "6#/&%\"cG6#\"#;*&\"\"\"F)\"#5!\"\"" } {TEXT -1 83 " to obtain the linking coefficients in the next stage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "eqs16 := \{seq(a[16,j]=add(expand(subs(\{op(d3),c[16]=1/10\},d[j, i]*c[16]^i)),i=1..7),j=1..15)\}:\ne4 := `union`(eqs16,\{c[16]=1/10\},e 3):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e4 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "e4;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#R!)y/& %\"cG6#\"\"##\"\"\"\"#;/&F66#\"\"'#\"\"(\"#:/&F66#F)#\"#R\"$+\"/&F66# \"\"%#\"#c\"$b$/&F66#\"\"$#\"$7\"\"%l5/&F&6$FLF:#\"#9FO/&F&6$FSF8#\"'_ .5\"(DU8\"/&F&6$FSF:#\"&G*=F[o/&F&6$F8F:F9/&F66#F0F:/&F66#\"#7F:/&F66# \"#6#\"#>\"#?/&F66#\"#5#F^pFjo/&F66#F(#\")oeV9\")h)yh\"/&F66#\"\")#\"# C\"#D/&F66#FA#FG\"$]#/&F&6$F?F8\"\"!/&F&6$F?F:#\"&*37\"'?FD/&F&6$F)FL# \"*fr2y\"\"*++WD\"/&F&6$F)FS#!*(f1GN\"*++)3D/&F&6$F)F8F\\r/&F&6$F)F:# \")za\\%*F^s/&F&6$FLFS#\"#UFO/&F&6$FLF8F\\r/&F&6$FAF)#!)ZRk@\"*v$4Ef/& F&6$FAFL#\".**)H)HV1$\"/+++]g7F/&F&6$FAFSF\\r/&F&6$FAF8F\\r/&F&6$FAF:# \")**3S@\"*+++]$/&F&6$F?F)#\"'+/'*\"(\">4_/&F&6$F?FL#\"(x`]#\")?bo5/&F &6$F?FSF\\r/&F&6$F`qFA#\".S#Q=K$*>\"-p!fM_!Q/&F&6$F`qF?#\"-)[+o$>M\",D i7^z%/&F&6$F`qF)#!.3Y;G\"R$)\"-v%Q+1R*/&F&6$F`qFL#!\"(F)/&F&6$F^pF(#\" VHA7Xl'o'*Q![:<0QoW_Lb4E>9xv\"[\"X++;7O%G&Q!*=T1_f;!))Gl9:Q^?ky*)>/&F& 6$F`qFSF\\r/&F&6$F`qF8F\\r/&F&6$F`qF:#!,6)eaO:\",vdc4O\"/&F&6$FAF?#\"* V>RC\"\"+++Dcn/&F&6$FepF:#!A\")Q,X[#[:!Ga_qOx2j\"AoVR!>7^&Qj'y`yNp$y/& F&6$F(F`q#\"F](y#e$\\bJ$H^j;*[$oUgO\"\"HTm$3=kkKb];=xEC#=9jW\"/&F&6$F( FA#\"U++v8NI**)HxM_'z@`[q^P6(>zM)>v\"U^t**R!RtjK)\\B\"yq!*yf'f-y3HDs$QOId'[S!3'\\w9\"O=X&=mjjg4#p7r6c(>-lD5#p\") H/&F&6$F(FSF\\r/&F&6$F(F8F\\r/&F&6$F(F:#!R^:1GZ&pU3ZYOt:QWewBG_7\"4%= \"R)*QtLb>i#=J[8QD'f@>:r^hB>*H/&F&6$FepFA#\"EDJXu-*[,o2!*\\8BvI-(=\"D[ [P`cd_2,y\"3)\\))pC_#/&F&6$FepF?#\"C&z?-X\"4)>\")fK$=Wye,5\"B'*G(Q&*p; q.Ds2UoKU=/&F&6$FepF)#!=v$*[`CHbUR0[gyL\"<_**\\IApC)p0nUq^/&F&6$FepFL# !8:sCq4#3^Y$[>$\"7#>Km-gp@4g&p/&F&6$FepFSF\\r/&F&6$FepF8F\\r/&F&6$F^pF A#\"Iv$f0t1D*o,c#\\N@>08bkA#\"HwdNZ+Q[cVxbDv'fJW!)[O/&F&6$F^pF?#\"FHXG x\\:,q1%)o*\\^Y#>hc'\"E?bJ#ej3*RZDj%=-([0.(*/&F&6$F^pF)#!:v8)oSHo9L$[L #[\"9;[%*R79MOB8kd/&F&6$F^pFL#!7H\"[y2BtJTNi*\"7SMd#zx\\c\\W'Q/&F&6$F^ pFSF\\r/&F&6$F^pF8F\\r/&F&6$F^pF:#!G.n6&o$*z:\"yM&4p#=f1h65\"F+Sur=TJA WIS*o.X*>Gc*/&F&6$FepF(#!Jl&3z([Kx+*\\/F#om:)G=o&H&\"L31%>&RN7WAA)\\j) RVx%f*zF\"H/&F&6$FepF`q#\"7D1&))*42]&e\"3>\"9wTRH**=Rq1(3<\"/&F&6$FjoF (#!Y-MYITbv&ymh:q&o&[TX&=Q_tGj^#GI#\"Yb!H;:/XUy*\\Lg0/C0RW.XW\">$=pJ,? 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'1x%)FM$!'mu5FX72$!'d6:FMF+F+F+F+$!'LDoFM$!'7j#*FM$!'\\gZFX$\"'6#o#FX$ !'.(G'FX$\"'\\!e)FX$!'+t8FM$!'+(f'F/$\"'5(4#FM$\"'oOnFM$\"'D[')FM72$\" '.mVF/F+F+F+F+$\"'6p@FM$\"'qkHFM$\"'O#e\"FX$!'C9\"*FM$\"'H8@FX$!'&y&GF X$\"'LOXF/$\"'8fAF/$!'KQ%)F/$!'`:?FM$!'OaEFMQ(pprint46\"" }}}{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(SO7_16) do\n e qn_group := convert(SO7_16[ct],'polynom_order_conditions',7):\n tt : = expand(subs(\{op(e4),op(d4)\},eqn_group));\n tt := map(_Z->`if`(lh s(_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\" p olynomials at the node " }{XPPEDIT 18 0 "c[17] = 13/40;" "6#/&%\"cG6# \"#<*&\"#8\"\"\"\"#S!\"\"" }{TEXT -1 83 " to obtain the linking coeff icients in the next stage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "eqs17 := \{seq(a[17,j]=add(expand( subs(\{op(d4),c[17]=13/40\},d[j,i]*c[17]^i)),i=1..7),j=1..16)\}:\ne5 : = `union`(eqs17,\{c[17]=13/40\},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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eL><6UQ\"FK" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calcula tion for stage 18" }}{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_17 := SimpleOrderConditi ons(7,17,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "whch := [1,2,3,6,7,8,12,15,16,21,2 7,31,32,33,36,64]:\nordeqns5 := []:\nfor ct in whch do\n eqn_group : = convert(SO7_17[ct],'polynom_order_conditions',7):\n ordeqns5 := [o p(ordeqns5),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 "eqns5 := []:\nfor ct t o 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 "" }}{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$\"#=\"#<\"\"!" }{TEXT -1 29 " as in the publi shed scheme. " }}{PARA 0 "" 0 "" {TEXT -1 22 "We use the fact that " }{XPPEDIT 18 0 "a[18,17] = Sum(d[17,i]*c[18]^i,i = 1 .. 7);" "6#/&%\"a G6$\"#=\"#<-%$SumG6$*&&%\"dG6$F(%\"iG\"\"\")&%\"cG6#F'F0F1/F0;F1\"\"( " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[18] = 2/5;" "6#/&%\"cG6#\"#=*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "c_18 := 2/5;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c_18G#\"\"#\"\"&" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "eq := add (subs(\{op(d5),c[18]=c_18\},d[17,i]*c[18]^i),i=1..7)=0:\ndd := \{d[1,1 ]=1,seq(d[1,i]=0,i=2..6)\}:\nsol := \{d[1,7]=expand(rationalize(solve( subs(dd,eq))))\}:\nsol;\ndd_5 := `union`(subs(sol,dd),sol):\nd_5 := `u nion`(subs(dd_5,d5),dd_5):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/&%\"d G6$\"\"\"\"\"(,&#\"8v*>Z#399S+(\\Q\"6#>3-aCc9@i " 0 "" {MPLTEXT 1 0 67 "subs(d_5,matrix([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+7 3F+F+F+F+F+$\"&WB%!\"#$\"&2d&F/$\"&tO#!\"\"$!&g;\"F4$\"&\"GHF4$!&(=UF4 $\"&(fq!\"$$\"&:'HF=$!&SK'F4$!&c#*)F/$!&#GBF/$\"&tX'F473F+F+F+F+F+$!&0 1#F4$!&&4FF4$!&t9\"F*$\"&Oj&F4$!&rT\"F*$\"&T/#F*$!&PU$F/$!&DV\"F/$\"&v ;$F*$\"&&)R%F4$\"&G4\"F4$!&SB$F*73F+F+F+F+F+$\"&`O&F4$\"&'\\qF4$\"&*pH F*$!&?X\"F*$\"&0m$F*$!&%*G&F*$\"&:())F/$\"&qp$F/$!&;Z)F*$!&A;\"F*$!&\\ y#F4$\"&_l)F*73F+F+F+F+F+$!&Oh(F4$!&S***F4$!&5=%F*$\"&?.#F*$!&!Q^F*$\" &EW(F*$!&0D\"F4$!&9=&F/$\"&(G7F)$\"&Tn\"F*$\"&q*QF4$!&mD\"F)73F+F+F+F+ F+$\"&1\\&F4$\"&#)>(F4$\"&U)HF*$!&)Q9F*$\"&[l$F*$!&!4`F*$\"&6%*)F/$\"& Sn$F/$!&h+*F*$!&LA\"F*$!&Ky#F4$\"&>A*F*73$!&gE#F/F+F+F+F+$!&@3\"F*$!&I U\"F*$!&z-'F*$\"&6'HF*$!&oW(F*$\"&S2\"F)$!&()z\"F4$!&=_(F/$\"&Kh\"F)$ \"&tG#F*$\"&qD'F4$!&)[;F)Q(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of o rder conditions are satisfied." }}{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;\ne nd 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] = 2/5;" "6#/&%\"cG6#\"#=*&\"\"#\"\"\"\"\"&!\"\"" }{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 "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..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#1\"*z3ql6F//&F&6$F9\"\"*, &*(\"StpKvN*3Y!y7s.\\ZPC-%3%pfBh**fF-\"U]K!z$3P8JU3FMAJZKD! 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g,#\\p9!*e)yB9$>FE/&F&6$F(\"#:$!?o4**>W$p-HW^;\"Q4CF,/&F&6$F(\"#;$\"?< %f],g\\`.+ZxYZZ\"FE/&F&6$F(\"# " 0 "" {MPLTEXT 1 0 62 "[seq(a[18,i]=expand(subs (ee,a[18,i])),i=1..17)]:\nevalf[30](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#73/&%\"aG6$\"#=\"\"\"$\"?*Gf!p!H%eu VEA8`Ki%\\QB!#I/&F&6$F(\"\"($\">tu1bp=z\"o1dMM.JFE/&F&6$F(\"\")$\"?Su, \"***f!o!o)zt_lR\"FE/&F&6$F(\"\"*$!?na:YvNa&z$=u)ej>(F,/&F&6$F(\"#5$\" ?7A@)p(4d1P2dv,lg,#\\p9!*e)yB9$>FE/&F&6$F(\"#:$!?o4**>W$p-HW^;\"Q4CF,/&F&6$F( \"#;$\"?<%f],g\\`.+ZxYZZ\"FE/&F&6$F(\"# " 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 -1 223 "Each standard (simple ) order condition gives rise to a group \{list) of equations to be sa tisfied by the \"d\" coefficients of the weight polynomials for a give n 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 0 205 "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[ct],'polynom_order_cond itions',7):\n ordeqns6 := [op(ordeqns6),op(eqn_group)];\nend do:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitu te for all known coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns6 := []:\nfor ct to nops(ordeqns6) do\n eqns6 := [op(eqns 6),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(e qns6) minus \{seq(d[1,i],i=1..7),seq(d[9,i],i=1..7)\}):\ninfolevel[sol ve]:=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 29 " as in the published scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[19 ] = 3/10;" "6#/&%\"cG6#\"#>*&\"\"$\"\"\"\"#5!\"\"" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "c_19 := 3/10;" }}{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),d d_6):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<$/&%\"dG6$\"\"*\"\"(, &#\"hn+AQ'o!*ery/ep/.$=A.!R6V$G0J)[wK68K\"fnTLO(z$*=/!4I2y4^#\\3e0Pn,h ^[\\Q)pL\"!\"\"*(\"TvohWlSHy*o#G,iyZ&Ru;kAX\"z\"[5\"\"\"\"Ryg^X)>\"3:$ =j#[(z7$z.&*>r5kb5\"F0F+#F3\"\"#F0/&F(6$F3F+,&#\"9+JK5EeF\"ovWo#\"6$*4 qe5V())HIAF0*(\"2D!\\wOQ'4-\"F3\"0ZvWTA@%zF0F+F5F0" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(d_6, matrix([seq([seq(d[j,i],j=1..10)],i=1..7)])):\nevalf[7](%);\nsubs(d_6, matrix([seq([seq(d[j,i],j=11..18)],i=1..7)])):\nevalf[7](%);" }}{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+$\"('H*)G!\"&$\"((H?NF/$\"(ATz'F/F+$\"(`;X%F/7,F+F+F+ F+F+$!(]Fa\"!\"%$!(w'z=F9$!(Zxi$F9F+$!(wpP#F97,F+F+F+F+F+$\"(o(=XF9$\" (Kc]&F9$\"(zD1\"!\"$F+$\"(XA'pF97,F+F+F+F+F+$!(efP(F9$!(,o)*)F9$!(UWt \"FGF+$!(Vk8\"FG7,F+F+F+F+F+$\"(c!=iF9$\"(Jgd(F9$\"(j@Y\"FGF+$\"()R!e* F97,$!(]wB\"FGF+F+F+F+$!(J')G$FG$!()\\*)RFG$!($GbrFG$!(x;\"\\F9$!(k\"3 TFGQ)pprint126\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*$ \"\"!F)F(F(F(F(F(F(F(7*$!(JU6\"!\"%$\"(Px[#!\"'$\"(G,)R!\"($\"(em4$!\" $$!(p&\\eF-$!(3ZK%!\"&$!(:l'RF6$\"(W+V\"F67*$\"(![\\fF-$!(O$G8F;$!(*4o ?F0$!(#Q%Q\"!\"#$\"(aB\"HF6$\"(VOFI$\"([[w\"FI7*$\"(*Hf6FI$!(`xo#F-$!(`P]$F;$!(mq+%!\"\"$\"(lH 6(FI$\"(zt>(F6$\"(5L3&Fhq$!(0(p " 0 " " {MPLTEXT 1 0 286 "nm := NULL:\nfor ct to nops(SO7_18) do\n eqn_gro up := convert(SO7_18[ct],'polynom_order_conditions',7):\n tt := expa nd(subs(\{op(e6),op(d_6)\},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\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9 \"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J\" #K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"# \\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate the \"weight\" polynomials a t the node " }{XPPEDIT 18 0 "c[19] = 3/10;" "6#/&%\"cG6#\"#>*&\"\"$\" \"\"\"#5!\"\"" }{TEXT -1 83 " to obtain the linking coefficients in t he next 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 [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#1\"*z3ql6F// &F&6$F9\"\"*,&*(\"StpKvN*3Y!y7s.\\ZPC-%3%pfBh**fF-\"U]K!z$3P8JU3FMAJZKD!Q\\\"pN<-UMAP>FW,&*(\"3DYJQ^XXM5F-\"4[)=V*e_Uww#F/F0F1F/#\"3v$fPKZbs]$ \"4KqcWhxaV7\"F-/&F&6$F9\"#7,&*(\"1h\\UW5RaSF-\"4+DZTZ\"\\.m8F/F0F1F/# \"36ErVv/P`C\"5+v3A6sB0\\?F-/&F&6$\"#9Ffo,&#\"4v8%)Rs`7I`\"\"4O.\"\\F) 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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO7_19 := SimpleOrderConditi ons(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 "" }}{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 \{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$\"#?\"#<\"\" !" }{TEXT -1 29 " as in the published scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[20] = 7/10;" "6#/&%\"cG6#\"#?* &\"\"(\"\"\"\"#5!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "c_20 := 7/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c _20G#\"\"(\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 382 "eq1 := add(subs(\{op(d7),c[20]=c_20\},d[17,i] *c[20]^i),i=1..7)=0:\neq2 := add(subs(\{op(d7),c[20]=c_20\},d[18,i]*c[ 20]^i),i=1..7)=0:\neq3 := add(subs(\{op(d7),c[20]=c_20\},d[19,i]*c[20] ^i),i=1..7)=0:\ndd := \{d[1,1]=1,seq(d[1,i]=0,i=2..6),seq(d[9,i]=0,i=1 ..6),seq(d[13,i]=0,i=1..6)\}:\nsol := solve(subs(dd,\{eq1,eq2,eq3\})); \ndd_7 := `union`(subs(sol,dd),sol):\nd_7 := `union`(subs(dd_7,d7),dd_ 7):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<%/&%\"dG6$\"\"*\"\"(,&# \"hn+ekVfzM`&or'zz6o;rr!yHro#oJs!*\\]%R\"gn4+pE1I_)*eg8!3B[A`G'>awz*=[ g_'>Z#*!\"\"*(\"VD\"[HOK()>S#3z#4c+H,]tKz()[/h-r\"\"\"\"W#Q>6Q*QJ\"QP? 6$zj(*Gen^M(3q*z**o@F0F+#F3\"\"#F3/&F(6$F3F+,&#\"8+2)3Hl5cK)fH$\"7h])) *eew`Hq?%F0*(\"4v()*[Xhc>=pF3\"6Vd**p@!)3f\"e:F0F+F5F3/&F(6$\"#8F+,&# \"++x#fp#\",.([te=F3*(\")+c%)oF3FHF0F+F5F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "subs(d_7,matrix([ seq([seq(d[j,i],j=1..11)],i=1..7)])):\nevalf[7](%);\nsubs(d_7,matrix([ seq([seq(d[j,i],j=12..19)],i=1..7)])):\nevalf[7](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7-$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+7 -F+F+F+F+F+$\"(OWD%!\"'$\"(sN=&F/$\"(A/+\"!\"&F+$\"(#)\\b'F/$!(ⓈF47 -F+F+F+F+F+$!(m^i#F4$!(#[)>$F4$!(DI<'F4F+$!()pWSF4$\"(sB,\"!\"%7-F+F+F +F+F+$\"(!>#=*F4$\"(^(=6FD$\"(u\"f@FDF+$\"(PZT\"FD$!(H5a$FD7-F+F+F+F+F +$!(T<&=FD$!([hD#FD$!(SVN%FDF+$!(]I&GFD$\"($3TrFD7-F+F+F+F+F+$\"(XD,#F D$\"(q?X#FD$\"(hCt%FDF+$\"(435$FD$!(07w(FD7-$!('*pr(F/F+F+F+F+$!(d*p)) F4$!(3&z5FD$!(`d/#FD$!(L)*R$F/$!(k-I\"FD$\"(8_M$FDQ)pprint166\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*$\"\"!F)F(F(F(F(F(F(F( 7*$\"(YJm$!\"(F($!(gN$f!\"#$\"(dq!R!\"%$!()G>F3$!(a/m\"FD $\"('p#4\"F0$!(IT`\"FD7*$\"(Kg!zFAF($!(buz)FD$\"(YoY'F;$!(msm%F3$\"(UU u%FD$!(caB$F0$\"(3YJ%FD7*$!(#Q%f\"F6F($\"($[e8F)$!(8'\\5F0$\"(=de'F3$! (sFQ(FD$\"(*G/_F0$!(%z>mFD7*$\"(QGt\"F6F($!(W%R5F)$\"(cv@)F;$!(?b$\\F3 $\"(*Q#o&FD$!(5W4%F0$\"([&R]FD7*$!(Xw`(FA$\"(1%[:F-$\"(w%pLFD$!(P<^#F; $\"(j\"\\>F3$!(P&Q=FD$\"(T:I\"F0$!(vdj\"FDQ)pprint176\"" }}}{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 e qn_group := convert(SO7_19[ct],'polynom_order_conditions',7):\n tt : = expand(subs(\{op(e7),op(d_7)\},eqn_group));\n tt := map(_Z->`if`(l hs(_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\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"# 6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G \"#H\"#I\"#J\"#K\"#L\"#M\"#N\"#O\"#P\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\" #Y\"#Z\"#[\"#\\\"#]\"#^\"#_\"#`\"#a\"#b\"#c\"#d\"#e\"#f\"#g\"#h\"#i\"# j\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate the \"weight\" p olynomials at the node " }{XPPEDIT 18 0 "c[20] = 7/10;" "6#/&%\"cG6# \"#?*&\"\"(\"\"\"\"#5!\"\"" }{TEXT -1 82 " to obtain the linking coeff icients in the next stage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "eqs20 := \{seq(a[20,j]=add(expand( subs(\{op(d_7),c[20]=c_20\},d[j,i]*c[20]^i)),i=1..7),j=1..19)\}:\ne8 : = `union`(eqs20,\{c[20]=c_20\},e7):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e8 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 3 "e8;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<]x/&% \"aG6$\"#<\"#:,&*(\"*HdT3*\"\"\"\",k/r\\['!\"\"\"\"(#F-\"\"#F-#\"+P%** zJ#\",'*Gr\"eBF//&F&6$\"#?\"#6,&*(\"4OfZoBI)3g:F-\"5D\\F![0TU$o@F/F0F1 F/#\"6KC0=vL&=IE7\"5vZ#3W;BF]]'F//&F&6$\"#=\"#5#\"0%QgH+Rof\"1v=Af/\\ 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d8 := solve(\{op(eqns8)\}):\ninfole vel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "d8 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33373 "d8 := \{d[1,1] = 1, d[13,3] = -602655454921832537181960288056 6514575441175091155923561220047/29764604421103659390479360657784111101 96314303465327138387653-2139317505131212408973113339484697976204874101 33479950210474/2705873129191241762770850968889464645633013003150297398 53423*7^(1/2), d[7,8] = -504934820595770118651483720615821616344457874 618213075439408117675781250000000/118310850065290181027779673026426112 288178048907015165910136386441056921975113*7^(1/2)-5829252743778141191 277392529471287085269196234842362636809291368688964843750000000/826992 8419563783653841799144547185248943645618600360097118533412229878846060 3987, d[4,5] = 0, d[1,8] = 1755557759904094373588398202354871858232742 54154528202609108731250/1567287515835716873348194891510140237022606105 01511298233884565093*7^(1/2)-55786798557415101320354184726993408339398 27654525857901613430487513750/9826892724289944795893181969768579286131 7402784447583992645622313311, d[12,8] = 398322888381933595831285599712 0149472135026200554177016609686308125/11700682913197398487424345218512 6870238578305849193259513447587858+80392145392847409094641048445807581 719640248385135864947859274375/390022763773246616247478173950422900795 26101949731086504482529286*7^(1/2), d[20,1] = 0, d[5,3] = 0, d[8,8] = \+ 4672983575772741830559064962212755834137930320117548086574952282714843 75/3160036509778104019125754770024038904705579234993390197579804715579 99+9431322828772428829902640294193634610457182926248640916490469970703 125/105334550325936800637525159000801296823519307833113006585993490519 333*7^(1/2), d[6,8] = -19185107058122785203535289160049564079494884688 82540636667169305833671875/1916933157181657006994397288454159304110238 1013121305728007311065105038-49854907736726005687427277190461458195347 9798642430508104866804921875/82271809321101159098471986628933875712885 755421121483811190176245086*7^(1/2), d[16,1] = 0, d[4,2] = 0, d[3,8] = 0, d[6,1] = 0, d[3,3] = 0, d[4,3] = 0, d[5,1] = 0, d[2,1] = 0, d[17,1 ] = 0, d[2,6] = 0, d[2,5] = 0, d[15,3] = 27150595984707922614116206747 40552998114611488544663872000000/4438204603435370087084464552146740953 1546774125745618706117*7^(1/2)-975620145254031770622577302338700408050 11233298292327296000000/1464607519133672128737873302208424514541043546 149605417301861, d[17,2] = 1393815010347086709454061781438896515045751 1215120796621209600/55835477269025623676223908881846095862268522287228 359017373-173069075495440698843906977750181786857481132028743640678400 /4295036713001971052017223760142007374020655560556027616721*7^(1/2), d [16,4] = 9157110678028170982401939084023751427627959028125420984500000 /12885110139005913156051671280426022122061966681668082850163-155557300 430237724477930535584876290931582349335611732000000/128851101390059131 56051671280426022122061966681668082850163*7^(1/2), d[13,5] = -16943893 24006523377473023508146745285244449384512671429450766/2705873129191241 76277085096888946464563301300315029739853423*7^(1/2)-14905395364072035 7553471138481703434532931554920520150727204840/29764604421103659390479 36065778411110196314303465327138387653, d[1,4] = -15942899352571795873 2435122909792054219844229114153042132964106856070471/78615141794319558 3671454557581486342890539222275580671941164978506488+13395534087691097 5521865058804930854478181657243056758022875707595/44779643309591910667 091282614575435343503031571860370923967018598*7^(1/2), d[13,8] = 69353 937077584493629916215748288775268139741309544613455050000/297646044211 0365939047936065778411110196314303465327138387653+30505470673633401643 2297948882625960092399514374513276935000/27058731291912417627708509688 8946464563301300315029739853423*7^(1/2), d[3,5] = 0, d[19,8] = -512461 01677545589535973599862632917488838190987880233485000000/1002175233033 7932454706855440331350539381529641297397772349+11344242745950932585133 76484761384930885032455880792216250000/3340584110112644151568951813443 783513127176547099132590783*7^(1/2), d[9,8] = -54357017459008758272536 5294967767996683441362351167162273537812273148111556941724552328895688 97360064327858125/3730883235311418648153371792128911382418899046603930 8844192456619345505657387191406228559216756605808267811-10970690809251 4677104590989638959580116175688861875077800455971354334916525114156693 3644183495378079259924375/12436277451038062160511239307096371274729663 488679769614730818873115168552462397135409519738918868602755937*7^(1/2 ), d[14,4] = -35700514888657337576645408405026489305210345903327408528 41768/14794015344784566956948215173822469843848924708581872902039*7^(1 /2)+8543707845705682942507623072572991013191256249879172243240468/1479 4015344784566956948215173822469843848924708581872902039, d[17,7] = -13 040552832797494056061335704744868473870997008694932232601600000/502519 295421230613086015179936614862760416700585055231156357+970398404805540 150632185310931084731489172095720641989836800000/502519295421230613086 015179936614862760416700585055231156357*7^(1/2), d[20,4] = -4281076147 419059649694705547949444587841860930558872690173400/100217523303379324 54706855440331350539381529641297397772349+3426759976387913287518433990 36669849568846218979805820800700/1431678904333990350672407920047335791 340218520185342538907*7^(1/2), d[8,5] = -20954054940897124105843926324 1027813186596877686456237303114804759765625/42133820130374720255010063 6003205187294077231332452026343973962077332*7^(1/2)-411804599204346385 2343792034438314420149017616045054996408305694060546875/12640146039112 41607650301908009615561882231693997356079031921886231996, d[5,8] = 0, \+ d[2,8] = 0, d[3,6] = 0, d[18,1] = 0, d[5,4] = 0, d[20,7] = 29017213838 17167216825184518040806692095561741960030898200000/1002175233033793245 4706855440331350539381529641297397772349-35787786010842250848335956162 65137525299387923370917980350000/1002175233033793245470685544033135053 9381529641297397772349*7^(1/2), d[20,5] = 7410578398906801657428051910 976918842569991849588522199953300/100217523303379324547068554403313505 39381529641297397772349-4988893657127186826296687829354717852873073088 222647736756900/100217523303379324547068554403313505393815296412973977 72349*7^(1/2), d[17,8] = -24354768876635010124733067870650225839790671 5390477320192000000/50251929542123061308601517993661486276041670058505 5231156357*7^(1/2)+330594867457186157461279967732034184772176970754959 5504640000000/50251929542123061308601517993661486276041670058505523115 6357, d[15,5] = -77271389330436736046341591203930142770502369122751232 1664000000/14646075191336721287378733022084245145410435461496054173018 61+21503904804481981635576759310429973127232607726738406848000000/4438 2046034353700870844645521467409531546774125745618706117*7^(1/2), d[15, 7] = -554305650946780364723806333094169233744998044820565696000000000/ 1464607519133672128737873302208424514541043546149605417301861+15425807 732760500780763672238300515681881597379952672000000000/443820460343537 00870844645521467409531546774125745618706117*7^(1/2), d[7,6] = -251936 4373823950685032974284884563547030699522842299536969973000488281250000 00/9100834620406937002136897925109700945244465300539628146933568187773 609382701*7^(1/2)-7303417208846689306832501286514466988244713997616442 372528485130509277343750000000/275664280652126121794726638151572841631 45485395334533657061778040766262820201329, d[1,7] = 163074924384235731 939807678090398452743117931030873070102421348689659750/687882490700296 135712522737883800550029221819491133087948519356193177-699489475093839 665173148832780546819469357457958881532054048398750/156728751583571687 334819489151014023702260610501511298233884565093*7^(1/2), d[9,5] = 243 7414794617643124169781328211525734456694174242413234093912632977672202 1558976935336200985921795322095810211/49745109804152248642044957228385 4850989186539547190784589232754924606742098495885416380789556754744110 23748*7^(1/2)+47901879871146737411537515322056486071550813309271882356 6820002396538239888287074784313198222024975036400066777/14923532941245 6745926134871685156455296755961864157235376769826477382022629548765624 914236867026423233071244, d[10,2] = 2815976631631622644232771544332426 36609627963296730630613584596992/9007423325867068321825065597305036458 4855703288525066641120617043+11087620167925462303787199386406600593938 2896536227671113630596096/81885666598791530198409687248227604168050639 35320460603738237913*7^(1/2), d[7,2] = -512593083799952678590586102627 18392599578938940008688345921713867187500000/1444576923874116984466174 27382693665797531195246660764237040764885295387027*7^(1/2)-91919029433 3874368789413455291388409919396531032571289112707441406250000000/11219 5474420889752460206201933892080436082561641573193557434994060912750590 97, d[10,8] = 22047100580764751308730110947717870343403344733525449061 074282421760000/819675522653903217286080969354758317722186899925578106 4341976150913+12135520524655026802175232617481328007309615391138686003 9155223040000/74515956604900292480552815395887119792926081811416191494 0179650083*7^(1/2), d[6,3] = 44721084360784434677784729801158958699023 144255148014082283092570062625/547695187766187716284113510986902658317 2108860891801636573517447172868+48224465738913401109111657520645193884 053404474223761391463039244625/113478357684277460825478602246805345810 87690402913308111888300171736*7^(1/2), d[19,6] = -46130182751302283817 918221879360970634138119727341303044612500/143167890433399035067240792 0047335791340218520185342538907+73582497806289497992903335512786543012 42533370906131902325000/3340584110112644151568951813443783513127176547 099132590783*7^(1/2), d[17,3] = -1469604360050531892734122746115860001 716171503446532235375411200/502519295421230613086015179936614862760416 700585055231156357+170797506941086482591181116160029856187748421507562 731687116800/502519295421230613086015179936614862760416700585055231156 357*7^(1/2), d[9,7] = 421523003793281873607124126014202633408966246385 3643029744516593998712667294578315821888672681938019401125/11992552990 3935025655846087821565778926997719273671838136266334359837690959135941 51793172361541821217701*7^(1/2)+11896114982465836977186512629614352109 3082405738927579797140416872237104277273046947314201554244828166835964 7125/26116182647179930537073602544902379676932293326227516190934719633 5418539601710339843599914517296240657874677, d[18,7] = 316143710529606 2675426098100171489277186422929136425922375000/47722630144466345022413 5973349111930446739506728447512969-14356107232061379289516947904289317 47906894457174721992937500/1431678904333990350672407920047335791340218 520185342538907*7^(1/2), d[13,6] = 32940231821852493147901757808184303 48500565191103210664134000/3865533041701773946815501384127806636618590 0045004248550489+19786853818194620715195780547250433584161183384451114 06746700/270587312919124176277085096888946464563301300315029739853423* 7^(1/2), d[16,3] = 408483912305568790490948742180409553333366096463537 88000000/12885110139005913156051671280426022122061966681668082850163*7 ^(1/2)-2404600995794184108553369816093489658400897946481484185500000/1 2885110139005913156051671280426022122061966681668082850163, d[12,6] = \+ 4990550874280040648638722246400872012250593517892524642718365021325/39 002276377324661624747817395042290079526101949731086504482529286+104289 9908700117841039313312353663742325293833338076584273444464975/78004552 754649323249495634790084580159052203899462173008965058572*7^(1/2), d[6 ,4] = -198843564303262219767550870816527521354664566489563466165166345 71272397625/3067093051490651211191035661526654886576380962099408916481 16977041680608-7608215819256113462967395406365033755315888444874428725 36857709497625/4701246246920066234198399235939078612164900309778370503 4965814997192*7^(1/2), d[13,7] = -212782850393317473657425990408971734 919709700004931017869430000/297646044211036593904793606577841111019631 4303465327138387653-12154687334329603894369194108866287748636092278505 63214449000/2705873129191241762770850968889464645633013003150297398534 23*7^(1/2), d[7,4] = -377607155541862436437229124747725244425078683080 9446983798340124914005126953125000/82699284195637836538417991445471852 489436456186003600971185334122298788460603987-192641670807867235686269 180225709354160799250354290899373597571538085937500000/169015500093271 68718254239003775158898311149843859309415733769491579560282159*7^(1/2) , d[8,2] = 17193471936928197999297729709829937388089628904716534461686 1328125/10003281132567597401474374074150170638510855444740076598859780 6761+26808225213594592443500811319552220037327838869838401763840251953 125/360118120772433506453077466669406142986390796010642757558952104339 6*7^(1/2), d[9,4] = -5633819537049559793410035100952088381039577274326 20573796804991228957665352906175788066319539947839934746531703/5969413 1764982698370453948674062582118702384745662894150707930590952809051819 5062499656947468105692932284976-16742059538830070367125095134382121545 3717089311180338142614388912369164429346425644199370551869060190111133 3/71064442577360355202921367469122121569883791363884397798461822132086 67744264226934519725565096496344431964*7^(1/2), d[11,8] = -18495495227 36736578425317868674896966708332754574645607920741018560000000/6500478 39315421139406191831136850859181556352527037679017120259217719-1964675 002696647740703375759826243832120046196602305650193859520000000/114043 48058165283147477049669067558933009760570649783842405618582767*7^(1/2) , d[11,5] = 1091254872061015387230549673942225748089579395236601842570 1905663872000/11404348058165283147477049669067558933009760570649783842 405618582767*7^(1/2)+4074768226599438636959251699100997766836376192247 637561931221314581376000/650047839315421139406191831136850859181556352 527037679017120259217719, d[17,4] = 6881214659525582008629854370912320 640539588082195871903822643200/502519295421230613086015179936614862760 416700585055231156357-650424614032174093791051728417050717482161451339 083515867955200/502519295421230613086015179936614862760416700585055231 156357*7^(1/2), d[16,5] = -1904500031433728925514060581816599561474819 4763205317544500000/12885110139005913156051671280426022122061966681668 082850163+323528778864697640409442763387912876880174943254587092000000 /12885110139005913156051671280426022122061966681668082850163*7^(1/2), \+ d[18,6] = -14706725244159097369839650045084389134466768834323339695143 750/1431678904333990350672407920047335791340218520185342538907+2337058 843108387223130118014187125966651720500656422395131250/143167890433399 0350672407920047335791340218520185342538907*7^(1/2), d[1,5] = 74243391 633422797921589354811480821232154726827981191218422498436501339/196537 854485798895917863639395371585722634805568895167985291244626622-195020 6477967455511058326696135743462758645751611098779858131152365/31345750 3167143374669638978302028047404521221003022596467769130186*7^(1/2), d[ 1,2] = -49913430094603751641566666553290435741449130095513853795522453 9656077/43675078774621976870636364310082574605029956793087815107842498 805916+35643680302198397392977791611006902844604115504781863583325615/ 382731994099076159547788740295516541397461808306498896786042894*7^(1/2 ), d[9,6] = -681034077919222653192393052878857407825529591170262848989 25632386305073324820674656222237927256276086704523325/1243627745103806 2160511239307096371274729663488679769614730818873115168552462397135409 519738918868602755937-142319033624938281765906796099925768908728962696 69391033131831084299927256021775791805841788147988818788330975/2487255 4902076124321022478614192742549459326977359539229461637746230337104924 794270819039477837737205511874*7^(1/2), d[6,5] = 169067902650530333296 48335601858388471864447291145441331568974896761472375/7667732628726628 0279775891538166372164409524052485222912029244260420152+11076521234134 088057149961100737269777954476399103782076984857286503375/329087237284 404636393887946515735502851543021684485935244760704980344*7^(1/2), d[1 7,6] = 204815956340976350229984779371978141007462827728521468931276800 00/502519295421230613086015179936614862760416700585055231156357-121517 737612379293186559956233074423082943425456607622266880000/386553304170 17739468155013841278066366185900045004248550489*7^(1/2), d[19,3] = 240 01196016558020249137334006257368893543757664440775397285500/1002175233 0337932454706855440331350539381529641297397772349-79556015865202888152 6157940792991314731592433374109190961500/33405841101126441515689518134 43783513127176547099132590783*7^(1/2), d[11,7] = 782810749810557137210 8920646280380631802076888100147265069689408000000/11404348058165283147 477049669067558933009760570649783842405618582767*7^(1/2)+4047766933649 9798751079880256441409547001320907230018857505974948288000000/45503348 75207947975843342817957956014270894467689263753119841814524033, d[9,1] = 0, d[4,4] = 0, d[18,4] = -64100320748810287723994815540776906119912 77986846048477664625/1908905205778653800896543893396447721786958026913 790051876+962240401383305759281068249126719609416913245186419069251125 /1431678904333990350672407920047335791340218520185342538907*7^(1/2), d [6,6] = -6467507300969516521025929921764741053468015851759795471956639 951946875/164543618642202318196943973257867751425771510842242967622380 352490172*7^(1/2)-7211053313381233086300879132909929222458082754180954 209603197468316165625/191693315718165700699439728845415930411023810131 21305728007311065105038, d[20,8] = -4349221682502648266218568851914987 64502971172424562510000000/1002175233033793245470685544033135053938152 9641297397772349+89819114766083809798011116695077425075397012034051613 5250000/10021752330337932454706855440331350539381529641297397772349*7^ (1/2), d[14,7] = -1274675079284016384193381413114386681183847167811776 6637062000/14794015344784566956948215173822469843848924708581872902039 +5326324058359793211972372472079770450825900701602014640012000/1479401 5344784566956948215173822469843848924708581872902039*7^(1/2), d[13,1] \+ = 0, d[12,1] = 0, d[5,2] = 0, d[11,3] = 150897521680799186986973522562 260953677284916446359064398733016553216000/650047839315421139406191831 136850859181556352527037679017120259217719+475105694989829181251346976 96975104131967515527218789537205684352000/3932533813160442464647258506 57502032172750364505164960082952364923*7^(1/2), d[8,4] = 1007501166082 04859903313329960826963417814591192968820314883670072265625/4213382013 03747202550100636003205187294077231332452026343973962077332*7^(1/2)+48 4330218915220688426360032319541473733045728142617871096157479453320312 5/50560584156449664306012076320384622475289267759894243161276875449279 84, d[4,8] = 0, d[15,8] = 13911797411215697627565458646656178555900194 5472146240000000000/14646075191336721287378733022084245145410435461496 05417301861-3871523079657954441500094362198449477079860886355680000000 000/44382046034353700870844645521467409531546774125745618706117*7^(1/2 ), d[5,7] = 0, d[4,7] = 0, d[8,1] = 0, d[19,1] = 0, d[14,6] = 20750685 530389736142396736277471776562099060814518578546774600/147940153447845 66956948215173822469843848924708581872902039-8670827363319521458391749 977399960880738754626900190802259600/147940153447845669569482151738224 69843848924708581872902039*7^(1/2), d[7,1] = 0, d[2,4] = 0, d[3,2] = 0 , d[2,7] = 0, d[12,7] = -871735628966902039227630330599693355576653687 96189760818312215457125/8190478039238178941197041652958880916700481409 44352816594133115006-1884216309611934219225257333643902673671649097641 5059145514438625/22942515516073330367498716114730758870309471735135933 23793089958*7^(1/2), d[15,6] = -25111974848881885321672674753908806295 730804842016137600000000/443820460343537008708446455214674095315467741 25745618706117*7^(1/2)+82033185179704454423090234313012641155003796319 965868800000000/133146138103061102612533936564402228594640322377236856 118351, d[13,4] = 4469818265075669076176973572725235246798406373189011 3472400110/29764604421103659390479360657784111101963143034653271383876 53+116383830142974352559205642901851452389500804492627501586498/386553 30417017739468155013841278066366185900045004248550489*7^(1/2), d[17,5] = -16284045126278290461296539094899248799774589651392237003204198400/ 502519295421230613086015179936614862760416700585055231156357+135275606 1845794935005032527459177157343737472884618209964851200/50251929542123 0613086015179936614862760416700585055231156357*7^(1/2), d[14,3] = -224 3525181240063021054547606023596942176560700713986922286012/14794015344 784566956948215173822469843848924708581872902039+937473583863200678290 023199702129127111060390461868310702712/147940153447845669569482151738 22469843848924708581872902039*7^(1/2), d[6,7] = 4198689519461862291563 8860421102159164607440878383392575567397620821546875/13418532100271599 0489607810191791151287716667091849140096051177455735266+32564479674342 770145680506638056725979974514653489394159326992515625/134871818559182 2280302819452933342224801405826575762029691642233526*7^(1/2), d[4,1] = 0, d[3,7] = 0, d[10,1] = 0, d[5,5] = 0, d[14,5] = -177692423219797321 13303486591852924208923200638318894675721108/1479401534478456695694821 5173822469843848924708581872902039+74250092762108075418415670310042506 86195270805576966863546408/1479401534478456695694821517382246984384892 4708581872902039*7^(1/2), d[9,3] = 12670790712923032331792565244718181 7497902186839500388646054412454304529085318728275770866029281645866603 0863/10659666386604053280438205120368318235482568704582659669769273319 813001616396340401779588347644744516647946+307745337126029212059142399 9734486315665303260173440817411443098308014249943618454887532557145872 390493628329/497451098041522486420449572283854850989186539547190784589 23275492460674209849588541638078955675474411023748*7^(1/2), d[3,4] = 0 , d[3,1] = 0, d[15,1] = 0, d[5,6] = 0, d[4,6] = 0, d[14,1] = 0, d[2,3] = 0, d[15,2] = 128517437682336804655701775914367552839359491871761062 4000000/162734168792630236526430366912047168282338171794400601922429-3 5765200672383149604976200334469073612125825077957568000000/49313384482 61522318982738391274156614616308236193957634013*7^(1/2), d[7,5] = 1284 2507633976844176721173398876707596683191966556931280741549480590463867 187500000/826992841956378365384179914454718524894364561860036009711853 34122298788460603987+2804599143838483635686420579755276860426016270238 510991238647118735351562500000/118310850065290181027779673026426112288 178048907015165910136386441056921975113*7^(1/2), d[14,8] = 31991413830 68294686005314463711435892226627541404474243530000/1479401534478456695 6948215173822469843848924708581872902039-13367848788808393357967508437 51432009304881437822766197780000/1479401534478456695694821517382246984 3848924708581872902039*7^(1/2), d[20,3] = 1285842974433897654229156181 372139355262370475712225168576900/100217523303379324547068554403313505 39381529641297397772349-6298922792250297070873755214202056839547373898 85991590309100/1002175233033793245470685544033135053938152964129739777 2349*7^(1/2), d[12,4] = 4128407656495197095864782641237351791512208292 8531250158904168376423/18721092661115837579878952349620299238172528935 87092152215161405728+1226841689391452206935525741914267005199301196723 05467038075627253/2228701507275694949985589565430988004544348682841776 3716847159592*7^(1/2), d[13,2] = 2721489311917062733130875756847927895 477861273414985588679/472454038430216815721894613615620811142272111661 16303783931+402585847555372253030773794622450357040040289344403163858/ 4295036713001971052017223760142007374020655560556027616721*7^(1/2), d[ 8,6] = 585473819099436616359792295616289997925144761821549862844906260 302734375/105334550325936800637525159000801296823519307833113006585993 490519333+941148748854609247806624242582970866506815265275359827432029 7900390625/16205315434759507790388486000123276434387585820478924090152 844695282*7^(1/2), d[7,7] = 127574072573303770128993440408331408200224 952046321093978563413855778808593750000000/578894989369464855768925940 118302967426055193302025206798297338856091519224227909+201187679882678 6379735123103598287827469131143296795192190628542480468750000000/11831 0850065290181027779673026426112288178048907015165910136386441056921975 113*7^(1/2), d[16,2] = 31675561449932759585814449048894682735398944250 970524500000/143167890433399035067240792004733579134021852018534253890 7-538091653799326086073602371256585993376203834697972000000/1431678904 333990350672407920047335791340218520185342538907*7^(1/2), d[19,7] = 20 3703904923247205940579809550396524495519753185149039998000000/10021752 330337932454706855440331350539381529641297397772349-452003265568192567 6944197548888486813595105051207128217750000/33405841101126441515689518 13443783513127176547099132590783*7^(1/2), d[11,6] = -23172836050705597 54766097807721904833622226626044503834275727836710400000/2166826131051 40379802063943712283619727185450842345893005706753072573-1274352967522 6432076498828545773668006242117239918967340627507526400000/11404348058 165283147477049669067558933009760570649783842405618582767*7^(1/2), d[7 ,3] = 3541056331199227679532598709973213822538697442497997014415902415 18554687500000/1183108500652901810277796730264261122881780489070151659 10136386441056921975113*7^(1/2)+67940851965291117811367357780127224374 270905887018262802541051081181640625000000/118141834565196909340597130 63638836069919493740857657281597904874614112637229141, d[19,4] = -1111 14488575406951697814647617112481576947418506724192036853000/1002175233 0337932454706855440331350539381529641297397772349+43280316339671395592 9400840527752756100563709091898459035500/47722630144466345022413597334 9111930446739506728447512969*7^(1/2), d[20,6] = -671362494647510761287 9670524364761895187151459653206454772500/10021752330337932454706855440 331350539381529641297397772349+582596385077990210493473562564800586764 5807120912036599905000/10021752330337932454706855440331350539381529641 297397772349*7^(1/2), d[2,2] = 0, d[1,3] = -24623094573511389233937258 5690030669766812810495003823744255113735/31345750316714337466963897830 2028047404521221003022596467769130186*7^(1/2)+323107495203126164523432 378313687072540527571660364270765036005492591/503943216630253579276573 4343471066300580379629971670973981826785298, d[10,6] = 828678293072444 49502424268554211128993861815964251993805443017305395200/8196755226539 032172860809693547583177221868999255781064341976150913+605499894650785 59411540852248961942552952505671499453948353424025600/5731996661915407 1138886781073759322917635447547243224226167665391*7^(1/2), d[9,2] = -3 7142425507719738213341950069205380303694158659572836293946045413460539 828251623048052653226120876208741/219334699312840602478152368731858399 90704873877742098085945006830890949827976009057159646805853383779111-5 7912823636946473834949306239340766013198183661281840867605716405127576 56160930124011293784825753160970893/7896049175262261689213485274346902 3966537545959871553109402024591207419380713632605774728501072181604799 6*7^(1/2), d[10,3] = -256962403700955597405150437397475457337271082383 744977126104318773248/117096503236271888183725852764965473960312414275 0825866334568021559-85105166119311678537257515346907811541623568055391 798062460169174016/745159566049002924805528153958871197929260818114161 914940179650083*7^(1/2), d[6,2] = -31855552542635478526726360177851346 9051754900650813057635058169612625/27384759388309385814205675549345132 91586054430445900818286758723586434-2368597088783713017630215327405106 8033533413543752892588902929763625/47012462469200662341983992359390786 121649003097783705034965814997192*7^(1/2), d[12,5] = -3510202737592865 6197360237373946948817888902208668958451100751617257/46802731652789593 9496973808740507480954313223396773038053790351432-17861136364023416484 14048196007301317823652463115059649161263392851/1560091055092986464989 91269580169160318104407798924346017930117144*7^(1/2), d[10,4] = 142816 64055712408136475245264915777099888212847111472750534674661140736/8196 755226539032172860809693547583177221868999255781064341976150913+462991 82679347465106460276860537873218520453497703926889996052319232/1064513 66578428989257932593422695885418465831159165987848597092869*7^(1/2), d [11,2] = -163347330891545876403903842900594992244726299935020978849222 045568000/114043480581652831474770496690675589330097605706497838424056 18582767*7^(1/2)-71657832081718310340617577294538331105648420163748795 1993884383744000/21668261310514037980206394371228361972718545084234589 3005706753072573, d[12,3] = -92850310609122940672342183074126380973594 493023226754287693593983/334305226091354242497838434814648200681652302 42626645575270739388-2255127622897123456876081777974926631981739820338 50914510537775289/1560091055092986464989912695801691603181044077989243 46017930117144*7^(1/2), d[19,2] = -10810719405586300269221788426869054 893442683433666843966500/530251446049626055804595525943457700496377229 69827501441+1497118870780096785653851352791263510405606027720535895500 /53025144604962605580459552594345770049637722969827501441*7^(1/2), d[1 0,5] = -67405275067629095544857313147675304862007044448615245204310881 6262144/74515956604900292480552815395887119792926081811416191494017965 0083*7^(1/2)-485722733188634615447532181733923039204280659310512959419 34257889156096/8196755226539032172860809693547583177221868999255781064 341976150913, d[18,8] = -242744046529228723481724648461884456189053668 5058664154375000/14316789043339903506724079200473357913402185201853425 38907+360305284791876976027312121944428132436664116668984431562500/143 1678904333990350672407920047335791340218520185342538907*7^(1/2), d[11, 1] = 0, d[18,2] = -415192595528324478554061654986779355006785628652362 77287875/636301735259551266965514631132149240595652675637930017292+332 8506778536405261654129899831216571480624909539443236125/15907543381488 7816741378657783037310148913168909482504323*7^(1/2), d[19,5] = 2596099 94539919112520381575091296551490549603905366764948523500/1002175233033 7932454706855440331350539381529641297397772349-63010218735262322883353 53065278679997660367485595369684528500/3340584110112644151568951813443 783513127176547099132590783*7^(1/2), d[14,2] = 29553726321805022820241 612807131610602365379038713689058228/164377948275384077299424613042471 8871538769412064652544671-12349243040856320027880115681345382618617915 842794662375528/164377948275384077299424613042471887153876941206465254 4671*7^(1/2), d[8,3] = -7625009568771142617570744472154927592785083145 7556362286280073529296875/63200730195562080382515095400480778094111584 6998678039515960943115998-26456361535933405820655294979290299863007510 302862110709158453404296875/421338201303747202550100636003205187294077 231332452026343973962077332*7^(1/2), d[15,4] = 37153197829623461490081 9651625346784235446910086141052544000000/14646075191336721287378733022 08424514541043546149605417301861-1033938740112198656813449730435356437 1909018441162307008000000/44382046034353700870844645521467409531546774 125745618706117*7^(1/2), d[11,4] = -1198100622862629546256079694150511 550870494101799806263079322013821216000/650047839315421139406191831136 850859181556352527037679017120259217719-524691072537160417643721547041 4409301918039054299216617679970826112000/11404348058165283147477049669 067558933009760570649783842405618582767*7^(1/2), d[10,7] = -4835311646 05213332827621048053052792169911733168059717563493500416000/7451595660 49002924805528153958871197929260818114161914940179650083*7^(1/2)-48250 4110415092660293138266356471191063590434946166327068843254107648000/57 377286585773225210025667854833082240553082994790467450393833056391, d[ 18,3] = -252678416666023204018532478629807794349483090715026166341375/ 1431678904333990350672407920047335791340218520185342538907*7^(1/2)+206 7355049882807580588801884762356866603197474794104605922875/28633578086 67980701344815840094671582680437040370685077814, d[16,7] = -1366191469 8306229195729216969711744974728895376228541750000000/12885110139005913 156051671280426022122061966681668082850163+232083093008364823648443315 110295386092801194235238000000000/128851101390059131560516712804260221 22061966681668082850163*7^(1/2), d[18,5] = 115171192587764373750584763 24737880745366423251156991761578500/1431678904333990350672407920047335 791340218520185342538907-200127195041823647282315828529801511539224593 8807932725791125/14316789043339903506724079200473357913402185201853425 38907*7^(1/2), d[16,8] = -58247520424623867073113342801624660418662229 380970000000000/128851101390059131560516712804260221220619666816680828 50163*7^(1/2)+34288264824201544624108500185364959092541434114322762500 00000/12885110139005913156051671280426022122061966681668082850163, d[2 0,2] = -18724388334826048809573294764906626932882278291570734642700/11 13528036704214717189650604481261171042392182366377530261+1185358024193 049429204894939437968202309486148704187724700/159075433814887816741378 657783037310148913168909482504323*7^(1/2), d[8,7] = -37578433507644725 022360307728310855961544565257002421026817336669921875/105334550325936 800637525159000801296823519307833113006585993490519333*7^(1/2)-1460984 971666740349835101113156446918665956503535718197130879330322265625/316 003650977810401912575477002403890470557923499339019757980471557999, d[ 16,6] = -377812617364563816781274652988438993015682333907315400000000/ 12885110139005913156051671280426022122061966681668082850163*7^(1/2)+22 240498794939982278338073459970930766377295148098911525000000/128851101 39005913156051671280426022122061966681668082850163, d[1,6] = -40182236 816895046261664896973126676995030163633026716312006139827990900/982689 27242899447958931819697685792861317402784447583992645622313311+8759327 9292444328573400503228921644809254715323653677309899691125/12056057814 120899025755345319308771054020046961654715248760351161*7^(1/2), d[12,2 ] = 220462599128642020502802009787550092329168455670656206543685061/55 71753768189237374963973913577470011360871707104440929211789898+3819415 383218199980396074693157583421412619088899955423132510917/222870150727 56949499855895654309880045443486828417763716847159592*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(d8,matrix([seq([seq(d[j,i],j= 1..11)],i=1..8)])):\nevalf[8](%);\nsubs(d8,matrix([seq([seq(d[j,i],j=1 2..20)],i=1..8)])):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7*7-$\"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+7-$!)d>=6!\"'F+F+F+F +$!)oJ\\9!\"($!)\\u?5F2$\")9XT@F/$!)D%)4@F/$\")G2&*QF/$!))z-7%F/7-$\") Jv.iF/F+F+F+F+$\")s)3%>F/$\")\\&pO\"F/$!)exnG!\"&$\")fWDGFF$!))zh@&FF$ \")7xFFF+F+F+F+$!)a[w5FF$!))>;e(F/$\")4d!f\"!\"%$!)K4n:F V$\")'zI*GFV$!)INgIFV7-$\")R&Hh$FFF+F+F+F+$\")AW&4$FF$\")35!=#FF$!)\"* ptXFV$\")!*=1XFV$!)M2>$)FV$\")\"p+!))FV7-$!)/y'*QFFF+F+F+F+$!)wp,[FF$! )f!=Q$FF$\")Dz%4(FV$!)(p+*pFV$\")uY!H\"!\"$$!).3l8Fcp7-$\")')f_AFFF+F+ F+F+$\")M$yw$FF$\")nbFV$\")A-&[&FV$!)Sh75Fcp$\")=;r5Fcp7-$! )[f!Q&F/F+F+F+F+$!)(\\6;\"FF$!)`!z<)F/$\")um:F9Q)pprint196\"" }}}{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(S O8_20[ct],'polynom_order_conditions',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 "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 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 37 "e_u := map(_U->lhs(_U)=rhs(_ U)/u,e8):" }}}{PARA 0 "" 0 "" {TEXT -1 43 " ... and the weight polynom ials (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(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 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_20 := PrincipalErr orTerms(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 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. 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0 "" {TEXT -1 28 "inte rpolation coefficients " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "cffs := [seq(seq(d[j,i]=subs(d8,d[ j,i]),j=1..20),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$\"\" \"\"\"#,&#\"`oxglRXAbz`Q^&4I\"\\9uN/H`lmm:k^Pg%4IM\"*\\\"_o;f!))\\Uy5: y3$zc*H]guD35VOO1(o(>iu(y]nV!\"\"*(\"in:cK$ej=y/b6/Y%G!p+6;zxHR(R)>-.o Vc$F'\"jn%*G/'y'*))\\1$3=Y(RTl^&HS()yZ&fh2*4%*>t#QF-\"\"(#F'F(F'" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"\"'\"\"#,&#\"`oDEhp\"e]jdI \"3l+\\v^!pM^y(*\\ \"e'\\.0Py(4.!\\;7'y!RfB*R)>Mi1?pCY7q%F-\"\"(#F0F(F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"\"(\"\"#,&*(\"eo++](=nQr@fM)o3+%*Q*y&*f#R =FE5'e!fyE&**z$3$f7&\"\"\"\"foFqQ&H&)[wSqBk2mY_>JvzlOp#QFuhY%)p6uQ#pdW 9!\"\"F'#F,F(F.#\"fo+++]iSTuq7\"*GrD.JlR>*4%)Q\"HbMT*yoV(QL%H!>>*\"ho( 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"6#/&%\"dG6$\"#?\"\"#,&#\"fn+FkM2 d\"HyA)G$pi1\\w%Ht&4)[g#[L)QC(=\"enh-`xjO#=#RU5C!e`=\"\"\"\"\"ZBV]#[4*oJ\"*[,JPIy d'y8u;y)[\"QVv!f\"F-\"\"(#F0F(F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/& %\"dG6$\"\"\"\"\"$,&*(\"]oNP6bUuBQ+&\\5G\"owp1.!p&es$RB*Q6Nd%4BY#F'\"] o'=I\"pxY'fAI+@7_/u/G?Iy*Q'pYPVr;.vX8$!\"\"\"\"(#F'\"\"#F-#\"`o\"f#\\0 g.l2Fk.mrv_SD2(o8$yBVBX;EJ?&\\2JK\"^o)H&yE=)R(4nr*H'z.e+j1rMMMdw#zNDIm @VR]F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"\"'\"\"$,&#\"boDE 1qD4$G#39![^DWJ-*pe*e6!)HZyxYV%ygV3@Z%\"aooGX1_d;646S8*QdYC#[F-\"_oO<<+ $))=63L\"HS!p(3\"eM0oC-'ya#3YxUod$yM6!\"\"\"\"(#F-\"\"#F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"\"(\"\"$,&*(\"io++](oa&=:C!fT9q*z\\U upQD#Q@t*4()fK&zwA*>Jj0TN\"\"\"\"io8^(>#p0TkQO,\"f;:q!*[!y\")G7hUEInzx -\"=!Hl+&3J=\"!\"\"F'#F,\"\"#F,#\"[p+++D1k\"=\"3^5a-GE=q)e!4FuVAF,ydtO 6y6\"Hl>&3%z'\"[pT\"Hsj7Thu[!zf\"Gdw&3u$\\>*pg$)QO18(fS$4p>lX$=9=\"F, " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"\")\"\"$,&#\"bovoHHN2!G 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"6#/&%\"dG6$\"#:\"\"&,&#\"jn+++k;K7vA\"pB]qF9IR?\"fTj/OnVI $*Qrs(\"hnh=Ivgk9!\"\"*(\"in+++[oSQns2EBFJ(* H/JfndN;)>[/[!R]@\"\"\"\"fn[Zh&*f;=egS^D*GPVJ+]/>\"fnj,&G3o;om>1A@-E/Gr;0cJ\"f+R,6&)G\"!\" \"*(\"gn+++#4(eaK%\\/K+P AR^'*eu(*z[#**[4RlHh/Hyi7XSG;\"gndj:J_0&e+nTgF'[hO*z^,'381B@aH>D]!\"\" *(\"[o+7&['*4#=Y)GZPPMdrr^6\"en2*QDM&=?&=-M\"zNt/?zSs1N!*RL/*y;V\"\"\"\"* (\"hnD6zDF$z!)QfC#R:^,)H&GeJ#GZO#=/&>F,?F-F,!\"\"\"\"(#F-\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"dG6$\"#>\"\"&,&#\"jn+N_[\\wm`!Rg \\0\\^lH\"4v:Q?D6>*RX**4'f#\"fn\\Bx(R(HT'H:QR0NJ.WboqaC$zLIBv@+\"\"\" \"*(\"hn+&GXop`f&[n.m(**z'y_1``L)GKi_t=-,jF-\"en$y!fK\"*4ZlA&)e\\=**pD%)=p(4\">0Gul,o!*)Ry0T(\"fn\\ Bx(R(HT'H:QR0NJ.WboqaC$zLIBv@+\"\"\"\"*(\"hn+pvOxkA#)3tI(G&yra$Hyo'HEo =Frl$*)))\\F-F,!\"\"\"\"(#F-\"\"#F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/&%\"dG6$\"\"\"\"\"',&#\"bo+4*z#)Rh+7jrEIjj,.&*pnEJ(p*[mhi/&*o\"oB#=S \"_o6LJAck#*ReZWy-uJhGz&o(p>=$*ezW**GCF*o#)*!\"\"*(\"\\oD6p**)4tn`OK:Z 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "seq( d[1,i]=subs(d8,d[1,i]),i=1..8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6*/&% \"dG6$\"\"\"F'F'/&F%6$F'\"\"#,&#\"`oxglRXAbz`Q^&4I\"\\9uN/H`lmm:k^Pg%4 IM\"*\\\"_o;f!))\\Uy5:y3$zc*H]guD35VOO1(o(>iu(y]nV!\"\"*(\"in:cK$ej=y/ b6/Y%G!p+6;zxHR(R)>-.oVc$F'\"jn%*G/'y'*))\\1$3=Y(RTl^&HS()yZ&fh2*4%*>t #QF0\"\"(#F'F+F'/&F%6$F'\"\"$,&*(\"]oNP6bUuBQ+&\\5G\"owp1.!p&es$RB*Q6N d%4BY#F'\"]o'=I\"pxY'fAI+@7_/u/G?Iy*Q'pYPVr;.vX8$F0F4F5F0#\"`o\"f#\\0g .l2Fk.mrv_SD2(o8$yBVBX;EJ?&\\2JK\"^o)H&yE=)R(4nr*H'z.e+j1rMMMdw#zNDIm@ VR]F'/&F%6$F'\"\"%,&#\"cor/2co5kH8UI:9\"HU%)>U0#z4H7NCtezrDN**G%f\"\"` o)[1&y\\;T>n!evAAR0*GM'[\"edXXrOe&>VzT^hyF0*(\"]o&f2d(G-en0Vsl\"=yW&3$ \\!)e]'=_v4\"p(3MbR8F'\"\\o)f=q'R#4Pg=dJI]V`VvXh#G\"4n1\">f4LkzZ%F0F4F 5F'/&F%6$F'\"\"&,&#\"boR8]O%)\\A%=7>\")z#osa@B@3[6[N*e@zzAMj\"RVU(\"`o AmiW7H&)z;&*)ob![jAder`RRO'y\"f*))z&[ay`'>F'*(\"^olB:J\"e)z()46;vX'eFY Vd8'pE$e5^bu'zZ1-&>F'F=F0F4F5F0/&F%6$F'\"\"',&#\"bo+4*z#)Rh+7jrEIjj,.& *pnEJ(p*[mhi/&*o\"oB#=S\"_o6LJAck#*ReZWy-uJhGz&o(p>=$*ezW**GCF*o#)*F0* (\"\\oD6p**)4tn`OK:ZD4[k@*GK]+MdGVW#HzKf()F'\"\\oh6Ng([_ra;'p/?S0r(3$> `Mbd-**379y0c?\"F0F4F5F'/&F%6$F'F4,&#\"co](f'*o[8U-,2t3.Jz6VFX)R!4yw!) 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[7 stage scheme] .. (shorter method)" }}{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 3978 "e1 := \{c[2]=1/16,c[3]=112/1065,c[4]=56 /355,c[5]=39/100,c[6]=7/15,c[7]=39/250,c[8]=24/25,\n c[9]=14435868/16 178861,c[10]=11/12,c[11]=19/20,c[12]=1,c[13]=1,\n a[2,1]=1/16,a[3,1]= 18928/1134225,a[3,2]=100352/1134225,a[4,1]=14/355,a[4,2]=0,\n a[4,3]= 42/355,\n a[5,1]=94495479/250880000,a[5,2]=0,a[5,3]=-352806597/250880 000,a[5,4]=178077159/125440000,\n a[6,1]=12089/252720,a[6,2]=0,a[6,3] =0,a[6,4]=2505377/10685520,a[6,5]=960400/5209191,\n a[7,1]=21400899/3 50000000,a[7,2]=0,a[7,3]=0,a[7,4]=3064329829899/27126050000000,\n a[7 ,5]=-21643947/592609375,a[7,6]=124391943/6756250000,\n a[8,1]=-153654 58811/13609565775,a[8,2]=0,a[8,3]=0,a[8,4]=-7/5,\n a[8,5]=-8339128164 608/939060038475,a[8,6]=341936800488/47951126225,\n a[8,7]=1993321838 240/380523459069,\n a[9,1]=-18409112522823765844381573364647084269547 28061551/\n 2991923615171151921596253813483118262 195533733898,a[9,2]=0,a[9,3]=0,\n a[9,4]=-147649608040486573036383722 52908780219281424435/\n 298169210256502197561171126920 9606363661854518,\n a[9,5]=-87532504850213044111861342178526674286269 4404520560000/\n 170212030428894418395571677575961339495 435011888324169,\n a[9,6]=7632051964154290925661849798370645637589377 834346780/\n 1734087257418811583049800347581865260479233 950396659,\n a[9,7]=7519834791971137517048532179652347729899303513750 000/\n 1045677303502317596597890707812349832637339039997 351,\n a[9,8]=1366042683489166351293315549358278750/\n \+ 144631418224267718165055326464180836641,\n a[10,1]=-6307773670525428 0154824845013881/78369357853786633855112190394368,\n a[10,2]=0,a[10,3 ]=0,a[10,4]=-31948346510820970247215/6956009216960026632192,\n a[10,5 ]=-3378604805394255292453489375/517042670569824692230499952,\n a[10,6 ]=1001587844183325981198091450220795/184232684207722503701669953872896 ,\n a[10,7]=187023075231349900768014890274453125/25224698849808178010 752575653374848,\n a[10,8]=1908158550070998850625/1170870670391899293 94176,\n a[10,9]=-52956818288156668227044990077324877908565/\n \+ 2912779959477433986349822224412353951940608,\n a[11,1]=-1 0116106591826909534781157993685116703/95628199450368940304422314118717 44000,\n a[11,2]=0,a[11,3]=0,a[11,4]=-9623541317323077848129/38644495 64977792573440,\n a[11,5]=-4823348333146829406881375/5764132336341412 39944816,\n a[11,6]=6566119246514996884067001154977284529/97030548702 1846325473990863582315520,\n a[11,7]=22264551305192135492560168925067 30559375/364880443159675255577435648380047355776,\n a[11,8]=397472627 82380466933662225/1756032802431424164410720256,\n a[11,9]=48175771419 260955335244683805171548038966866545122229/\n 1989786420513 815146528880165952064118903852843612160000,\n a[11,10]=-2378292068163 246/47768728487211875,\n a[12,1]=-3218022174758599831659045535578571/ 1453396753634469525663775847094384,\n a[12,2]=0,a[12,3]=0,a[12,4]=262 90092604284231996745/5760876126062860430544,\n a[12,5]=-6970692975609 26452045586710000/41107967755245430594036502319,\n a[12,6]=1827357820 434213461438077550902273440/139381013914245317709567680839641697,\n a [12,7]=643504802814241550941949227194107500000/24212460911883655086049 4007545333945331,\n a[12,8]=162259938151380266113750/5909108283524418 3497007,\n a[12,9]=-2302825163287352381854541485685701561667857555413 0463402/\n 2001316918319144450344390524040560334997842450 4151629055,\n a[12,10]=7958341351371843889152/3284467988443203581305, \n a[12,11]=-507974327957860843878400/121555654819179042718967,\n a[ 13,1]=4631674879841/103782082379976,a[13,2]=0,a[13,3]=0,a[13,4]=0,a[13 ,5]=0,\n a[13,6]=14327219974204125/40489566827933216,\n a[13,7]=2720 762324010009765625000/10917367480696813922225349,\n a[13,8]=-49853300 5859375/95352091037424,\n a[13,9]=40593203046377724792670503059617543 7402459637909765779/\n 788039194363218410832018860412015 37229769115088303952,\n a[13,10]=-10290327637248/1082076946951,\n a[ 13,11]=863264105888000/85814662253313,\n a[13,12]=-29746300739/247142 463456\}:" }}}{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%'matrixG6#7.7/$\"++++]i!#6F(%!GF+F+F+F+F+F+F+F+F+F+7/$\"+#>V;0 \"!#5$\"+!p/)o;F*$\"+MsiZ))F*F+F+F+F+F+F+F+F+F+F+7/$\"+*ykud\"F/$\"+s> mVRF*$\"\"!F:$\"+#f)4$=\"F/F+F+F+F+F+F+F+F+F+7/$\"+++++RF/$\"+m3cmPF/F 9$!+&HwiS\"!\"*$\"+4-i>9FDF+F+F+F+F+F+F+F+7/$\"+nmmmYF/$\"+A\\b$y%F*F9 F9$\"+unkWBF/$\"++WmV=F/F+F+F+F+F+F+F+7/$\"++++g:F/$\"+rDa9hF*F9F9$\"+ @JmH6F/$!+AEJ_OF*$\"+U)Q6%=F*F+F+F+F+F+F+7/$\"+++++'*F/$!+M!>!H6FDF9F9 $!+++++9FD$!+qGH!)))FD$\"+(4W48(FD$\"+3yOQ_FDF+F+F+F+F+7/$\"+;EnA*)F/$ !+K`$H:'F/F9F9$!+jI(=&\\FD$!+#)obU^FD$\"+wD>,WFD$\"+rJN\">(FD$\"+gE*\\ W*!#7F+F+F+F+7/$\"+nmmm\"*F/$!+&\\v([!)F/F9F9$!+%>8Hf%FD$!+V'zW`'FD$\" +;s`OaFD$\"+oPG9uFD$\"+))>pH;F*$!+&4&3==F*F+F+F+7/$\"+++++&*F/$!+1\"ey 0\"FDF9F9$!+(yu-\\#FD$!+k`'yO)FD$\"+9R1nnFD$\"+ed(=5'FD$\"+m#pME#F*$\" +!*G:@CF*$!+/Owy\\F*F+F+7/$\"\"\"F:$!+@)QT@#FDF9F9$\"+Z!eNc%FD$!+@Oq&p \"!\")$\"+z@068F_s$\"+ZAudEFD$\"+T$Hfu#FD$!+A\\l]6FD$\"+hH-BCFD$!+TW%* yTFDF+7/Fgr$\"+(*[)GY%F*F9F9F9F9$\"+Gn\\QNF/$\"+y89#\\#F/$!+r%Q$G_FD$ \"+#\\l6:&FD$!+sGz4&*FD$\"+9M'f+\"F_s$!+i%4O?\"F/Q)pprint206\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "convert(ListTools[Enumerate](SimpleOrderConditions(7)),matrix):\n linalg[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(F2F(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#*(FF(#F(FTQ)pprint236\"" }}}{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 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_13 := SimpleOrderConditions(7,1 3,'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_ord er_eqns14 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_13[ct] ,'interpolation_order_condition'):\n interp_order_eqns14 := [op(inte rp_order_eqns14),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions c an 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],s eq(op(StageOrderConditions(i,14..14,'expanded')),i=2..7),\n add(a[1 4,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)):\nnops(eqs_14);\nin dets(eqs_14);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<,&%\"aG6$\"#9\"\"(&F%6$F'\"\"'&F%6$F' \"#8&F%6$F'\"#6&%\"cG6#F'&F%6$F'\"\"\"&F%6$F'\"#7&F%6$F'\"\")&F%6$F'\" #5&F%6$F'\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We solve for the lin king coefficients in terms of " }{XPPEDIT 18 0 "c[14];" "6#&%\"cG6#\" #9" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infole vel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "sol_14 := solve(\{op(eqs_14)\},indets(eqs_14) minus \{c[14]\}):\ninfolevel[solv e]:=0:" }}}{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 sati sfied." }}{PARA 0 "" 0 "" {TEXT -1 6 "EITHER" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "extra_eqn := 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:" }}}{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 "ex pand(subs(e2,extra_eqn)):\neq_14 := subs(sol_14,%);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%&eq_14G/,,*&#\"?,#pC'y1TRIf4-a]L\"@?hDAsv^`H=7a)3!) p\"\"\"*$)&%\"cG6#\"#9\"\"'F+F+F+*&#\"M;lQSW\\\">,)=Nkk7Y_,^/u ;eF+*$)F.\"\"#F+F+F+*&#\"=+on+WcQoKIx!))))*\"?.kbIRz$Qd/`8A]u\"F+*$)F. \"\"$F+F+!\"\"*&#\"=+#p,5T'4?sC\">ni]9#)3([F+*$)F.\"\"%F+F +F+*&#\"=gPZ!3&*pyG7Tl@#pF6F+*$)F.\"\"&F+F+FA,$*&#F+\"$?\"F+F,F+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#,,*&#\"=gPZ!3&*pyG7Tl@#p\"?.kbIRz$Qd/`8A]u\"\"\"\"*$)&%\"cG6#\"# 9\"\"'F(F(F(*&#\"M;lQSW\\\">,)=Nkk7Y_,^/u;eF(*$)F+\"\"#F(F(F(* &#\"=+on+WcQoKIx!))))*F'F(*$)F+\"\"$F(F(!\"\"*&#\"=+#p,5T'4?sC\">n i]9#)3([F(*$)F+\"\"%F(F(F(*&#\"=gPZ!3&*pyG7Tl@#pF3F(*$)F+\"\"& F(F(F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"M;lQSW\\\"?.k bIRz$Qd/`8A]u\"\"\"\"*()&%\"cG6#\"#9\"\"#F(,(*&F.F(F*F(F(*&F.F(F+F(!\" \"\"\"$F(F(),&F+F(F(F3F/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;\nsolv e(%);" }}{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 coefficients for this stage." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "e3 := \{c[14]=1/2-1/14*7^(1/2)\}:\ne4 := solve(\{o p(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]=s ubs(e5,c[14]),seq(a[14,i]=subs(e5,a[14,i]),i=1..13)]:\nevalf[20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&%\"cG6#\"#9$\"5R'Q'Q&\\jx,6$!#?/ &%\"aG6$F(\"\"\"$\"5s%4+P#p+yjY!#@/&F.6$F(\"\"#$\"\"!F9/&F.6$F(\"\"$F8 /&F.6$F(\"\"%F8/&F.6$F(\"\"&F8/&F.6$F(\"\"'$\"41!e([t6p:G%F+/&F.6$F(\" \"($\"5;/C/@93]VBF+/&F.6$F(\"\")$\"5D:;Y(RxXC!R!#>/&F.6$F(\"\"*$!5d\"3 Ml^\"Qr`!4'F X/&F.6$F(\"#7$!5n!>dR2HG>D$F3/&F.6$F(\"#8$!5$oi;'QaN]f7F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "These linking coe fficients can be compared 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 which of the (adapted) simpl e 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 := convert(SO7_13[ct],'interpolatio n_order_condition'):\n if expand(subs(e5,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop(recd);\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 4966 "e5 := \{a[14,2] = 0, a[9,3] = 0, a[14,12] = 1100613127343/484399 22837376-29746300739/1424703612864*7^(1/2), a[7,6] = 124391943/6756250 000, a[9,8] = 1366042683489166351293315549358278750/144631418224267718 165055326464180836641, a[9,2] = 0, a[8,2] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[10,7] = \+ 187023075231349900768014890274453125/252246988498081780107525756533748 48, a[12,5] = -697069297560926452045586710000/411079677552454305940365 02319, a[7,2] = 0, a[11,4] = -9623541317323077848129/38644495649777925 73440, a[11,5] = -4823348333146829406881375/576413233634141239944816, \+ a[12,1] = -3218022174758599831659045535578571/145339675363446952566377 5847094384, a[12,3] = 0, a[12,4] = 26290092604284231996745/57608761260 62860430544, a[12,7] = 643504802814241550941949227194107500000/2421246 09118836550860494007545333945331, a[11,1] = -1011610659182690953478115 7993685116703/9562819945036894030442231411871744000, a[11,2] = 0, a[11 ,3] = 0, a[12,6] = 1827357820434213461438077550902273440/1393810139142 45317709567680839641697, a[10,8] = 1908158550070998850625/117087067039 189929394176, a[10,9] = -52956818288156668227044990077324877908565/291 2779959477433986349822224412353951940608, a[12,8] = 162259938151380266 113750/59091082835244183497007, a[12,9] = -230282516328735238185454148 56857015616678575554130463402/2001316918319144450344390524040560334997 8424504151629055, a[12,2] = 0, a[12,10] = 7958341351371843889152/32844 67988443203581305, a[12,11] = -507974327957860843878400/12155565481917 9042718967, a[11,8] = 39747262782380466933662225/175603280243142416441 0720256, a[11,9] = 481757714192609553352446838051715480389668665451222 29/1989786420513815146528880165952064118903852843612160000, a[11,10] = -2378292068163246/47768728487211875, a[11,6] = 6566119246514996884067 001154977284529/970305487021846325473990863582315520, a[11,7] = 222645 5130519213549256016892506730559375/36488044315967525557743564838004735 5776, a[3,2] = 100352/1134225, a[13,3] = 0, a[3,1] = 18928/1134225, a[ 13,6] = 14327219974204125/40489566827933216, a[13,7] = 272076232401000 9765625000/10917367480696813922225349, a[13,8] = -498533005859375/9535 2091037424, a[10,1] = -63077736705254280154824845013881/78369357853786 633855112190394368, a[10,2] = 0, a[13,11] = 863264105888000/8581466225 3313, a[6,1] = 12089/252720, a[6,2] = 0, a[5,4] = 178077159/125440000, a[13,12] = -29746300739/247142463456, c[11] = 19/20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[10,6] = 1001587844183325981198091450220795/1842 32684207722503701669953872896, a[10,3] = 0, a[7,5] = -21643947/5926093 75, a[7,4] = 3064329829899/27126050000000, a[9,4] = -14764960804048657 303638372252908780219281424435/298169210256502197561171126920960636366 1854518, a[8,7] = 1993321838240/380523459069, a[9,1] = -18409112522823 76584438157336464708426954728061551/2991923615171151921596253813483118 262195533733898, a[13,4] = 0, a[13,5] = 0, a[9,5] = -87532504850213044 1118613421785266742862694404520560000/17021203042889441839557167757596 1339495435011888324169, a[8,1] = -15365458811/13609565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8,6] = 3419368 00488/47951126225, a[10,4] = -31948346510820970247215/6956009216960026 632192, a[10,5] = -3378604805394255292453489375/5170426705698246922304 99952, a[13,9] = 40593203046377724792670503059617543740245963790976577 9/78803919436321841083201886041201537229769115088303952, a[13,10] = -1 0290327637248/1082076946951, a[4,3] = 42/355, a[5,1] = 94495479/250880 000, a[4,1] = 14/355, a[9,6] = 763205196415429092566184979837064563758 9377834346780/1734087257418811583049800347581865260479233950396659, a[ 9,7] = 7519834791971137517048532179652347729899303513750000/1045677303 502317596597890707812349832637339039997351, a[6,3] = 0, a[6,4] = 25053 77/10685520, a[6,5] = 960400/5209191, a[7,1] = 21400899/350000000, a[7 ,3] = 0, c[2] = 1/16, a[5,2] = 0, a[5,3] = -352806597/250880000, a[13, 1] = 4631674879841/103782082379976, a[13,2] = 0, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, a[14,3] = 0, a[14,5] \+ = 0, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[14, 7] = 129834778101757666015625000/534951006554143882189042101-827664115 29674389648437500/26212599321153050227263062949*7^(1/2), a[14,1] = 923 507123432989/20341288146475296+3728619917660047/7973784953418316032*7^ (1/2), a[14,11] = -5071676622092000/4204918450412337-38037899804633200 0/206041004070204513*7^(1/2), a[14,6] = 1533012537239841375/7935955098 274910336-11049456415304617875/194430899907735303232*7^(1/2), a[14,8] \+ = 13061564753515625/18689009843335104+31047286856168359375/25641321505 055762688*7^(1/2), a[14,9] = -3211799428826566849882311586102611111468 427562764815533/154455682095190808523075696640755012970347465573075745 92-35609926058546725508440519811766382833790503793377686129/6359939850 9784450568325286852075593576025427000678248320*7^(1/2), a[14,10] = 373 02437685024/53021770400599+15742143124443936/12990333748146755*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 conditi ons can be adapted to give a method of stage by stage construction for an interpolation scheme that avoids dealing with 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 0 219 "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],'interpolati on_order_condition'):\n interp_order_eqns15 := [op(interp_order_eqns 15),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 0 439 "interp_order_eqns15 := [add(a[15,i],i=1..14)=c[15],seq(op(StageOr derConditions(i,15..15,'expanded')),i=2..7),\n add(a[15,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..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[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" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "We specify the node for this stage immedi ately, namely " }{XPPEDIT 18 0 "c[15] = 9/20;" "6#/&%\"cG6#\"#:*&\"\" *\"\"\"\"#?!\"\"" }{TEXT -1 80 ", and have enough equations to determi ne the corresponding 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$\"#:\"#6&F%6$F'\"\"*&F%6$F' \"#9&F%6$F'\"#7&F%6$F'\"#8&F%6$F'\"\"\"&F%6$F'\"\"'&F%6$F'\"\"(&F%6$F' \"\")&F%6$F'\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infol evel[solve] := 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[20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&% \"aG6$\"#:\"\"\"$\"5=AnYLDQJi\\!#@/&F&6$F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1 /&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F&6$F(\"\"'$\"5AZ=&*feiwIvF,/&F&6$F(\" \"($\"5XQM_`1&eV7#!#?/&F&6$F(\"\")$!4K6[5@OD<(**!#>/&F&6$F(\"\"*$\"5Eo xr*4kFY:$FK/&F&6$F(\"#5$!4o]a, " 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 6015 "e8 := \{a[14,2] = 0, a[9,3] = 0, a[14,12] = \+ 1100613127343/48439922837376-29746300739/1424703612864*7^(1/2), a[7,6] = 124391943/6756250000, a[9,8] = 136604268348916635129331554935827875 0/144631418224267718165055326464180836641, a[9,2] = 0, a[8,2] = 0, a[1 4,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[4,2] = 0, c[14] = 1/2-1/14 *7^(1/2), a[10,7] = 187023075231349900768014890274453125/2522469884980 8178010752575653374848, a[12,5] = -697069297560926452045586710000/4110 7967755245430594036502319, a[7,2] = 0, a[11,4] = -96235413173230778481 29/3864449564977792573440, a[11,5] = -4823348333146829406881375/576413 233634141239944816, a[12,1] = -3218022174758599831659045535578571/1453 396753634469525663775847094384, a[12,3] = 0, a[12,4] = 262900926042842 31996745/5760876126062860430544, a[12,7] = 643504802814241550941949227 194107500000/242124609118836550860494007545333945331, a[11,1] = -10116 106591826909534781157993685116703/956281994503689403044223141187174400 0, a[11,2] = 0, a[11,3] = 0, a[12,6] = 1827357820434213461438077550902 273440/139381013914245317709567680839641697, a[10,8] = 190815855007099 8850625/117087067039189929394176, a[10,9] = -5295681828815666822704499 0077324877908565/2912779959477433986349822224412353951940608, a[12,8] \+ = 162259938151380266113750/59091082835244183497007, a[12,9] = -2302825 1632873523818545414856857015616678575554130463402/20013169183191444503 443905240405603349978424504151629055, a[12,2] = 0, a[12,10] = 79583413 51371843889152/3284467988443203581305, a[12,11] = -5079743279578608438 78400/121555654819179042718967, a[11,8] = 39747262782380466933662225/1 756032802431424164410720256, a[11,9] = 4817577141926095533524468380517 1548038966866545122229/19897864205138151465288801659520641189038528436 12160000, a[11,10] = -2378292068163246/47768728487211875, a[11,6] = 65 66119246514996884067001154977284529/9703054870218463254739908635823155 20, a[11,7] = 2226455130519213549256016892506730559375/364880443159675 255577435648380047355776, a[3,2] = 100352/1134225, a[13,3] = 0, a[3,1] = 18928/1134225, a[13,6] = 14327219974204125/40489566827933216, a[13, 7] = 2720762324010009765625000/10917367480696813922225349, a[13,8] = - 498533005859375/95352091037424, a[10,1] = -630777367052542801548248450 13881/78369357853786633855112190394368, a[10,2] = 0, a[13,11] = 863264 105888000/85814662253313, a[6,1] = 12089/252720, a[6,2] = 0, a[5,4] = \+ 178077159/125440000, a[13,12] = -29746300739/247142463456, c[11] = 19/ 20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[10,6] = 1001587844183325981 198091450220795/184232684207722503701669953872896, a[10,3] = 0, a[7,5] = -21643947/592609375, a[7,4] = 3064329829899/27126050000000, a[9,4] \+ = -14764960804048657303638372252908780219281424435/2981692102565021975 611711269209606363661854518, a[8,7] = 1993321838240/380523459069, a[9, 1] = -1840911252282376584438157336464708426954728061551/29919236151711 51921596253813483118262195533733898, a[13,4] = 0, a[13,5] = 0, a[9,5] \+ = -875325048502130441118613421785266742862694404520560000/170212030428 894418395571677575961339495435011888324169, a[8,1] = -15365458811/1360 9565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -8339128164608/9390600384 75, a[8,6] = 341936800488/47951126225, a[10,4] = -31948346510820970247 215/6956009216960026632192, a[10,5] = -3378604805394255292453489375/51 7042670569824692230499952, a[13,9] = 405932030463777247926705030596175 437402459637909765779/788039194363218410832018860412015372297691150883 03952, a[13,10] = -10290327637248/1082076946951, a[4,3] = 42/355, a[5, 1] = 94495479/250880000, a[4,1] = 14/355, a[9,6] = 7632051964154290925 661849798370645637589377834346780/173408725741881158304980034758186526 0479233950396659, a[9,7] = 7519834791971137517048532179652347729899303 513750000/1045677303502317596597890707812349832637339039997351, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a[7,1] = 214 00899/350000000, a[7,3] = 0, c[2] = 1/16, a[5,2] = 0, a[5,3] = -352806 597/250880000, a[13,1] = 4631674879841/103782082379976, a[13,2] = 0, c [7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, a [14,3] = 0, a[14,5] = 0, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100 , c[6] = 7/15, a[14,7] = 129834778101757666015625000/53495100655414388 2189042101-82766411529674389648437500/26212599321153050227263062949*7^ (1/2), a[15,2] = 0, a[15,3] = 0, a[15,4] = 0, a[14,1] = 92350712343298 9/20341288146475296+3728619917660047/7973784953418316032*7^(1/2), a[14 ,11] = -5071676622092000/4204918450412337-380378998046332000/206041004 070204513*7^(1/2), a[14,6] = 1533012537239841375/7935955098274910336-1 1049456415304617875/194430899907735303232*7^(1/2), a[14,8] = 130615647 53515625/18689009843335104+31047286856168359375/25641321505055762688*7 ^(1/2), a[14,9] = -321179942882656684988231158610261111146842756276481 5533/15445568209519080852307569664075501297034746557307574592-35609926 058546725508440519811766382833790503793377686129/635993985097844505683 25286852075593576025427000678248320*7^(1/2), a[14,10] = 37302437685024 /53021770400599+15742143124443936/12990333748146755*7^(1/2), c[15] = 9 /20, a[15,5] = 0, a[15,9] = -24605573253403773914515339074815619398214 07244475387801507/2594367718068208759940802832632149373819559344470912 000000+199091459817160514703569453039285921263305239484481767/41676757 2250203215264447300211189502579457643520000000*7^(1/2), a[15,8] = 1552 0930467726675/5625380975771648-223623196267702275/157510667321606144*7 ^(1/2), a[15,7] = 253006492725129434765625/1078258516612031004664232-4 6193539913722265625/5503324547053749203536*7^(1/2), a[15,6] = 76845735 1210057567857/5182664553975451648000-62532827932224226599/226741574236 4260096000*7^(1/2), a[15,1] = 3791756097508153347/82000657682944000000 +821104665636321/642242913320960000*7^(1/2), a[15,12] = -1539638779949 901/263618627686400000, a[15,11] = -61485359208535917/1430244370888550 0+887648262380253891/400468423848794000*7^(1/2), a[15,10] = 4260591521 3080224/16907452296109375-1106650521763256349/860743025983750000*7^(1/ 2), a[15,14] = 708939/16000000*7^(1/2), a[15,13] = 101277/64000000*7^( 1/2)-79893/16000000\}: " }{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 (simpl e) 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 \"approximat e\" 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] = 1/10;" "6#/&%\"cG6#\"#;*&\"\"\"F)\"#5!\"\"" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "e9 := `uni on`(e8,\{c[16]=1/10,seq(a[16,i]=0,i=2..5)\}):\neqs_16 := expand(subs(e 9,interp_order_eqns16)):\nnops(eqs_16);\nindets(eqs_16);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<-&%\"aG6$\"#;\"#8&F%6$F'\"#5&F%6$F'\"\")&F%6$F'\"#7&F%6$F'\"\"*&F%6 $F'\"#9&F%6$F'\"\"(&F%6$F'\"#:&F%6$F'\"#6&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[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "e11 := `union`(e9,e10):\n[seq(a[16, i]=subs(e11,a[16,i]),i=1..15)]:\nevalf[20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#71/&%\"aG6$\"#;\"\"\"$\"5c'\\\"3=$Rr9y%!#@/&F&6$F(\"\"# $\"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F&6$F(\"\"'$\"5 -(4^FDu%*G<\"F,/&F&6$F(\"\"($\"5KK$G*=^FsU\"*F,/&F&6$F(\"\")$!5ln:)))) fc[@Q!#?/&F&6$F(\"\"*$\"5Feg\"=D+q(>7FQ/&F&6$F(\"#5$!5*\\AP5P2e0v$FQ/& F&6$F(\"#6$\"5mqTxknkaukFQ/&F&6$F(\"#7$!5\"R8*>$4lE0>\"F,/&F&6$F(\"#8$ !5F.s,\"zzF," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 " These linking coefficients can be compared 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 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 8639 "e11 := \{a[14,2] = 0, a[9,3] = 0, a[14,12] = 1100613127343/48439 922837376-29746300739/1424703612864*7^(1/2), a[7,6] = 124391943/675625 0000, a[9,8] = 1366042683489166351293315549358278750/14463141822426771 8165055326464180836641, a[9,2] = 0, a[8,2] = 0, a[14,4] = 0, a[14,13] \+ = 3/392-3/392*7^(1/2), a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[10,7] = 187023075231349900768014890274453125/25224698849808178010752575653374 848, a[12,5] = -697069297560926452045586710000/41107967755245430594036 502319, a[7,2] = 0, a[11,4] = -9623541317323077848129/3864449564977792 573440, a[11,5] = -4823348333146829406881375/576413233634141239944816, a[12,1] = -3218022174758599831659045535578571/14533967536344695256637 75847094384, a[12,3] = 0, a[12,4] = 26290092604284231996745/5760876126 062860430544, a[12,7] = 643504802814241550941949227194107500000/242124 609118836550860494007545333945331, a[11,1] = -101161065918269095347811 57993685116703/9562819945036894030442231411871744000, a[11,2] = 0, a[1 1,3] = 0, a[12,6] = 1827357820434213461438077550902273440/139381013914 245317709567680839641697, a[10,8] = 1908158550070998850625/11708706703 9189929394176, a[10,9] = -52956818288156668227044990077324877908565/29 12779959477433986349822224412353951940608, a[12,8] = 16225993815138026 6113750/59091082835244183497007, a[12,9] = -23028251632873523818545414 856857015616678575554130463402/200131691831914445034439052404056033499 78424504151629055, a[12,2] = 0, a[12,10] = 7958341351371843889152/3284 467988443203581305, a[12,11] = -507974327957860843878400/1215556548191 79042718967, a[11,8] = 39747262782380466933662225/17560328024314241644 10720256, a[11,9] = 48175771419260955335244683805171548038966866545122 229/1989786420513815146528880165952064118903852843612160000, a[11,10] \+ = -2378292068163246/47768728487211875, a[11,6] = 656611924651499688406 7001154977284529/970305487021846325473990863582315520, a[11,7] = 22264 55130519213549256016892506730559375/3648804431596752555774356483800473 55776, a[3,2] = 100352/1134225, a[13,3] = 0, a[3,1] = 18928/1134225, a [13,6] = 14327219974204125/40489566827933216, a[13,7] = 27207623240100 09765625000/10917367480696813922225349, a[13,8] = -498533005859375/953 52091037424, a[10,1] = -63077736705254280154824845013881/7836935785378 6633855112190394368, a[10,2] = 0, a[13,11] = 863264105888000/858146622 53313, a[6,1] = 12089/252720, a[6,2] = 0, a[5,4] = 178077159/125440000 , a[13,12] = -29746300739/247142463456, c[11] = 19/20, c[12] = 1, c[13 ] = 1, a[2,1] = 1/16, a[10,6] = 1001587844183325981198091450220795/184 232684207722503701669953872896, a[10,3] = 0, a[7,5] = -21643947/592609 375, a[7,4] = 3064329829899/27126050000000, a[9,4] = -1476496080404865 7303638372252908780219281424435/29816921025650219756117112692096063636 61854518, a[8,7] = 1993321838240/380523459069, a[9,1] = -1840911252282 376584438157336464708426954728061551/299192361517115192159625381348311 8262195533733898, a[13,4] = 0, a[13,5] = 0, a[9,5] = -8753250485021304 41118613421785266742862694404520560000/1702120304288944183955716775759 61339495435011888324169, a[8,1] = -15365458811/13609565775, a[8,3] = 0 , a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8,6] = 341936 800488/47951126225, a[10,4] = -31948346510820970247215/695600921696002 6632192, a[10,5] = -3378604805394255292453489375/517042670569824692230 499952, a[13,9] = 4059320304637772479267050305961754374024596379097657 79/78803919436321841083201886041201537229769115088303952, a[13,10] = - 10290327637248/1082076946951, a[4,3] = 42/355, a[5,1] = 94495479/25088 0000, a[4,1] = 14/355, a[9,6] = 76320519641542909256618497983706456375 89377834346780/1734087257418811583049800347581865260479233950396659, a [9,7] = 7519834791971137517048532179652347729899303513750000/104567730 3502317596597890707812349832637339039997351, a[6,3] = 0, a[6,4] = 2505 377/10685520, a[6,5] = 960400/5209191, a[7,1] = 21400899/350000000, a[ 7,3] = 0, c[2] = 1/16, a[5,2] = 0, a[5,3] = -352806597/250880000, a[13 ,1] = 4631674879841/103782082379976, a[13,2] = 0, c[7] = 39/250, c[8] \+ = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, a[14,3] = 0, a[14,5] = 0, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[14 ,7] = 129834778101757666015625000/534951006554143882189042101-82766411 529674389648437500/26212599321153050227263062949*7^(1/2), a[15,2] = 0, a[15,3] = 0, a[16,5] = 0, a[16,2] = 0, a[15,4] = 0, a[16,3] = 0, a[14 ,1] = 923507123432989/20341288146475296+3728619917660047/7973784953418 316032*7^(1/2), a[14,11] = -5071676622092000/4204918450412337-38037899 8046332000/206041004070204513*7^(1/2), a[14,6] = 1533012537239841375/7 935955098274910336-11049456415304617875/194430899907735303232*7^(1/2), a[14,8] = 13061564753515625/18689009843335104+31047286856168359375/25 641321505055762688*7^(1/2), a[16,11] = -144523340211331802403253407277 74770157932021057436/4240045924489341440085090167766501187355651467497 5*7^(1/2)+1379482776995886509031984497778742508172264966358576/8904096 44142761702417868935230965249344686808174475, a[16,12] = 2400779388266 9277531292945295458719370616706273/30527865734984387083739134000796184 70033568180000*7^(1/2)-83884827511307050547839026717499224825563955775 903/2564340721738688515034087256066879514828197271200000, a[14,9] = -3 211799428826566849882311586102611111468427562764815533/154455682095190 80852307569664075501297034746557307574592-3560992605854672550844051981 1766382833790503793377686129/63599398509784450568325286852075593576025 427000678248320*7^(1/2), a[14,10] = 37302437685024/53021770400599+1574 2143124443936/12990333748146755*7^(1/2), a[16,13] = -41144367294168192 86145536389728774047607/456542312392180862139685668388139491300000+167 3093837200937887039405670571696062733/91308462478436172427937133677627 8982600000*7^(1/2), a[16,4] = 0, a[16,15] = -2975918446967643161447796 202231979904/141528116841576067263302557200323242303*7^(1/2)+533032087 36809836098042063038756642432/707640584207880336316512786001616211515, a[16,14] = 1001373308041744989114984611143231300353/40206851375447746 381620044659182739290625-892131449472993412180461043706539941657/25732 384880286557684236828581876953146000*7^(1/2), c[16] = 1/10, c[15] = 9/ 20, a[15,5] = 0, a[15,9] = -246055732534037739145153390748156193982140 7244475387801507/25943677180682087599408028326321493738195593444709120 00000+199091459817160514703569453039285921263305239484481767/416767572 250203215264447300211189502579457643520000000*7^(1/2), a[15,8] = 15520 930467726675/5625380975771648-223623196267702275/157510667321606144*7^ (1/2), a[15,7] = 253006492725129434765625/1078258516612031004664232-46 193539913722265625/5503324547053749203536*7^(1/2), a[15,6] = 768457351 210057567857/5182664553975451648000-62532827932224226599/2267415742364 260096000*7^(1/2), a[15,1] = 3791756097508153347/82000657682944000000+ 821104665636321/642242913320960000*7^(1/2), a[15,12] = -15396387799499 01/263618627686400000, a[15,11] = -61485359208535917/14302443708885500 +887648262380253891/400468423848794000*7^(1/2), a[15,10] = 42605915213 080224/16907452296109375-1106650521763256349/860743025983750000*7^(1/2 ), a[16,1] = -4215121951102178392285859687667763860057798131121/552490 7006530535028430669612520227037472745107200000*7^(1/2)+214649483202843 017900825928522094099475577072470507941/430735562496636837154026079666 1082004089688904200800000, a[15,14] = 708939/16000000*7^(1/2), a[15,13 ] = 101277/64000000*7^(1/2)-79893/16000000, a[16,8] = 4482425539475296 955559304996688796936033039629875/208459775873845042444195383874932059 79265982789632*7^(1/2)-46323379005207505717635146937831732889161714866 875/48707428401292639724941806040008163970784940556544, a[16,7] = 3993 2181887340468446859878465078752005970844232914062500/79914064371853798 13952210414813050100306894094759398308201*7^(1/2)+35436451724188622846 9307408685840332599095120809109472656250/45311274498841103545109033051 98999406874008951728578840749967, a[16,6] = 11859134302486743174348706 18267722069953058737510283/3000844231717917181895561450392578605178513 1637358800*7^(1/2)-311994431911337356840024036100680775990247322848978 459/3360945539524067243723028824439688037799934743384185600, a[16,9] = 225742768490093552473689023053975168948914960175403840172271750955829 3702749265180053129/15382319963668643757680340043617345728033389807810 5274095131111501836720940560681990528000*7^(1/2)+424927747007115794173 5499442073130545124562465471930167206290592193363892663828255136238663 /511041528480475840465122234652209083659859298107142717257457071610594 26234354242200368775000, a[16,10] = 8267488680937620786773066839850328 7366098540852/729064214207911394139200255820908786902329141125*7^(1/2) -72880049498632524878182125169775730823179168857424/107957585565402264 132150807111942262675921815128125\}: " }{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 1 7" }}{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 scheme that avoids d ealing 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 := SimpleOr derConditions(7,16,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,17,25,2 7,32,61,63,64]:\ninterp_order_eqns17 := []:\nfor ct in whch do\n tem p_eqn := convert(SO7_16[ct],'interpolation_order_condition'):\n inte rp_order_eqns17 := [op(interp_order_eqns17),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 0 585 "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]*a dd(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, ##6 1\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] = 13/40;" "6#/&%\"cG6#\"# <*&\"#8\"\"\"\"#S!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "e12 := `union`(e11,\{c[17]=13/40,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$\"#<\"\"'&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 80 "e14 := `union`(e12,e13):\n[seq(a[17,i]=subs(e14,a[17,i]),i=1.. 16)]:\nevalf[20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#72/&%\"aG6$\"# <\"\"\"$\"5EW5a>Z(=>$H!#@/&F&6$F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1/&F&6$F( \"\"%F1/&F&6$F(\"\"&F1/&F&6$F(\"\"'$\"5n^A\\_&z%[!#?/&F&6$F(\"\")$\"5&y+X*Rg8C@@FK/&F&6$F(\"\"*$!5.\\]Wr$=(o p5FK/&F&6$F(\"#5$\"5=X:#Q`v'H`EFK/&F&6$F(\"#6$!5/X@UnzcB(y$FK/&F&6$F( \"#7$\"53(RN*3:\")4\"H'!#A/&F&6$F(\"#8$\"5#Q/8\\A/#H)o#Fdo/&F&6$F(\"#9 $\"5pWIbx]p%eK\"FK/&F&6$F(\"#:$!5]Xp))[HeZBhF,/&F&6$F(\"#;$\"5lJ!eL><6 UQ\"FK" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "These linking coefficients can be compared with those of the publi shed 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 (ada pted) simple order conditions are satisfied at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "rec d := []:\nfor ct to nops(SO7_16) do\n tt := convert(SO7_16[ct],'inte rpolation_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 "#-------- -------------------------------------------------------------" }} {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 10044 "e14 := \{c[8] = 24/25, c[9] = 14435868/161 78861, a[14,10] = 37302437685024/53021770400599+15742143124443936/1299 0333748146755*7^(1/2), a[9,3] = 0, a[17,4] = 0, a[17,3] = 0, a[11,4] = -9623541317323077848129/3864449564977792573440, a[14,9] = -3211799428 826566849882311586102611111468427562764815533/154455682095190808523075 69664075501297034746557307574592-3560992605854672550844051981176638283 3790503793377686129/63599398509784450568325286852075593576025427000678 248320*7^(1/2), a[11,9] = 48175771419260955335244683805171548038966866 545122229/1989786420513815146528880165952064118903852843612160000, a[9 ,4] = -14764960804048657303638372252908780219281424435/298169210256502 1975611711269209606363661854518, a[6,5] = 960400/5209191, a[14,12] = 1 100613127343/48439922837376-29746300739/1424703612864*7^(1/2), a[14,11 ] = -5071676622092000/4204918450412337-380378998046332000/206041004070 204513*7^(1/2), a[11,10] = -2378292068163246/47768728487211875, a[14,2 ] = 0, a[4,1] = 14/355, a[9,6] = 7632051964154290925661849798370645637 589377834346780/1734087257418811583049800347581865260479233950396659, \+ a[12,4] = 26290092604284231996745/5760876126062860430544, a[9,2] = 0, \+ a[8,2] = 0, a[10,9] = -52956818288156668227044990077324877908565/29127 79959477433986349822224412353951940608, a[7,6] = 124391943/6756250000, a[9,8] = 1366042683489166351293315549358278750/1446314182242677181650 55326464180836641, a[11,2] = 0, a[11,3] = 0, a[14,4] = 0, a[14,13] = 3 /392-3/392*7^(1/2), a[12,9] = -230282516328735238185454148568570156166 78575554130463402/2001316918319144450344390524040560334997842450415162 9055, a[12,2] = 0, a[7,2] = 0, a[10,7] = 18702307523134990076801489027 4453125/25224698849808178010752575653374848, a[12,5] = -69706929756092 6452045586710000/41107967755245430594036502319, a[4,2] = 0, c[14] = 1/ 2-1/14*7^(1/2), a[12,8] = 162259938151380266113750/5909108283524418349 7007, a[12,10] = 7958341351371843889152/3284467988443203581305, a[12,1 1] = -507974327957860843878400/121555654819179042718967, a[11,8] = 397 47262782380466933662225/1756032802431424164410720256, a[11,5] = -48233 48333146829406881375/576413233634141239944816, a[12,1] = -321802217475 8599831659045535578571/1453396753634469525663775847094384, a[12,6] = 1 827357820434213461438077550902273440/139381013914245317709567680839641 697, a[10,8] = 1908158550070998850625/117087067039189929394176, a[11,6 ] = 6566119246514996884067001154977284529/9703054870218463254739908635 82315520, a[12,7] = 643504802814241550941949227194107500000/2421246091 18836550860494007545333945331, a[11,1] = -1011610659182690953478115799 3685116703/9562819945036894030442231411871744000, a[12,3] = 0, a[2,1] \+ = 1/16, a[10,6] = 1001587844183325981198091450220795/18423268420772250 3701669953872896, a[10,3] = 0, a[7,4] = 3064329829899/27126050000000, \+ a[7,5] = -21643947/592609375, a[14,3] = 0, a[14,5] = 0, c[10] = 11/12, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, c[3] = 112/1065, a[13,5] = 0, a[8,3] = 0, a[8,7] = 1993321838240/380523459069, a[14,6] = 1533012 537239841375/7935955098274910336-11049456415304617875/1944308999077353 03232*7^(1/2), a[14,8] = 13061564753515625/18689009843335104+310472868 56168359375/25641321505055762688*7^(1/2), c[11] = 19/20, c[12] = 1, c[ 13] = 1, c[2] = 1/16, a[10,5] = -3378604805394255292453489375/51704267 0569824692230499952, a[13,9] = 405932030463777247926705030596175437402 459637909765779/78803919436321841083201886041201537229769115088303952, a[13,10] = -10290327637248/1082076946951, a[7,1] = 21400899/350000000 , a[14,1] = 923507123432989/20341288146475296+3728619917660047/7973784 953418316032*7^(1/2), a[14,7] = 129834778101757666015625000/5349510065 54143882189042101-82766411529674389648437500/2621259932115305022726306 2949*7^(1/2), a[6,3] = 0, a[6,4] = 2505377/10685520, a[9,7] = 75198347 91971137517048532179652347729899303513750000/1045677303502317596597890 707812349832637339039997351, a[5,3] = -352806597/250880000, a[5,2] = 0 , a[5,1] = 94495479/250880000, a[4,3] = 42/355, a[9,1] = -184091125228 2376584438157336464708426954728061551/29919236151711519215962538134831 18262195533733898, a[13,4] = 0, a[9,5] = -8753250485021304411186134217 85266742862694404520560000/1702120304288944183955716775759613394954350 11888324169, a[8,1] = -15365458811/13609565775, a[7,3] = 0, a[13,1] = \+ 4631674879841/103782082379976, a[13,2] = 0, c[7] = 39/250, a[10,4] = - 31948346510820970247215/6956009216960026632192, a[8,6] = 341936800488/ 47951126225, a[8,5] = -8339128164608/939060038475, a[8,4] = -7/5, a[11 ,7] = 2226455130519213549256016892506730559375/36488044315967525557743 5648380047355776, a[6,2] = 0, a[5,4] = 178077159/125440000, a[13,12] = -29746300739/247142463456, a[13,11] = 863264105888000/85814662253313, a[6,1] = 12089/252720, a[13,7] = 2720762324010009765625000/1091736748 0696813922225349, a[13,8] = -498533005859375/95352091037424, a[13,6] = 14327219974204125/40489566827933216, a[3,2] = 100352/1134225, a[13,3] = 0, a[3,1] = 18928/1134225, a[10,1] = -63077736705254280154824845013 881/78369357853786633855112190394368, a[10,2] = 0, a[17,10] = 17668130 783823819/66589350581600000, a[17,5] = 0, a[16,3] = 0, a[16,4] = 0, a[ 17,13] = 3949144173/1143910400000-908415729/3145753600000*7^(1/2), a[1 7,12] = 62747303504529774530387/5371467850349828505600000-729594201713 541940737/358097856689988567040000*7^(1/2), a[17,16] = 112099727493/17 64824965120+9992573019/352964993024*7^(1/2), a[17,6] = 860764742109302 032039902111/7040098361067108075752652800-227711699316765384611310459/ 7040098361067108075752652800*7^(1/2), a[17,9] = -355361387898928797854 6743631019160506304244781237959476464169103/90144623003970691762221902 742145266933120080348307809053900800000-399866634050793590138577034430 419123328121213083504583/156622849751717676376323883963632206247641290 71529984000*7^(1/2), a[17,1] = 6676290817642855930112293/1735100510764 70358835200000-674107528569903747657/194736308727800627200000*7^(1/2), a[17,8] = 17032572234499325505625/51013187037284085006336-96613381108 97182125/209930810853020925952*7^(1/2), a[17,7] = 54888690084350082907 540673828125/365048175129272014088413829892096-30419486646169688917382 8125/7899680270269138271354212352*7^(1/2), a[17,14] = 319287235181733/ 9769566771200000+29511701788023/781565341696000*7^(1/2), a[17,11] = -6 5155243311676871271269/116570087991061957996800+661671021433877476047/ 9714173999255163166400*7^(1/2), a[17,15] = -2317999437/23581712896+908 415729/64849710464*7^(1/2), a[16,2] = 0, a[16,5] = 0, a[15,2] = 0, a[1 5,4] = 0, a[15,5] = 0, a[15,3] = 0, c[17] = 13/40, a[17,2] = 0, a[16,1 1] = -14452334021133180240325340727774770157932021057436/4240045924489 3414400850901677665011873556514674975*7^(1/2)+137948277699588650903198 4497778742508172264966358576/89040964414276170241786893523096524934468 6808174475, a[16,12] = 24007793882669277531292945295458719370616706273 /3052786573498438708373913400079618470033568180000*7^(1/2)-83884827511 307050547839026717499224825563955775903/256434072173868851503408725606 6879514828197271200000, a[16,13] = -4114436729416819286145536389728774 047607/456542312392180862139685668388139491300000+16730938372009378870 39405670571696062733/913084624784361724279371336776278982600000*7^(1/2 ), a[16,15] = -2975918446967643161447796202231979904/14152811684157606 7263302557200323242303*7^(1/2)+53303208736809836098042063038756642432/ 707640584207880336316512786001616211515, a[16,14] = 100137330804174498 9114984611143231300353/40206851375447746381620044659182739290625-89213 1449472993412180461043706539941657/25732384880286557684236828581876953 146000*7^(1/2), c[16] = 1/10, a[15,1] = 3791756097508153347/8200065768 2944000000+821104665636321/642242913320960000*7^(1/2), a[15,12] = -153 9638779949901/263618627686400000, a[15,6] = 768457351210057567857/5182 664553975451648000-62532827932224226599/2267415742364260096000*7^(1/2) , a[15,7] = 253006492725129434765625/1078258516612031004664232-4619353 9913722265625/5503324547053749203536*7^(1/2), a[15,8] = 15520930467726 675/5625380975771648-223623196267702275/157510667321606144*7^(1/2), a[ 15,9] = -2460557325340377391451533907481561939821407244475387801507/25 94367718068208759940802832632149373819559344470912000000+1990914598171 60514703569453039285921263305239484481767/4167675722502032152644473002 11189502579457643520000000*7^(1/2), c[15] = 9/20, a[16,1] = -421512195 1102178392285859687667763860057798131121/55249070065305350284306696125 20227037472745107200000*7^(1/2)+21464948320284301790082592852209409947 5577072470507941/43073556249663683715402607966610820040896889042008000 00, a[15,10] = 42605915213080224/16907452296109375-1106650521763256349 /860743025983750000*7^(1/2), a[15,11] = -61485359208535917/14302443708 885500+887648262380253891/400468423848794000*7^(1/2), a[16,7] = 399321 81887340468446859878465078752005970844232914062500/7991406437185379813 952210414813050100306894094759398308201*7^(1/2)+3543645172418862284693 07408685840332599095120809109472656250/4531127449884110354510903305198 999406874008951728578840749967, a[16,6] = 1185913430248674317434870618 267722069953058737510283/300084423171791718189556145039257860517851316 37358800*7^(1/2)-31199443191133735684002403610068077599024732284897845 9/3360945539524067243723028824439688037799934743384185600, a[16,8] = 4 482425539475296955559304996688796936033039629875/208459775873845042444 19538387493205979265982789632*7^(1/2)-46323379005207505717635146937831 732889161714866875/48707428401292639724941806040008163970784940556544, a[15,13] = 101277/64000000*7^(1/2)-79893/16000000, a[15,14] = 708939/ 16000000*7^(1/2), a[16,9] = 225742768490093552473689023053975168948914 9601754038401722717509558293702749265180053129/15382319963668643757680 3400436173457280333898078105274095131111501836720940560681990528000*7^ (1/2)+4249277470071157941735499442073130545124562465471930167206290592 193363892663828255136238663/511041528480475840465122234652209083659859 29810714271725745707161059426234354242200368775000, a[16,10] = 8267488 6809376207867730668398503287366098540852/72906421420791139413920025582 0908786902329141125*7^(1/2)-728800494986325248781821251697757308231791 68857424/107957585565402264132150807111942262675921815128125\}: " } {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 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_17 := SimpleOrderConditions(7,17,'expanded'):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,17,25,27,32,61,63,64]:\ninterp_order_eqns18 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_17[ct],'interpo lation_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 spec ified 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(StageOr derConditions(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]*ad d(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]^6/6 0, ##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] = 2/5;" "6#/&%\"cG6#\"#=*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT -1 10 " a nd also " }{XPPEDIT 18 0 "a[18,17]=0" "6#/&%\"aG6$\"#=\"#<\"\"!" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "e15 := `u nion`(e14,\{c[18]=2/5,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$\"#=\"#9&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&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 80 "e17 := `u nion`(e15,e16):\n[seq(a[18,i]=subs(e17,a[18,i]),i=1..17)]:\nevalf[20]( %);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#73/&%\"aG6$\"#=\"\"\"$\"5!)[$[c Z8zN(G!#@/&F&6$F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\" \"&F1/&F&6$F(\"\"'$\"4EA8`Ki%\\QB!#?/&F&6$F(\"\"($\"4>z\"o1dMM.JFE/&F& 6$F(\"\")$\"5g!o!o)zt_lR\"FE/&F&6$F(\"\"*$!5Oa&z$=u)ej>(F,/&F&6$F(\"#5 $\"55d1P2dv,lFE/ &F&6$F(\"#:$!5&p-HW^;\"Q4CF,/&F&6$F(\"#;$\"5'\\`.+ZxYZZ\"FE/&F&6$F(\"# " 0 "" {MPLTEXT 1 0 205 "recd : = []:\nfor ct to nops(SO7_17) do\n tt := convert(SO7_17[ct],'interpo lation_order_condition'):\n if expand(subs(e17,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 "e17" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11315 "e17 := \{ a[14,2] = 0, a[9,3] = 0, a[14,12] = 1100613127343/48439922837376-29746 300739/1424703612864*7^(1/2), a[7,6] = 124391943/6756250000, a[9,8] = \+ 1366042683489166351293315549358278750/14463141822426771816505532646418 0836641, a[9,2] = 0, a[8,2] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7 ^(1/2), a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[10,7] = 18702307523134 9900768014890274453125/25224698849808178010752575653374848, a[12,5] = \+ -697069297560926452045586710000/41107967755245430594036502319, a[7,2] \+ = 0, a[11,4] = -9623541317323077848129/3864449564977792573440, a[11,5] = -4823348333146829406881375/576413233634141239944816, a[12,1] = -321 8022174758599831659045535578571/1453396753634469525663775847094384, a[ 12,3] = 0, a[12,4] = 26290092604284231996745/5760876126062860430544, a [12,7] = 643504802814241550941949227194107500000/242124609118836550860 494007545333945331, a[11,1] = -10116106591826909534781157993685116703/ 9562819945036894030442231411871744000, a[11,2] = 0, a[11,3] = 0, a[12, 6] = 1827357820434213461438077550902273440/139381013914245317709567680 839641697, a[10,8] = 1908158550070998850625/117087067039189929394176, \+ a[10,9] = -52956818288156668227044990077324877908565/29127799594774339 86349822224412353951940608, a[12,8] = 162259938151380266113750/5909108 2835244183497007, a[12,9] = -23028251632873523818545414856857015616678 575554130463402/200131691831914445034439052404056033499784245041516290 55, a[12,2] = 0, a[12,10] = 7958341351371843889152/3284467988443203581 305, a[12,11] = -507974327957860843878400/121555654819179042718967, a[ 11,8] = 39747262782380466933662225/1756032802431424164410720256, a[11, 9] = 48175771419260955335244683805171548038966866545122229/19897864205 13815146528880165952064118903852843612160000, a[11,10] = -237829206816 3246/47768728487211875, a[11,6] = 656611924651499688406700115497728452 9/970305487021846325473990863582315520, a[11,7] = 22264551305192135492 56016892506730559375/364880443159675255577435648380047355776, a[3,2] = 100352/1134225, a[13,3] = 0, a[3,1] = 18928/1134225, a[13,6] = 143272 19974204125/40489566827933216, a[13,7] = 2720762324010009765625000/109 17367480696813922225349, a[13,8] = -498533005859375/95352091037424, a[ 10,1] = -63077736705254280154824845013881/7836935785378663385511219039 4368, a[10,2] = 0, a[13,11] = 863264105888000/85814662253313, a[6,1] = 12089/252720, a[6,2] = 0, a[5,4] = 178077159/125440000, a[13,12] = -2 9746300739/247142463456, c[11] = 19/20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[10,6] = 1001587844183325981198091450220795/184232684207722503 701669953872896, a[10,3] = 0, a[7,5] = -21643947/592609375, a[7,4] = 3 064329829899/27126050000000, a[9,4] = -1476496080404865730363837225290 8780219281424435/2981692102565021975611711269209606363661854518, a[8,7 ] = 1993321838240/380523459069, a[9,1] = -1840911252282376584438157336 464708426954728061551/299192361517115192159625381348311826219553373389 8, a[13,4] = 0, a[13,5] = 0, a[9,5] = -8753250485021304411186134217852 66742862694404520560000/1702120304288944183955716775759613394954350118 88324169, a[8,1] = -15365458811/13609565775, a[8,3] = 0, a[8,4] = -7/5 , a[8,5] = -8339128164608/939060038475, a[8,6] = 341936800488/47951126 225, a[10,4] = -31948346510820970247215/6956009216960026632192, a[10,5 ] = -3378604805394255292453489375/517042670569824692230499952, a[13,9] = 405932030463777247926705030596175437402459637909765779/788039194363 21841083201886041201537229769115088303952, a[13,10] = -10290327637248/ 1082076946951, a[4,3] = 42/355, a[5,1] = 94495479/250880000, a[4,1] = \+ 14/355, a[9,6] = 7632051964154290925661849798370645637589377834346780/ 1734087257418811583049800347581865260479233950396659, a[9,7] = 7519834 791971137517048532179652347729899303513750000/104567730350231759659789 0707812349832637339039997351, a[6,3] = 0, a[6,4] = 2505377/10685520, a [6,5] = 960400/5209191, a[7,1] = 21400899/350000000, a[7,3] = 0, c[2] \+ = 1/16, a[5,2] = 0, a[5,3] = -352806597/250880000, a[13,1] = 463167487 9841/103782082379976, a[13,2] = 0, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, a[14,3] = 0, a[14,5] = 0, c[3] = 11 2/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[18,1] = -14609590 5498393119/28971542512452028125*7^(1/2)+362058929348034963652/86045481 26198252353125, a[14,7] = 129834778101757666015625000/5349510065541438 82189042101-82766411529674389648437500/26212599321153050227263062949*7 ^(1/2), a[18,6] = -1265405169411103125627/26855843967693741133700*7^(1 /2)+3975969853211105248983/26855843967693741133700, a[17,13] = 3949144 173/1143910400000-908415729/3145753600000*7^(1/2), a[15,2] = 0, a[18,1 7] = 0, a[17,3] = 0, a[15,3] = 0, a[17,2] = 0, a[16,5] = 0, a[16,2] = \+ 0, a[15,4] = 0, a[17,16] = 9992573019/352964993024*7^(1/2)+11209972749 3/1764824965120, a[17,11] = 661671021433877476047/97141739992551631664 00*7^(1/2)-65155243311676871271269/116570087991061957996800, a[17,12] \+ = -729594201713541940737/358097856689988567040000*7^(1/2)+627473035045 29774530387/5371467850349828505600000, a[17,14] = 319287235181733/9769 566771200000+29511701788023/781565341696000*7^(1/2), a[17,15] = 908415 729/64849710464*7^(1/2)-2317999437/23581712896, a[17,10] = 17668130783 823819/66589350581600000, a[17,9] = -399866634050793590138577034430419 123328121213083504583/156622849751717676376323883963632206247641290715 29984000*7^(1/2)-35536138789892879785467436310191605063042447812379594 76464169103/9014462300397069176222190274214526693312008034830780905390 0800000, a[17,6] = -227711699316765384611310459/7040098361067108075752 652800*7^(1/2)+860764742109302032039902111/704009836106710807575265280 0, a[17,7] = -304194866461696889173828125/7899680270269138271354212352 *7^(1/2)+54888690084350082907540673828125/3650481751292720140884138298 92096, a[17,8] = -9661338110897182125/209930810853020925952*7^(1/2)+17 032572234499325505625/51013187037284085006336, a[16,3] = 0, a[17,1] = \+ -674107528569903747657/194736308727800627200000*7^(1/2)+66762908176428 55930112293/173510051076470358835200000, a[14,1] = 923507123432989/203 41288146475296+3728619917660047/7973784953418316032*7^(1/2), a[17,4] = 0, a[14,11] = -5071676622092000/4204918450412337-380378998046332000/2 06041004070204513*7^(1/2), a[18,4] = 0, a[18,2] = 0, a[17,5] = 0, a[14 ,6] = 1533012537239841375/7935955098274910336-11049456415304617875/194 430899907735303232*7^(1/2), a[14,8] = 13061564753515625/18689009843335 104+31047286856168359375/25641321505055762688*7^(1/2), a[18,5] = 0, a[ 16,11] = -14452334021133180240325340727774770157932021057436/424004592 44893414400850901677665011873556514674975*7^(1/2)+13794827769958865090 31984497778742508172264966358576/8904096441427617024178689352309652493 44686808174475, a[16,12] = 2400779388266927753129294529545871937061670 6273/3052786573498438708373913400079618470033568180000*7^(1/2)-8388482 7511307050547839026717499224825563955775903/25643407217386885150340872 56066879514828197271200000, a[14,9] = -3211799428826566849882311586102 611111468427562764815533/154455682095190808523075696640755012970347465 57307574592-35609926058546725508440519811766382833790503793377686129/6 3599398509784450568325286852075593576025427000678248320*7^(1/2), a[14, 10] = 37302437685024/53021770400599+15742143124443936/1299033374814675 5*7^(1/2), a[16,13] = -4114436729416819286145536389728774047607/456542 312392180862139685668388139491300000+167309383720093788703940567057169 6062733/913084624784361724279371336776278982600000*7^(1/2), a[16,4] = \+ 0, a[16,15] = -2975918446967643161447796202231979904/14152811684157606 7263302557200323242303*7^(1/2)+53303208736809836098042063038756642432/ 707640584207880336316512786001616211515, a[16,14] = 100137330804174498 9114984611143231300353/40206851375447746381620044659182739290625-89213 1449472993412180461043706539941657/25732384880286557684236828581876953 146000*7^(1/2), c[18] = 2/5, c[16] = 1/10, c[17] = 13/40, c[15] = 9/20 , a[15,5] = 0, a[18,3] = 0, a[18,16] = 15992416/387778923*7^(1/2)+2125 088/55396989, a[18,15] = 93046784/4559745267*7^(1/2)-254314496/3256960 905, a[18,14] = 28388719184/596287034375+3935951656/71554444125*7^(1/2 ), a[18,13] = 10035998/3456028125-1453856/3456028125*7^(1/2), a[18,12] = -4054391044424961/1366034914741472500*7^(1/2)+245337047543712611/20 490523721122087500, a[18,10] = 596839002960384/3381490459221875, a[18, 9] = -59996123596940840224374749037212780460893575326973/1613165643042 136101593301722342708423113370837903250*7^(1/2)+4545331033882832793172 679358463262481859874938464112848008/171937223442021735691493802532473 119608154450127235048396875, a[18,11] = 15060740399158849536/151783968 738361924475*7^(1/2)-233279804085211463168/455351906215085773425, a[15 ,9] = -2460557325340377391451533907481561939821407244475387801507/2594 367718068208759940802832632149373819559344470912000000+199091459817160 514703569453039285921263305239484481767/416767572250203215264447300211 189502579457643520000000*7^(1/2), a[15,8] = 15520930467726675/56253809 75771648-223623196267702275/157510667321606144*7^(1/2), a[15,7] = 2530 06492725129434765625/1078258516612031004664232-46193539913722265625/55 03324547053749203536*7^(1/2), a[15,6] = 768457351210057567857/51826645 53975451648000-62532827932224226599/2267415742364260096000*7^(1/2), a[ 15,1] = 3791756097508153347/82000657682944000000+821104665636321/64224 2913320960000*7^(1/2), a[18,8] = -1449590403533375/21622207233549366*7 ^(1/2)+401012292446311250/1264899123162637911, a[15,12] = -15396387799 49901/263618627686400000, a[15,11] = -61485359208535917/14302443708885 500+887648262380253891/400468423848794000*7^(1/2), a[15,10] = 42605915 213080224/16907452296109375-1106650521763256349/860743025983750000*7^( 1/2), a[16,1] = -4215121951102178392285859687667763860057798131121/552 4907006530535028430669612520227037472745107200000*7^(1/2)+214649483202 843017900825928522094099475577072470507941/430735562496636837154026079 6661082004089688904200800000, a[15,14] = 708939/16000000*7^(1/2), a[15 ,13] = 101277/64000000*7^(1/2)-79893/16000000, a[18,7] = -126579092138 74869625000000/225650046782590131286240929*7^(1/2)+4158162077129298437 32812500000/2317200330410418058178408099901, a[16,8] = 448242553947529 6955559304996688796936033039629875/20845977587384504244419538387493205 979265982789632*7^(1/2)-4632337900520750571763514693783173288916171486 6875/48707428401292639724941806040008163970784940556544, a[16,7] = 399 32181887340468446859878465078752005970844232914062500/7991406437185379 813952210414813050100306894094759398308201*7^(1/2)+3543645172418862284 69307408685840332599095120809109472656250/4531127449884110354510903305 198999406874008951728578840749967, a[16,6] = 1185913430248674317434870 618267722069953058737510283/300084423171791718189556145039257860517851 31637358800*7^(1/2)-31199443191133735684002403610068077599024732284897 8459/3360945539524067243723028824439688037799934743384185600, a[16,9] \+ = 22574276849009355247368902305397516894891496017540384017227175095582 93702749265180053129/1538231996366864375768034004361734572803338980781 05274095131111501836720940560681990528000*7^(1/2)+42492774700711579417 3549944207313054512456246547193016720629059219336389266382825513623866 3/51104152848047584046512223465220908365985929810714271725745707161059 426234354242200368775000, a[16,10] = 826748868093762078677306683985032 87366098540852/729064214207911394139200255820908786902329141125*7^(1/2 )-72880049498632524878182125169775730823179168857424/10795758556540226 4132150807111942262675921815128125\}: " }{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 1 9" }}{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 scheme that avoids d ealing 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 := SimpleOr derConditions(7,18,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,17,25,2 7,32,61,63,64]:\ninterp_order_eqns19 := []:\nfor ct in whch do\n tem p_eqn := convert(SO7_18[ct],'interpolation_order_condition'):\n inte rp_order_eqns19 := [op(interp_order_eqns19),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 0 584 "interp_order_eqns19 := [add (a[19,i],i=1..18)=c[19],seq(op(StageOrderConditions(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 add(a[19,i]*add(a[i,j]*a dd(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..18)=c[19]^6/120, ##27\n a dd(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 "W e specify " }{XPPEDIT 18 0 "c[19] = 3/10;" "6#/&%\"cG6#\"#>*&\"\"$\" \"\"\"#5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[19,17]=0" "6#/&%\"a G6$\"#>\"#<\"\"!" }{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]=3/10,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'\"#;&F%6$F'\"#5&F%6$F'\"#6&F%6$F'\"\"'&F%6$F' \"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$F'\"#7&F%6$F'\"#8&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 80 "e20 := `union`(e18,e19):\n[seq(a[19 ,i]=subs(e20,a[19,i]),i=1..18)]:\nevalf[20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#74/&%\"aG6$\"#>\"\"\"$\"5hN6Es=\"zD$H!#@/&F&6$F(\"\"#$ \"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F&6$F(\"\"'$\"5% \\TZr(3)3co$F,/&F&6$F(\"\"($\"5^#33tepl)p[F,/&F&6$F(\"\")$\"5yW#\\i%zQ XI@!#?/&F&6$F(\"\"*$!5#fPrh!*y$=u5FQ/&F&6$F(\"#5$\"5wrb/yiRkkEFQ/&F&6$ F(\"#6$!5wJWn==1k.QFQ/&F&6$F(\"#7$\"5NqL$3A!fi=j!#A/&F&6$F(\"#8$\"51Ia ZsB)e(*p#Fdo/&F&6$F(\"#9$\"5sVbv9+nzw5FQ/&F&6$F(\"#:$!5B+IWx<([:;'F,/& F&6$F(\"#;$\"59'*\\$Rq_&4$Q\"FQ/&F&6$F(\"# " 0 "" {MPLTEXT 1 0 205 "recd := []: \nfor ct to nops(SO7_18) do\n tt := convert(SO7_18[ct],'interpolatio n_order_condition'):\n if expand(subs(e20,lhs(tt)=rhs(tt))) then rec d := [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 12599 "e20 := \{a[14,2] = 0 , a[9,3] = 0, a[14,12] = 1100613127343/48439922837376-29746300739/1424 703612864*7^(1/2), a[7,6] = 124391943/6756250000, a[9,8] = 13660426834 89166351293315549358278750/144631418224267718165055326464180836641, a[ 9,2] = 0, a[8,2] = 0, a[14,4] = 0, a[19,12] = -5904670442992239/356841 7736467520000*7^(1/2)+2337899411211529353/218565586358635600000, a[14, 13] = 3/392-3/392*7^(1/2), a[19,11] = 1199511517980117216/216834241054 80274925*7^(1/2)-79948431067612848912/151783968738361924475, a[4,2] = \+ 0, c[14] = 1/2-1/14*7^(1/2), a[10,7] = 1870230752313499007680148902744 53125/25224698849808178010752575653374848, a[12,5] = -6970692975609264 52045586710000/41107967755245430594036502319, a[7,2] = 0, a[11,4] = -9 623541317323077848129/3864449564977792573440, a[11,5] = -4823348333146 829406881375/576413233634141239944816, a[12,1] = -32180221747585998316 59045535578571/1453396753634469525663775847094384, a[12,3] = 0, a[12,4 ] = 26290092604284231996745/5760876126062860430544, a[12,7] = 64350480 2814241550941949227194107500000/24212460911883655086049400754533394533 1, a[11,1] = -10116106591826909534781157993685116703/95628199450368940 30442231411871744000, a[11,2] = 0, a[11,3] = 0, a[12,6] = 182735782043 4213461438077550902273440/139381013914245317709567680839641697, a[10,8 ] = 1908158550070998850625/117087067039189929394176, a[10,9] = -529568 18288156668227044990077324877908565/2912779959477433986349822224412353 951940608, a[12,8] = 162259938151380266113750/59091082835244183497007, a[12,9] = -23028251632873523818545414856857015616678575554130463402/2 0013169183191444503443905240405603349978424504151629055, a[12,2] = 0, \+ a[12,10] = 7958341351371843889152/3284467988443203581305, a[12,11] = - 507974327957860843878400/121555654819179042718967, a[11,8] = 397472627 82380466933662225/1756032802431424164410720256, a[11,9] = 481757714192 60955335244683805171548038966866545122229/1989786420513815146528880165 952064118903852843612160000, a[11,10] = -2378292068163246/477687284872 11875, a[11,6] = 6566119246514996884067001154977284529/970305487021846 325473990863582315520, a[11,7] = 2226455130519213549256016892506730559 375/364880443159675255577435648380047355776, a[3,2] = 100352/1134225, \+ a[13,3] = 0, a[3,1] = 18928/1134225, a[13,6] = 14327219974204125/40489 566827933216, a[13,7] = 2720762324010009765625000/10917367480696813922 225349, a[13,8] = -498533005859375/95352091037424, a[10,1] = -63077736 705254280154824845013881/78369357853786633855112190394368, a[10,2] = 0 , a[13,11] = 863264105888000/85814662253313, a[6,1] = 12089/252720, a[ 6,2] = 0, a[5,4] = 178077159/125440000, a[13,12] = -29746300739/247142 463456, c[11] = 19/20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[10,6] = \+ 1001587844183325981198091450220795/184232684207722503701669953872896, \+ a[10,3] = 0, a[7,5] = -21643947/592609375, a[7,4] = 3064329829899/2712 6050000000, a[9,4] = -14764960804048657303638372252908780219281424435/ 2981692102565021975611711269209606363661854518, a[8,7] = 1993321838240 /380523459069, a[9,1] = -184091125228237658443815733646470842695472806 1551/2991923615171151921596253813483118262195533733898, a[13,4] = 0, a [13,5] = 0, a[9,5] = -875325048502130441118613421785266742862694404520 560000/170212030428894418395571677575961339495435011888324169, a[8,1] \+ = -15365458811/13609565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -83391 28164608/939060038475, a[8,6] = 341936800488/47951126225, a[10,4] = -3 1948346510820970247215/6956009216960026632192, a[10,5] = -337860480539 4255292453489375/517042670569824692230499952, a[13,9] = 40593203046377 7247926705030596175437402459637909765779/78803919436321841083201886041 201537229769115088303952, a[13,10] = -10290327637248/1082076946951, a[ 4,3] = 42/355, a[5,1] = 94495479/250880000, a[4,1] = 14/355, a[9,6] = \+ 7632051964154290925661849798370645637589377834346780/17340872574188115 83049800347581865260479233950396659, a[9,7] = 751983479197113751704853 2179652347729899303513750000/10456773035023175965978907078123498326373 39039997351, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/52 09191, a[7,1] = 21400899/350000000, a[7,3] = 0, c[2] = 1/16, a[5,2] = \+ 0, a[5,3] = -352806597/250880000, a[13,1] = 4631674879841/103782082379 976, a[13,2] = 0, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/1617886 1, c[10] = 11/12, a[14,3] = 0, a[14,5] = 0, c[3] = 112/1065, c[4] = 56 /355, c[5] = 39/100, c[6] = 7/15, a[18,1] = -146095905498393119/289715 42512452028125*7^(1/2)+362058929348034963652/8604548126198252353125, a [14,7] = 129834778101757666015625000/534951006554143882189042101-82766 411529674389648437500/26212599321153050227263062949*7^(1/2), a[19,13] \+ = 367198237/110592900000-3242183/13824112500*7^(1/2), a[18,6] = -12654 05169411103125627/26855843967693741133700*7^(1/2)+39759698532111052489 83/26855843967693741133700, a[17,13] = 3949144173/1143910400000-908415 729/3145753600000*7^(1/2), a[19,4] = 0, a[19,3] = 0, a[15,2] = 0, a[18 ,17] = 0, a[17,3] = 0, a[19,2] = 0, a[19,18] = 0, a[15,3] = 0, a[17,2] = 0, a[16,5] = 0, a[16,2] = 0, a[15,4] = 0, a[17,16] = 9992573019/352 964993024*7^(1/2)+112099727493/1764824965120, a[17,11] = 6616710214338 77476047/9714173999255163166400*7^(1/2)-65155243311676871271269/116570 087991061957996800, a[17,12] = -729594201713541940737/3580978566899885 67040000*7^(1/2)+62747303504529774530387/5371467850349828505600000, a[ 17,14] = 319287235181733/9769566771200000+29511701788023/7815653416960 00*7^(1/2), a[17,15] = 908415729/64849710464*7^(1/2)-2317999437/235817 12896, a[17,10] = 17668130783823819/66589350581600000, a[17,9] = -3998 66634050793590138577034430419123328121213083504583/1566228497517176763 7632388396363220624764129071529984000*7^(1/2)-355361387898928797854674 3631019160506304244781237959476464169103/90144623003970691762221902742 145266933120080348307809053900800000, a[19,9] = -167708663236226679032 764579220523036527050471387/808826050019168087288740728034261734335506 5504000*7^(1/2)-160655663824415526417155695016109151523479023935794534 31911/305666175008038641229322315613285545970052355781751197150000, a[ 17,6] = -227711699316765384611310459/7040098361067108075752652800*7^(1 /2)+860764742109302032039902111/7040098361067108075752652800, a[17,7] \+ = -304194866461696889173828125/7899680270269138271354212352*7^(1/2)+54 888690084350082907540673828125/365048175129272014088413829892096, a[17 ,8] = -9661338110897182125/209930810853020925952*7^(1/2)+1703257223449 9325505625/51013187037284085006336, a[16,3] = 0, a[17,1] = -6741075285 69903747657/194736308727800627200000*7^(1/2)+6676290817642855930112293 /173510051076470358835200000, a[14,1] = 923507123432989/20341288146475 296+3728619917660047/7973784953418316032*7^(1/2), a[19,16] = 5094859/2 21587956*7^(1/2)+42920003/553969890, a[17,4] = 0, a[14,11] = -50716766 22092000/4204918450412337-380378998046332000/206041004070204513*7^(1/2 ), a[19,15] = 7410704/651392181*7^(1/2)-298713632/3256960905, a[18,4] \+ = 0, a[18,2] = 0, a[19,14] = 126616972699/4770296275000+35109599707/11 44871106000*7^(1/2), a[17,5] = 0, a[14,6] = 1533012537239841375/793595 5098274910336-11049456415304617875/194430899907735303232*7^(1/2), a[14 ,8] = 13061564753515625/18689009843335104+31047286856168359375/2564132 1505055762688*7^(1/2), a[18,5] = 0, a[16,11] = -1445233402113318024032 5340727774770157932021057436/42400459244893414400850901677665011873556 514674975*7^(1/2)+1379482776995886509031984497778742508172264966358576 /890409644142761702417868935230965249344686808174475, a[16,12] = 24007 793882669277531292945295458719370616706273/305278657349843870837391340 0079618470033568180000*7^(1/2)-838848275113070505478390267174992248255 63955775903/2564340721738688515034087256066879514828197271200000, a[14 ,9] = -3211799428826566849882311586102611111468427562764815533/1544556 8209519080852307569664075501297034746557307574592-35609926058546725508 440519811766382833790503793377686129/635993985097844505683252868520755 93576025427000678248320*7^(1/2), a[14,10] = 37302437685024/53021770400 599+15742143124443936/12990333748146755*7^(1/2), a[19,17] = 0, a[16,13 ] = -4114436729416819286145536389728774047607/456542312392180862139685 668388139491300000+1673093837200937887039405670571696062733/9130846247 84361724279371336776278982600000*7^(1/2), a[16,4] = 0, a[16,15] = -297 5918446967643161447796202231979904/14152811684157606726330255720032324 2303*7^(1/2)+53303208736809836098042063038756642432/707640584207880336 316512786001616211515, a[16,14] = 100137330804174498911498461114323130 0353/40206851375447746381620044659182739290625-89213144947299341218046 1043706539941657/25732384880286557684236828581876953146000*7^(1/2), a[ 19,10] = 901046813736528/3381490459221875, a[19,8] = -1034454551383146 25/2767642525894318848*7^(1/2)+350725547323759375/1124354776144567032, c[19] = 3/10, c[18] = 2/5, c[16] = 1/10, c[17] = 13/40, c[15] = 9/20, a[19,7] = -2016276423759027541015625/64471441937882894653211694*7^(1/ 2)+33841871953716766022861328125/257466703378935339797600899989, a[19, 6] = -1842890934880471474773/70154041384995895206400*7^(1/2)+914025742 92269720401167/859387006966199716278400, a[15,5] = 0, a[19,5] = 0, a[1 9,1] = -408385534705961/145260583712400000*7^(1/2)+3749192080391080667 69/10197982964383113900000, a[18,3] = 0, a[18,16] = 15992416/387778923 *7^(1/2)+2125088/55396989, a[18,15] = 93046784/4559745267*7^(1/2)-2543 14496/3256960905, a[18,14] = 28388719184/596287034375+3935951656/71554 444125*7^(1/2), a[18,13] = 10035998/3456028125-1453856/3456028125*7^(1 /2), a[18,12] = -4054391044424961/1366034914741472500*7^(1/2)+24533704 7543712611/20490523721122087500, a[18,10] = 596839002960384/3381490459 221875, a[18,9] = -59996123596940840224374749037212780460893575326973/ 1613165643042136101593301722342708423113370837903250*7^(1/2)+454533103 3882832793172679358463262481859874938464112848008/17193722344202173569 1493802532473119608154450127235048396875, a[18,11] = 15060740399158849 536/151783968738361924475*7^(1/2)-233279804085211463168/45535190621508 5773425, a[15,9] = -24605573253403773914515339074815619398214072444753 87801507/2594367718068208759940802832632149373819559344470912000000+19 9091459817160514703569453039285921263305239484481767/41676757225020321 5264447300211189502579457643520000000*7^(1/2), a[15,8] = 1552093046772 6675/5625380975771648-223623196267702275/157510667321606144*7^(1/2), a [15,7] = 253006492725129434765625/1078258516612031004664232-4619353991 3722265625/5503324547053749203536*7^(1/2), a[15,6] = 76845735121005756 7857/5182664553975451648000-62532827932224226599/226741574236426009600 0*7^(1/2), a[15,1] = 3791756097508153347/82000657682944000000+82110466 5636321/642242913320960000*7^(1/2), a[18,8] = -1449590403533375/216222 07233549366*7^(1/2)+401012292446311250/1264899123162637911, a[15,12] = -1539638779949901/263618627686400000, a[15,11] = -61485359208535917/1 4302443708885500+887648262380253891/400468423848794000*7^(1/2), a[15,1 0] = 42605915213080224/16907452296109375-1106650521763256349/860743025 983750000*7^(1/2), a[16,1] = -4215121951102178392285859687667763860057 798131121/5524907006530535028430669612520227037472745107200000*7^(1/2) +214649483202843017900825928522094099475577072470507941/43073556249663 68371540260796661082004089688904200800000, a[15,14] = 708939/16000000* 7^(1/2), a[15,13] = 101277/64000000*7^(1/2)-79893/16000000, a[18,7] = \+ -12657909213874869625000000/225650046782590131286240929*7^(1/2)+415816 207712929843732812500000/2317200330410418058178408099901, a[16,8] = 44 82425539475296955559304996688796936033039629875/2084597758738450424441 9538387493205979265982789632*7^(1/2)-463233790052075057176351469378317 32889161714866875/48707428401292639724941806040008163970784940556544, \+ a[16,7] = 39932181887340468446859878465078752005970844232914062500/799 1406437185379813952210414813050100306894094759398308201*7^(1/2)+354364 517241886228469307408685840332599095120809109472656250/453112744988411 0354510903305198999406874008951728578840749967, a[16,6] = 118591343024 8674317434870618267722069953058737510283/30008442317179171818955614503 925786051785131637358800*7^(1/2)-3119944319113373568400240361006807759 90247322848978459/3360945539524067243723028824439688037799934743384185 600, a[16,9] = 2257427684900935524736890230539751689489149601754038401 722717509558293702749265180053129/153823199636686437576803400436173457 280333898078105274095131111501836720940560681990528000*7^(1/2)+4249277 4700711579417354994420731305451245624654719301672062905921933638926638 28255136238663/5110415284804758404651222346522090836598592981071427172 5745707161059426234354242200368775000, a[16,10] = 82674886809376207867 730668398503287366098540852/729064214207911394139200255820908786902329 141125*7^(1/2)-72880049498632524878182125169775730823179168857424/1079 57585565402264132150807111942262675921815128125\}: " }{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] = 7/10;" "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]=7/10,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'\"#6&F%6$F' \"\"'&F%6$F'\"#7&F%6$F'\"#8&F%6$F'\"\"*&F%6$F'\"#5&F%6$F'\"\"\"&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 80 "e23 : = `union`(e21,e22):\n[seq(a[20,i]=subs(e23,a[20,i]),i=1..19)]:\nevalf[ 20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#75/&%\"aG6$\"#?\"\"\"$\"5uX ih=l7AZk!#@/&F&6$F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F( \"\"&F1/&F&6$F(\"\"'$\"5m*[qRCC'yrP!#?/&F&6$F(\"\"($\"5)Rif+\"HTR%p%FE /&F&6$F(\"\")$\"5]**zeIz1M;7!#>/&F&6$F(\"\"*$!5'*RPq7,'3**z#FE/&F&6$F( \"#5$\"5Xn]7yQ7nF6FR/&F&6$F(\"#6$!59[G$\\b'y^v?FR/&F&6$F(\"#7$\"5f**e] ,!y*RnSF,/&F&6$F(\"#8$\"5e.j\\mQ)y^F\"F,/&F&6$F(\"#9$!5xuP=uR][+9FE/&F &6$F(\"#:$\"54M\\@a2wtg\\F,/&F&6$F(\"#;$!5CD\"G!>05yD;FE/&F&6$F(\"#F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "These linking coefficients can be compare d 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 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 0 205 "recd := []:\nfor ct to nops(SO7_19) do\n tt : = convert(SO7_19[ct],'interpolation_order_condition'):\n if expand(s ubs(e23,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 13877 "e23 := \{a[20,5] = 0, a[14,2] = 0, a[9,3] = 0, a[14,12] = 110 0613127343/48439922837376-29746300739/1424703612864*7^(1/2), a[7,6] = \+ 124391943/6756250000, a[9,8] = 1366042683489166351293315549358278750/1 44631418224267718165055326464180836641, a[9,2] = 0, a[8,2] = 0, a[14,4 ] = 0, a[20,8] = 134541471320113375/2767642525894318848*7^(1/2)+555006 824763125/510245713256409, a[19,12] = -5904670442992239/35684177364675 20000*7^(1/2)+2337899411211529353/218565586358635600000, a[14,13] = 3/ 392-3/392*7^(1/2), a[19,11] = 1199511517980117216/21683424105480274925 *7^(1/2)-79948431067612848912/151783968738361924475, a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[10,7] = 187023075231349900768014890274453125/25 224698849808178010752575653374848, a[12,5] = -697069297560926452045586 710000/41107967755245430594036502319, a[7,2] = 0, a[11,4] = -962354131 7323077848129/3864449564977792573440, a[11,5] = -482334833314682940688 1375/576413233634141239944816, a[12,1] = -3218022174758599831659045535 578571/1453396753634469525663775847094384, a[12,3] = 0, a[12,4] = 2629 0092604284231996745/5760876126062860430544, a[12,7] = 6435048028142415 50941949227194107500000/242124609118836550860494007545333945331, a[11, 1] = -10116106591826909534781157993685116703/9562819945036894030442231 411871744000, a[11,2] = 0, a[11,3] = 0, a[12,6] = 18273578204342134614 38077550902273440/139381013914245317709567680839641697, a[10,8] = 1908 158550070998850625/117087067039189929394176, a[10,9] = -52956818288156 668227044990077324877908565/291277995947743398634982222441235395194060 8, a[12,8] = 162259938151380266113750/59091082835244183497007, a[12,9] = -23028251632873523818545414856857015616678575554130463402/200131691 83191444503443905240405603349978424504151629055, a[12,2] = 0, a[12,10] = 7958341351371843889152/3284467988443203581305, a[12,11] = -50797432 7957860843878400/121555654819179042718967, a[11,8] = 39747262782380466 933662225/1756032802431424164410720256, a[11,9] = 48175771419260955335 244683805171548038966866545122229/198978642051381514652888016595206411 8903852843612160000, a[11,10] = -2378292068163246/47768728487211875, a [11,6] = 6566119246514996884067001154977284529/97030548702184632547399 0863582315520, a[11,7] = 2226455130519213549256016892506730559375/3648 80443159675255577435648380047355776, a[3,2] = 100352/1134225, a[13,3] \+ = 0, a[3,1] = 18928/1134225, a[13,6] = 14327219974204125/4048956682793 3216, a[13,7] = 2720762324010009765625000/10917367480696813922225349, \+ a[13,8] = -498533005859375/95352091037424, a[10,1] = -6307773670525428 0154824845013881/78369357853786633855112190394368, a[10,2] = 0, a[13,1 1] = 863264105888000/85814662253313, a[6,1] = 12089/252720, a[6,2] = 0 , a[5,4] = 178077159/125440000, a[13,12] = -29746300739/247142463456, \+ c[11] = 19/20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[10,6] = 10015878 44183325981198091450220795/184232684207722503701669953872896, a[10,3] \+ = 0, a[7,5] = -21643947/592609375, a[7,4] = 3064329829899/271260500000 00, a[9,4] = -14764960804048657303638372252908780219281424435/29816921 02565021975611711269209606363661854518, a[8,7] = 1993321838240/3805234 59069, a[9,1] = -1840911252282376584438157336464708426954728061551/299 1923615171151921596253813483118262195533733898, a[13,4] = 0, a[13,5] = 0, a[9,5] = -875325048502130441118613421785266742862694404520560000/1 70212030428894418395571677575961339495435011888324169, a[8,1] = -15365 458811/13609565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -8339128164608 /939060038475, a[8,6] = 341936800488/47951126225, a[10,4] = -319483465 10820970247215/6956009216960026632192, a[10,5] = -33786048053942552924 53489375/517042670569824692230499952, a[13,9] = 4059320304637772479267 05030596175437402459637909765779/7880391943632184108320188604120153722 9769115088303952, a[13,10] = -10290327637248/1082076946951, a[4,3] = 4 2/355, a[5,1] = 94495479/250880000, a[4,1] = 14/355, a[9,6] = 76320519 64154290925661849798370645637589377834346780/1734087257418811583049800 347581865260479233950396659, a[9,7] = 75198347919711375170485321796523 47729899303513750000/1045677303502317596597890707812349832637339039997 351, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a [7,1] = 21400899/350000000, a[7,3] = 0, c[2] = 1/16, a[5,2] = 0, a[5,3 ] = -352806597/250880000, a[13,1] = 4631674879841/103782082379976, a[1 3,2] = 0, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, a[14,3] = 0, a[14,5] = 0, c[3] = 112/1065, c[4] = 56/355, c[ 5] = 39/100, c[6] = 7/15, a[18,1] = -146095905498393119/28971542512452 028125*7^(1/2)+362058929348034963652/8604548126198252353125, a[14,7] = 129834778101757666015625000/534951006554143882189042101-8276641152967 4389648437500/26212599321153050227263062949*7^(1/2), a[19,13] = 367198 237/110592900000-3242183/13824112500*7^(1/2), a[18,6] = -1265405169411 103125627/26855843967693741133700*7^(1/2)+3975969853211105248983/26855 843967693741133700, a[17,13] = 3949144173/1143910400000-908415729/3145 753600000*7^(1/2), a[20,12] = 7679632208088369/3568417736467520000*7^( 1/2)+468087818175204739/13381566511753200000, a[19,4] = 0, a[20,14] = \+ -164678417029/4770296275000-45663651397/1144871106000*7^(1/2), a[20,19 ] = 0, a[19,3] = 0, a[15,2] = 0, a[18,17] = 0, a[17,3] = 0, a[19,2] = \+ 0, a[19,18] = 0, a[15,3] = 0, a[17,2] = 0, a[16,5] = 0, a[16,2] = 0, a [20,10] = 7074600250608/6273637215625, a[20,13] = 1321004573/110592900 000+4216793/13824112500*7^(1/2), a[20,7] = 2622375205154089396484375/6 4471441937882894653211694*7^(1/2)+255381525950382596240234375/70581795 0170703033255683247, a[20,15] = -9638384/651392181*7^(1/2)+57814624/65 1392181, a[20,16] = -6626389/221587956*7^(1/2)-46233929/553969890, a[1 5,4] = 0, a[20,6] = 2396869514758244044683/70154041384995895206400*7^( 1/2)+135939757794763166091/474013793141864156800, a[17,16] = 999257301 9/352964993024*7^(1/2)+112099727493/1764824965120, a[17,11] = 66167102 1433877476047/9714173999255163166400*7^(1/2)-65155243311676871271269/1 16570087991061957996800, a[17,12] = -729594201713541940737/35809785668 9988567040000*7^(1/2)+62747303504529774530387/537146785034982850560000 0, a[17,14] = 319287235181733/9769566771200000+29511701788023/78156534 1696000*7^(1/2), a[17,15] = 908415729/64849710464*7^(1/2)-2317999437/2 3581712896, a[17,10] = 17668130783823819/66589350581600000, a[17,9] = \+ -399866634050793590138577034430419123328121213083504583/15662284975171 767637632388396363220624764129071529984000*7^(1/2)-3553613878989287978 546743631019160506304244781237959476464169103/901446230039706917622219 02742145266933120080348307809053900800000, a[19,9] = -1677086632362266 79032764579220523036527050471387/8088260500191680872887407280342617343 355065504000*7^(1/2)-1606556638244155264171556950161091515234790239357 9453431911/30566617500803864122932231561328554597005235578175119715000 0, a[17,6] = -227711699316765384611310459/7040098361067108075752652800 *7^(1/2)+860764742109302032039902111/7040098361067108075752652800, a[1 7,7] = -304194866461696889173828125/7899680270269138271354212352*7^(1/ 2)+54888690084350082907540673828125/365048175129272014088413829892096, a[17,8] = -9661338110897182125/209930810853020925952*7^(1/2)+17032572 234499325505625/51013187037284085006336, a[16,3] = 0, a[17,1] = -67410 7528569903747657/194736308727800627200000*7^(1/2)+66762908176428559301 12293/173510051076470358835200000, a[20,1] = 276727826458195951/756807 64114160400000*7^(1/2)+27993284828754981193/510845157770582700000, a[2 0,9] = 113641767182046923760020640897484653184317971780717/42139837205 99865734774339193058503635887989127584000*7^(1/2)-19725249536158413435 648903585583405898983584276673979718229/561427668382111790013040987861 13671708785126572158383150000, a[14,1] = 923507123432989/2034128814647 5296+3728619917660047/7973784953418316032*7^(1/2), a[19,16] = 5094859/ 221587956*7^(1/2)+42920003/553969890, a[17,4] = 0, a[14,11] = -5071676 622092000/4204918450412337-380378998046332000/206041004070204513*7^(1/ 2), a[19,15] = 7410704/651392181*7^(1/2)-298713632/3256960905, a[18,4] = 0, a[18,2] = 0, a[19,14] = 126616972699/4770296275000+35109599707/1 144871106000*7^(1/2), a[17,5] = 0, a[20,3] = 0, a[14,6] = 153301253723 9841375/7935955098274910336-11049456415304617875/194430899907735303232 *7^(1/2), a[14,8] = 13061564753515625/18689009843335104+31047286856168 359375/25641321505055762688*7^(1/2), a[18,5] = 0, a[20,17] = 0, a[16,1 1] = -14452334021133180240325340727774770157932021057436/4240045924489 3414400850901677665011873556514674975*7^(1/2)+137948277699588650903198 4497778742508172264966358576/89040964414276170241786893523096524934468 6808174475, a[16,12] = 24007793882669277531292945295458719370616706273 /3052786573498438708373913400079618470033568180000*7^(1/2)-83884827511 307050547839026717499224825563955775903/256434072173868851503408725606 6879514828197271200000, a[14,9] = -32117994288265668498823115861026111 11468427562764815533/1544556820951908085230756966407550129703474655730 7574592-35609926058546725508440519811766382833790503793377686129/63599 398509784450568325286852075593576025427000678248320*7^(1/2), a[14,10] \+ = 37302437685024/53021770400599+15742143124443936/12990333748146755*7^ (1/2), a[19,17] = 0, a[16,13] = -4114436729416819286145536389728774047 607/456542312392180862139685668388139491300000+16730938372009378870394 05670571696062733/913084624784361724279371336776278982600000*7^(1/2), \+ a[16,4] = 0, a[16,15] = -2975918446967643161447796202231979904/1415281 16841576067263302557200323242303*7^(1/2)+53303208736809836098042063038 756642432/707640584207880336316512786001616211515, a[16,14] = 10013733 08041744989114984611143231300353/4020685137544774638162004465918273929 0625-892131449472993412180461043706539941657/2573238488028655768423682 8581876953146000*7^(1/2), a[19,10] = 901046813736528/3381490459221875, a[19,8] = -103445455138314625/2767642525894318848*7^(1/2)+35072554732 3759375/1124354776144567032, c[19] = 3/10, c[20] = 7/10, c[18] = 2/5, \+ c[16] = 1/10, c[17] = 13/40, c[15] = 9/20, a[20,4] = 0, a[19,7] = -201 6276423759027541015625/64471441937882894653211694*7^(1/2)+338418719537 16766022861328125/257466703378935339797600899989, a[19,6] = -184289093 4880471474773/70154041384995895206400*7^(1/2)+91402574292269720401167/ 859387006966199716278400, a[15,5] = 0, a[19,5] = 0, a[19,1] = -4083855 34705961/145260583712400000*7^(1/2)+374919208039108066769/101979829643 83113900000, a[18,3] = 0, a[20,2] = 0, a[18,16] = 15992416/387778923*7 ^(1/2)+2125088/55396989, a[18,15] = 93046784/4559745267*7^(1/2)-254314 496/3256960905, a[18,14] = 28388719184/596287034375+3935951656/7155444 4125*7^(1/2), a[20,18] = 0, a[18,13] = 10035998/3456028125-1453856/345 6028125*7^(1/2), a[20,11] = -1560088302368475936/21683424105480274925* 7^(1/2)-122630185337518052432/65050272316440824775, a[18,12] = -405439 1044424961/1366034914741472500*7^(1/2)+245337047543712611/204905237211 22087500, a[18,10] = 596839002960384/3381490459221875, a[18,9] = -5999 6123596940840224374749037212780460893575326973/16131656430421361015933 01722342708423113370837903250*7^(1/2)+45453310338828327931726793584632 62481859874938464112848008/1719372234420217356914938025324731196081544 50127235048396875, a[18,11] = 15060740399158849536/1517839687383619244 75*7^(1/2)-233279804085211463168/455351906215085773425, a[15,9] = -246 0557325340377391451533907481561939821407244475387801507/25943677180682 08759940802832632149373819559344470912000000+1990914598171605147035694 53039285921263305239484481767/4167675722502032152644473002111895025794 57643520000000*7^(1/2), a[15,8] = 15520930467726675/5625380975771648-2 23623196267702275/157510667321606144*7^(1/2), a[15,7] = 25300649272512 9434765625/1078258516612031004664232-46193539913722265625/550332454705 3749203536*7^(1/2), a[15,6] = 768457351210057567857/518266455397545164 8000-62532827932224226599/2267415742364260096000*7^(1/2), a[15,1] = 37 91756097508153347/82000657682944000000+821104665636321/642242913320960 000*7^(1/2), a[18,8] = -1449590403533375/21622207233549366*7^(1/2)+401 012292446311250/1264899123162637911, a[15,12] = -1539638779949901/2636 18627686400000, a[15,11] = -61485359208535917/14302443708885500+887648 262380253891/400468423848794000*7^(1/2), a[15,10] = 42605915213080224/ 16907452296109375-1106650521763256349/860743025983750000*7^(1/2), a[16 ,1] = -4215121951102178392285859687667763860057798131121/5524907006530 535028430669612520227037472745107200000*7^(1/2)+2146494832028430179008 25928522094099475577072470507941/4307355624966368371540260796661082004 089688904200800000, a[15,14] = 708939/16000000*7^(1/2), a[15,13] = 101 277/64000000*7^(1/2)-79893/16000000, a[18,7] = -1265790921387486962500 0000/225650046782590131286240929*7^(1/2)+41581620771292984373281250000 0/2317200330410418058178408099901, a[16,8] = 4482425539475296955559304 996688796936033039629875/208459775873845042444195383874932059792659827 89632*7^(1/2)-46323379005207505717635146937831732889161714866875/48707 428401292639724941806040008163970784940556544, a[16,7] = 3993218188734 0468446859878465078752005970844232914062500/79914064371853798139522104 14813050100306894094759398308201*7^(1/2)+35436451724188622846930740868 5840332599095120809109472656250/45311274498841103545109033051989994068 74008951728578840749967, a[16,6] = 11859134302486743174348706182677220 69953058737510283/3000844231717917181895561450392578605178513163735880 0*7^(1/2)-311994431911337356840024036100680775990247322848978459/33609 45539524067243723028824439688037799934743384185600, a[16,9] = 22574276 8490093552473689023053975168948914960175403840172271750955829370274926 5180053129/15382319963668643757680340043617345728033389807810527409513 1111501836720940560681990528000*7^(1/2)+424927747007115794173549944207 3130545124562465471930167206290592193363892663828255136238663/51104152 8480475840465122234652209083659859298107142717257457071610594262343542 42200368775000, a[16,10] = 8267488680937620786773066839850328736609854 0852/729064214207911394139200255820908786902329141125*7^(1/2)-72880049 498632524878182125169775730823179168857424/107957585565402264132150807 111942262675921815128125\}: " }{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 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 \"approximat e\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO8_20 := SimpleOrderConditions(8,2 0,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 223 "whch := [1,2,4,8,16,17,25,27,32,58,61,63,64,1 02,117,121,123,125,127,128]:\nordeqns := []:\nfor ct in whch do\n eq n_group := convert(SO8_20[ct],'polynom_order_conditions',8):\n ordeq ns := [op(ordeqns),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 0 131 "eqns := []:\nfor ct to nops(ordeqns) do\n eqns := [op(eqns),expand(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 "in folevel[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 3 "dd " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33374 "dd := \{d[1,1] = 1, d[13, 3] = -6026554549218325371819602880566514575441175091155923561220047/29 76460442110365939047936065778411110196314303465327138387653-2139317505 13121240897311333948469797620487410133479950210474/2705873129191241762 77085096888946464563301300315029739853423*7^(1/2), d[7,8] = -504934820 595770118651483720615821616344457874618213075439408117675781250000000/ 1183108500652901810277796730264261122881780489070151659101363864410569 21975113*7^(1/2)-58292527437781411912773925294712870852691962348423626 36809291368688964843750000000/8269928419563783653841799144547185248943 6456186003600971185334122298788460603987, d[4,5] = 0, d[1,8] = 1755557 75990409437358839820235487185823274254154528202609108731250/1567287515 83571687334819489151014023702260610501511298233884565093*7^(1/2)-55786 79855741510132035418472699340833939827654525857901613430487513750/9826 8927242899447958931819697685792861317402784447583992645622313311, d[12 ,8] = 3983228883819335958312855997120149472135026200554177016609686308 125/117006829131973984874243452185126870238578305849193259513447587858 +80392145392847409094641048445807581719640248385135864947859274375/390 02276377324661624747817395042290079526101949731086504482529286*7^(1/2) , d[20,1] = 0, d[5,3] = 0, d[8,8] = 4672983575772741830559064962212755 83413793032011754808657495228271484375/3160036509778104019125754770024 03890470557923499339019757980471557999+9431322828772428829902640294193 634610457182926248640916490469970703125/105334550325936800637525159000 801296823519307833113006585993490519333*7^(1/2), d[6,8] = -19185107058 12278520353528916004956407949488468882540636667169305833671875/1916933 1571816570069943972884541593041102381013121305728007311065105038-49854 9077367260056874272771904614581953479798642430508104866804921875/82271 809321101159098471986628933875712885755421121483811190176245086*7^(1/2 ), d[16,1] = 0, d[4,2] = 0, d[3,8] = 0, d[6,1] = 0, d[3,3] = 0, d[4,3] = 0, d[5,1] = 0, d[2,1] = 0, d[17,1] = 0, d[2,6] = 0, d[2,5] = 0, d[1 5,3] = 2715059598470792261411620674740552998114611488544663872000000/4 4382046034353700870844645521467409531546774125745618706117*7^(1/2)-975 62014525403177062257730233870040805011233298292327296000000/1464607519 133672128737873302208424514541043546149605417301861, d[17,2] = 1393815 0103470867094540617814388965150457511215120796621209600/55835477269025 623676223908881846095862268522287228359017373-173069075495440698843906 977750181786857481132028743640678400/429503671300197105201722376014200 7374020655560556027616721*7^(1/2), d[16,4] = 9157110678028170982401939 084023751427627959028125420984500000/128851101390059131560516712804260 22122061966681668082850163-1555573004302377244779305355848762909315823 49335611732000000/1288511013900591315605167128042602212206196668166808 2850163*7^(1/2), d[13,5] = -169438932400652337747302350814674528524444 9384512671429450766/27058731291912417627708509688894646456330130031502 9739853423*7^(1/2)-149053953640720357553471138481703434532931554920520 150727204840/297646044211036593904793606577841111019631430346532713838 7653, d[1,4] = -159428993525717958732435122909792054219844229114153042 132964106856070471/786151417943195583671454557581486342890539222275580 671941164978506488+133955340876910975521865058804930854478181657243056 758022875707595/447796433095919106670912826145754353435030315718603709 23967018598*7^(1/2), d[13,8] = 693539370775844936299162157482887752681 39741309544613455050000/2976460442110365939047936065778411110196314303 465327138387653+305054706736334016432297948882625960092399514374513276 935000/270587312919124176277085096888946464563301300315029739853423*7^ (1/2), d[3,5] = 0, d[19,8] = -5124610167754558953597359986263291748883 8190987880233485000000/10021752330337932454706855440331350539381529641 297397772349+113442427459509325851337648476138493088503245588079221625 0000/3340584110112644151568951813443783513127176547099132590783*7^(1/2 ), d[9,8] = -543570174590087582725365294967767996683441362351167162273 53781227314811155694172455232889568897360064327858125/3730883235311418 6481533717921289113824188990466039308844192456619345505657387191406228 559216756605808267811-109706908092514677104590989638959580116175688861 8750778004559713543349165251141566933644183495378079259924375/12436277 4510380621605112393070963712747296634886797696147308188731151685524623 97135409519738918868602755937*7^(1/2), d[14,4] = -35700514888657337576 64540840502648930521034590332740852841768/1479401534478456695694821517 3822469843848924708581872902039*7^(1/2)+854370784570568294250762307257 2991013191256249879172243240468/14794015344784566956948215173822469843 848924708581872902039, d[17,7] = -130405528327974940560613357047448684 73870997008694932232601600000/5025192954212306130860151799366148627604 16700585055231156357+9703984048055401506321853109310847314891720957206 41989836800000/5025192954212306130860151799366148627604167005850552311 56357*7^(1/2), d[20,4] = -42810761474190596496947055479494445878418609 30558872690173400/1002175233033793245470685544033135053938152964129739 7772349+342675997638791328751843399036669849568846218979805820800700/1 431678904333990350672407920047335791340218520185342538907*7^(1/2), d[8 ,5] = -209540549408971241058439263241027813186596877686456237303114804 759765625/421338201303747202550100636003205187294077231332452026343973 962077332*7^(1/2)-4118045992043463852343792034438314420149017616045054 996408305694060546875/126401460391124160765030190800961556188223169399 7356079031921886231996, d[5,8] = 0, d[2,8] = 0, d[3,6] = 0, d[18,1] = \+ 0, d[5,4] = 0, d[20,7] = 290172138381716721682518451804080669209556174 1960030898200000/10021752330337932454706855440331350539381529641297397 772349-3578778601084225084833595616265137525299387923370917980350000/1 0021752330337932454706855440331350539381529641297397772349*7^(1/2), d[ 20,5] = 7410578398906801657428051910976918842569991849588522199953300/ 10021752330337932454706855440331350539381529641297397772349-4988893657 127186826296687829354717852873073088222647736756900/100217523303379324 54706855440331350539381529641297397772349*7^(1/2), d[17,8] = -24354768 8766350101247330678706502258397906715390477320192000000/50251929542123 0613086015179936614862760416700585055231156357*7^(1/2)+330594867457186 1574612799677320341847721769707549595504640000000/50251929542123061308 6015179936614862760416700585055231156357, d[15,5] = -77271389330436736 0463415912039301427705023691227512321664000000/14646075191336721287378 73302208424514541043546149605417301861+2150390480448198163557675931042 9973127232607726738406848000000/44382046034353700870844645521467409531 546774125745618706117*7^(1/2), d[15,7] = -5543056509467803647238063330 94169233744998044820565696000000000/1464607519133672128737873302208424 514541043546149605417301861+154258077327605007807636722383005156818815 97379952672000000000/4438204603435370087084464552146740953154677412574 5618706117*7^(1/2), d[7,6] = -2519364373823950685032974284884563547030 69952284229953696997300048828125000000/9100834620406937002136897925109 700945244465300539628146933568187773609382701*7^(1/2)-7303417208846689 306832501286514466988244713997616442372528485130509277343750000000/275 6642806521261217947266381515728416314548539533453365706177804076626282 0201329, d[1,7] = 1630749243842357319398076780903984527431179310308730 70102421348689659750/6878824907002961357125227378838005500292218194911 33087948519356193177-6994894750938396651731488327805468194693574579588 81532054048398750/1567287515835716873348194891510140237022606105015112 98233884565093*7^(1/2), d[9,5] = 2437414794617643124169781328211525734 4566941742424132340939126329776722021558976935336200985921795322095810 211/497451098041522486420449572283854850989186539547190784589232754924 60674209849588541638078955675474411023748*7^(1/2)+47901879871146737411 5375153220564860715508133092718823566820002396538239888287074784313198 222024975036400066777/149235329412456745926134871685156455296755961864 157235376769826477382022629548765624914236867026423233071244, d[10,2] \+ = 281597663163162264423277154433242636609627963296730630613584596992/9 0074233258670683218250655973050364584855703288525066641120617043+11087 6201679254623037871993864066005939382896536227671113630596096/81885666 59879153019840968724822760416805063935320460603738237913*7^(1/2), d[7, 2] = -5125930837999526785905861026271839259957893894000868834592171386 7187500000/14445769238741169844661742738269366579753119524666076423704 0764885295387027*7^(1/2)-919190294333874368789413455291388409919396531 032571289112707441406250000000/112195474420889752460206201933892080436 08256164157319355743499406091275059097, d[10,8] = 22047100580764751308 730110947717870343403344733525449061074282421760000/819675522653903217 2860809693547583177221868999255781064341976150913+12135520524655026802 1752326174813280073096153911386860039155223040000/74515956604900292480 5528153958871197929260818114161914940179650083*7^(1/2), d[6,3] = 44721 084360784434677784729801158958699023144255148014082283092570062625/547 6951877661877162841135109869026583172108860891801636573517447172868+48 224465738913401109111657520645193884053404474223761391463039244625/113 47835768427746082547860224680534581087690402913308111888300171736*7^(1 /2), d[19,6] = -461301827513022838179182218793609706341381197273413030 44612500/1431678904333990350672407920047335791340218520185342538907+73 58249780628949799290333551278654301242533370906131902325000/3340584110 112644151568951813443783513127176547099132590783*7^(1/2), d[17,3] = -1 469604360050531892734122746115860001716171503446532235375411200/502519 295421230613086015179936614862760416700585055231156357+170797506941086 482591181116160029856187748421507562731687116800/502519295421230613086 015179936614862760416700585055231156357*7^(1/2), d[9,7] = 421523003793 2818736071241260142026334089662463853643029744516593998712667294578315 821888672681938019401125/119925529903935025655846087821565778926997719 27367183813626633435983769095913594151793172361541821217701*7^(1/2)+11 8961149824658369771865126296143521093082405738927579797140416872237104 2772730469473142015542448281668359647125/26116182647179930537073602544 9023796769322933262275161909347196335418539601710339843599914517296240 657874677, d[18,7] = 3161437105296062675426098100171489277186422929136 425922375000/477226301444663450224135973349111930446739506728447512969 -1435610723206137928951694790428931747906894457174721992937500/1431678 904333990350672407920047335791340218520185342538907*7^(1/2), d[13,6] = 3294023182185249314790175780818430348500565191103210664134000/3865533 0417017739468155013841278066366185900045004248550489+19786853818194620 71519578054725043358416118338445111406746700/2705873129191241762770850 96888946464563301300315029739853423*7^(1/2), d[16,3] = 408483912305568 79049094874218040955333336609646353788000000/1288511013900591315605167 1280426022122061966681668082850163*7^(1/2)-240460099579418410855336981 6093489658400897946481484185500000/12885110139005913156051671280426022 122061966681668082850163, d[12,6] = 4990550874280040648638722246400872 012250593517892524642718365021325/390022763773246616247478173950422900 79526101949731086504482529286+1042899908700117841039313312353663742325 293833338076584273444464975/780045527546493232494956347900845801590522 03899462173008965058572*7^(1/2), d[6,4] = -198843564303262219767550870 81652752135466456648956346616516634571272397625/3067093051490651211191 03566152665488657638096209940891648116977041680608-7608215819256113462 96739540636503375531588844487442872536857709497625/4701246246920066234 1983992359390786121649003097783705034965814997192*7^(1/2), d[13,7] = - 212782850393317473657425990408971734919709700004931017869430000/297646 0442110365939047936065778411110196314303465327138387653-12154687334329 60389436919410886628774863609227850563214449000/2705873129191241762770 85096888946464563301300315029739853423*7^(1/2), d[7,4] = -377607155541 8624364372291247477252444250786830809446983798340124914005126953125000 /826992841956378365384179914454718524894364561860036009711853341222987 88460603987-1926416708078672356862691802257093541607992503542908993735 97571538085937500000/1690155000932716871825423900377515889831114984385 9309415733769491579560282159*7^(1/2), d[8,2] = 17193471936928197999297 7297098299373880896289047165344616861328125/10003281132567597401474374 0741501706385108554447400765988597806761+26808225213594592443500811319 552220037327838869838401763840251953125/360118120772433506453077466669 4061429863907960106427575589521043396*7^(1/2), d[9,4] = -5633819537049 5597934100351009520883810395772743262057379680499122895766535290617578 8066319539947839934746531703/59694131764982698370453948674062582118702 3847456628941507079305909528090518195062499656947468105692932284976-16 7420595388300703671250951343821215453717089311180338142614388912369164 4293464256441993705518690601901111333/71064442577360355202921367469122 1215698837913638843977984618221320866774426422693451972556509649634443 1964*7^(1/2), d[11,8] = -184954952273673657842531786867489696670833275 4574645607920741018560000000/65004783931542113940619183113685085918155 6352527037679017120259217719-19646750026966477407033757598262438321200 46196602305650193859520000000/1140434805816528314747704966906755893300 9760570649783842405618582767*7^(1/2), d[11,5] = 1091254872061015387230 5496739422257480895793952366018425701905663872000/11404348058165283147 477049669067558933009760570649783842405618582767*7^(1/2)+4074768226599 438636959251699100997766836376192247637561931221314581376000/650047839 315421139406191831136850859181556352527037679017120259217719, d[17,4] \+ = 6881214659525582008629854370912320640539588082195871903822643200/502 519295421230613086015179936614862760416700585055231156357-650424614032 174093791051728417050717482161451339083515867955200/502519295421230613 086015179936614862760416700585055231156357*7^(1/2), d[16,5] = -1904500 0314337289255140605818165995614748194763205317544500000/12885110139005 913156051671280426022122061966681668082850163+323528778864697640409442 763387912876880174943254587092000000/128851101390059131560516712804260 22122061966681668082850163*7^(1/2), d[18,6] = -14706725244159097369839 650045084389134466768834323339695143750/143167890433399035067240792004 7335791340218520185342538907+23370588431083872231301180141871259666517 20500656422395131250/1431678904333990350672407920047335791340218520185 342538907*7^(1/2), d[1,5] = 742433916334227979215893548114808212321547 26827981191218422498436501339/1965378544857988959178636393953715857226 34805568895167985291244626622-1950206477967455511058326696135743462758 645751611098779858131152365/313457503167143374669638978302028047404521 221003022596467769130186*7^(1/2), d[1,2] = -49913430094603751641566666 5532904357414491300955138537955224539656077/43675078774621976870636364 310082574605029956793087815107842498805916+356436803021983973929777916 11006902844604115504781863583325615/3827319940990761595477887402955165 41397461808306498896786042894*7^(1/2), d[9,6] = -681034077919222653192 3930528788574078255295911702628489892563238630507332482067465622223792 7256276086704523325/12436277451038062160511239307096371274729663488679 769614730818873115168552462397135409519738918868602755937-142319033624 9382817659067960999257689087289626966939103313183108429992725602177579 1805841788147988818788330975/24872554902076124321022478614192742549459 326977359539229461637746230337104924794270819039477837737205511874*7^( 1/2), d[6,5] = 1690679026505303332964833560185838847186444729114544133 1568974896761472375/76677326287266280279775891538166372164409524052485 222912029244260420152+110765212341340880571499611007372697779544763991 03782076984857286503375/3290872372844046363938879465157355028515430216 84485935244760704980344*7^(1/2), d[17,6] = 204815956340976350229984779 37197814100746282772852146893127680000/5025192954212306130860151799366 14862760416700585055231156357-1215177376123792931865599562330744230829 43425456607622266880000/3865533041701773946815501384127806636618590004 5004248550489*7^(1/2), d[19,3] = 2400119601655802024913733400625736889 3543757664440775397285500/10021752330337932454706855440331350539381529 641297397772349-795560158652028881526157940792991314731592433374109190 961500/3340584110112644151568951813443783513127176547099132590783*7^(1 /2), d[11,7] = 7828107498105571372108920646280380631802076888100147265 069689408000000/114043480581652831474770496690675589330097605706497838 42405618582767*7^(1/2)+40477669336499798751079880256441409547001320907 230018857505974948288000000/455033487520794797584334281795795601427089 4467689263753119841814524033, d[9,1] = 0, d[4,4] = 0, d[18,4] = -64100 32074881028772399481554077690611991277986846048477664625/1908905205778 653800896543893396447721786958026913790051876+962240401383305759281068 249126719609416913245186419069251125/143167890433399035067240792004733 5791340218520185342538907*7^(1/2), d[6,6] = -6467507300969516521025929 921764741053468015851759795471956639951946875/164543618642202318196943 973257867751425771510842242967622380352490172*7^(1/2)-7211053313381233 086300879132909929222458082754180954209603197468316165625/191693315718 16570069943972884541593041102381013121305728007311065105038, d[20,8] = -434922168250264826621856885191498764502971172424562510000000/1002175 2330337932454706855440331350539381529641297397772349+89819114766083809 7980111166950774250753970120340516135250000/10021752330337932454706855 440331350539381529641297397772349*7^(1/2), d[14,7] = -1274675079284016 3841933814131143866811838471678117766637062000/14794015344784566956948 215173822469843848924708581872902039+532632405835979321197237247207977 0450825900701602014640012000/14794015344784566956948215173822469843848 924708581872902039*7^(1/2), d[13,1] = 0, d[12,1] = 0, d[5,2] = 0, d[11 ,3] = 1508975216807991869869735225622609536772849164463590643987330165 53216000/6500478393154211394061918311368508591815563525270376790171202 59217719+4751056949898291812513469769697510413196751552721878953720568 4352000/39325338131604424646472585065750203217275036450516496008295236 4923*7^(1/2), d[8,4] = 10075011660820485990331332996082696341781459119 2968820314883670072265625/42133820130374720255010063600320518729407723 1332452026343973962077332*7^(1/2)+484330218915220688426360032319541473 7330457281426178710961574794533203125/50560584156449664306012076320384 62247528926775989424316127687544927984, d[4,8] = 0, d[15,8] = 13911797 4112156976275654586466561785559001945472146240000000000/14646075191336 72128737873302208424514541043546149605417301861-3871523079657954441500 094362198449477079860886355680000000000/443820460343537008708446455214 67409531546774125745618706117*7^(1/2), d[5,7] = 0, d[4,7] = 0, d[8,1] \+ = 0, d[19,1] = 0, d[14,6] = 207506855303897361423967362774717765620990 60814518578546774600/1479401534478456695694821517382246984384892470858 1872902039-86708273633195214583917499773999608807387546269001908022596 00/14794015344784566956948215173822469843848924708581872902039*7^(1/2) , d[7,1] = 0, d[2,4] = 0, d[3,2] = 0, d[2,7] = 0, d[12,7] = -871735628 96690203922763033059969335557665368796189760818312215457125/8190478039 23817894119704165295888091670048140944352816594133115006-1884216309611 9342192252573336439026736716490976415059145514438625/22942515516073330 36749871611473075887030947173513593323793089958*7^(1/2), d[15,6] = -25 111974848881885321672674753908806295730804842016137600000000/443820460 34353700870844645521467409531546774125745618706117*7^(1/2)+82033185179 704454423090234313012641155003796319965868800000000/133146138103061102 612533936564402228594640322377236856118351, d[13,4] = 4469818265075669 0761769735727252352467984063731890113472400110/29764604421103659390479 36065778411110196314303465327138387653+1163838301429743525592056429018 51452389500804492627501586498/3865533041701773946815501384127806636618 5900045004248550489*7^(1/2), d[17,5] = -162840451262782904612965390948 99248799774589651392237003204198400/5025192954212306130860151799366148 62760416700585055231156357+1352756061845794935005032527459177157343737 472884618209964851200/502519295421230613086015179936614862760416700585 055231156357*7^(1/2), d[14,3] = -2243525181240063021054547606023596942 176560700713986922286012/147940153447845669569482151738224698438489247 08581872902039+9374735838632006782900231997021291271110603904618683107 02712/14794015344784566956948215173822469843848924708581872902039*7^(1 /2), d[6,7] = 41986895194618622915638860421102159164607440878383392575 567397620821546875/134185321002715990489607810191791151287716667091849 140096051177455735266+325644796743427701456805066380567259799745146534 89394159326992515625/1348718185591822280302819452933342224801405826575 762029691642233526*7^(1/2), d[4,1] = 0, d[3,7] = 0, d[10,1] = 0, d[5,5 ] = 0, d[14,5] = -1776924232197973211330348659185292420892320063831889 4675721108/14794015344784566956948215173822469843848924708581872902039 +7425009276210807541841567031004250686195270805576966863546408/1479401 5344784566956948215173822469843848924708581872902039*7^(1/2), d[9,3] = 126707907129230323317925652447181817497902186839500388646054412454304 5290853187282757708660292816458666030863/10659666386604053280438205120 3683182354825687045826596697692733198130016163963404017795883476447445 16647946+3077453371260292120591423999734486315665303260173440817411443 098308014249943618454887532557145872390493628329/497451098041522486420 4495722838548509891865395471907845892327549246067420984958854163807895 5675474411023748*7^(1/2), d[3,4] = 0, d[3,1] = 0, d[15,1] = 0, d[5,6] \+ = 0, d[4,6] = 0, d[14,1] = 0, d[2,3] = 0, d[15,2] = 128517437682336804 6557017759143675528393594918717610624000000/16273416879263023652643036 6912047168282338171794400601922429-35765200672383149604976200334469073 612125825077957568000000/493133844826152231898273839127415661461630823 6193957634013*7^(1/2), d[7,5] = 12842507633976844176721173398876707596 683191966556931280741549480590463867187500000/826992841956378365384179 91445471852489436456186003600971185334122298788460603987+2804599143838 483635686420579755276860426016270238510991238647118735351562500000/118 3108500652901810277796730264261122881780489070151659101363864410569219 75113*7^(1/2), d[14,8] = 319914138306829468600531446371143589222662754 1404474243530000/14794015344784566956948215173822469843848924708581872 902039-1336784878880839335796750843751432009304881437822766197780000/1 4794015344784566956948215173822469843848924708581872902039*7^(1/2), d[ 20,3] = 1285842974433897654229156181372139355262370475712225168576900/ 10021752330337932454706855440331350539381529641297397772349-6298922792 25029707087375521420205683954737389885991590309100/1002175233033793245 4706855440331350539381529641297397772349*7^(1/2), d[12,4] = 4128407656 4951970958647826412373517915122082928531250158904168376423/18721092661 11583757987895234962029923817252893587092152215161405728+1226841689391 45220693552574191426700519930119672305467038075627253/2228701507275694 9499855895654309880045443486828417763716847159592*7^(1/2), d[13,2] = 2 721489311917062733130875756847927895477861273414985588679/472454038430 21681572189461361562081114227211166116303783931+4025858475553722530307 73794622450357040040289344403163858/4295036713001971052017223760142007 374020655560556027616721*7^(1/2), d[8,6] = 585473819099436616359792295 616289997925144761821549862844906260302734375/105334550325936800637525 159000801296823519307833113006585993490519333+941148748854609247806624 2425829708665068152652753598274320297900390625/16205315434759507790388 486000123276434387585820478924090152844695282*7^(1/2), d[7,7] = 127574 0725733037701289934404083314082002249520463210939785634138557788085937 50000000/5788949893694648557689259401183029674260551933020252067982973 38856091519224227909+2011876798826786379735123103598287827469131143296 795192190628542480468750000000/118310850065290181027779673026426112288 178048907015165910136386441056921975113*7^(1/2), d[16,2] = 31675561449 932759585814449048894682735398944250970524500000/143167890433399035067 2407920047335791340218520185342538907-53809165379932608607360237125658 5993376203834697972000000/14316789043339903506724079200473357913402185 20185342538907*7^(1/2), d[19,7] = 203703904923247205940579809550396524 495519753185149039998000000/100217523303379324547068554403313505393815 29641297397772349-4520032655681925676944197548888486813595105051207128 217750000/3340584110112644151568951813443783513127176547099132590783*7 ^(1/2), d[11,6] = -231728360507055975476609780772190483362222662604450 3834275727836710400000/21668261310514037980206394371228361972718545084 2345893005706753072573-12743529675226432076498828545773668006242117239 918967340627507526400000/114043480581652831474770496690675589330097605 70649783842405618582767*7^(1/2), d[7,3] = 3541056331199227679532598709 97321382253869744249799701441590241518554687500000/1183108500652901810 27779673026426112288178048907015165910136386441056921975113*7^(1/2)+67 9408519652911178113673577801272243742709058870182628025410510811816406 25000000/1181418345651969093405971306363883606991949374085765728159790 4874614112637229141, d[19,4] = -11111448857540695169781464761711248157 6947418506724192036853000/10021752330337932454706855440331350539381529 641297397772349+432803163396713955929400840527752756100563709091898459 035500/477226301444663450224135973349111930446739506728447512969*7^(1/ 2), d[20,6] = -6713624946475107612879670524364761895187151459653206454 772500/10021752330337932454706855440331350539381529641297397772349+582 5963850779902104934735625648005867645807120912036599905000/10021752330 337932454706855440331350539381529641297397772349*7^(1/2), d[2,2] = 0, \+ d[1,3] = -246230945735113892339372585690030669766812810495003823744255 113735/313457503167143374669638978302028047404521221003022596467769130 186*7^(1/2)+3231074952031261645234323783136870725405275716603642707650 36005492591/5039432166302535792765734343471066300580379629971670973981 826785298, d[10,6] = 8286782930724444950242426855421112899386181596425 1993805443017305395200/81967552265390321728608096935475831772218689992 55781064341976150913+6054998946507855941154085224896194255295250567149 9453948353424025600/57319966619154071138886781073759322917635447547243 224226167665391*7^(1/2), d[9,2] = -37142425507719738213341950069205380 303694158659572836293946045413460539828251623048052653226120876208741/ 2193346993128406024781523687318583999070487387774209808594500683089094 9827976009057159646805853383779111-57912823636946473834949306239340766 0131981836612818408676057164051275765616093012401129378482575316097089 3/78960491752622616892134852743469023966537545959871553109402024591207 4193807136326057747285010721816047996*7^(1/2), d[10,3] = -256962403700 955597405150437397475457337271082383744977126104318773248/117096503236 2718881837258527649654739603124142750825866334568021559-85105166119311 678537257515346907811541623568055391798062460169174016/745159566049002 924805528153958871197929260818114161914940179650083*7^(1/2), d[6,2] = \+ -318555525426354785267263601778513469051754900650813057635058169612625 /273847593883093858142056755493451329158605443044590081828675872358643 4-23685970887837130176302153274051068033533413543752892588902929763625 /47012462469200662341983992359390786121649003097783705034965814997192* 7^(1/2), d[12,5] = -35102027375928656197360237373946948817888902208668 958451100751617257/468027316527895939496973808740507480954313223396773 038053790351432-178611363640234164841404819600730131782365246311505964 9161263392851/15600910550929864649899126958016916031810440779892434601 7930117144*7^(1/2), d[10,4] = 1428166405571240813647524526491577709988 8212847111472750534674661140736/81967552265390321728608096935475831772 21868999255781064341976150913+4629918267934746510646027686053787321852 0453497703926889996052319232/10645136657842898925793259342269588541846 5831159165987848597092869*7^(1/2), d[11,2] = -163347330891545876403903 842900594992244726299935020978849222045568000/114043480581652831474770 49669067558933009760570649783842405618582767*7^(1/2)-71657832081718310 3406175772945383311056484201637487951993884383744000/21668261310514037 9802063943712283619727185450842345893005706753072573, d[12,3] = -92850 310609122940672342183074126380973594493023226754287693593983/334305226 09135424249783843481464820068165230242626645575270739388-2255127622897 12345687608177797492663198173982033850914510537775289/1560091055092986 46498991269580169160318104407798924346017930117144*7^(1/2), d[19,2] = \+ -10810719405586300269221788426869054893442683433666843966500/530251446 04962605580459552594345770049637722969827501441+1497118870780096785653 851352791263510405606027720535895500/530251446049626055804595525943457 70049637722969827501441*7^(1/2), d[10,5] = -67405275067629095544857313 1476753048620070444486152452043108816262144/74515956604900292480552815 3958871197929260818114161914940179650083*7^(1/2)-485722733188634615447 53218173392303920428065931051295941934257889156096/8196755226539032172 860809693547583177221868999255781064341976150913, d[18,8] = -242744046 5292287234817246484618844561890536685058664154375000/14316789043339903 50672407920047335791340218520185342538907+3603052847918769760273121219 44428132436664116668984431562500/1431678904333990350672407920047335791 340218520185342538907*7^(1/2), d[11,1] = 0, d[18,2] = -415192595528324 47855406165498677935500678562865236277287875/6363017352595512669655146 31132149240595652675637930017292+3328506778536405261654129899831216571 480624909539443236125/159075433814887816741378657783037310148913168909 482504323*7^(1/2), d[19,5] = 25960999453991911252038157509129655149054 9603905366764948523500/10021752330337932454706855440331350539381529641 297397772349-630102187352623228833535306527867999766036748559536968452 8500/3340584110112644151568951813443783513127176547099132590783*7^(1/2 ), d[14,2] = 295537263218050228202416128071316106023653790387136890582 28/1643779482753840772994246130424718871538769412064652544671-12349243 040856320027880115681345382618617915842794662375528/164377948275384077 2994246130424718871538769412064652544671*7^(1/2), d[8,3] = -7625009568 7711426175707444721549275927850831457556362286280073529296875/63200730 1955620803825150954004807780941115846998678039515960943115998-26456361 535933405820655294979290299863007510302862110709158453404296875/421338 201303747202550100636003205187294077231332452026343973962077332*7^(1/2 ), d[15,4] = 371531978296234614900819651625346784235446910086141052544 000000/1464607519133672128737873302208424514541043546149605417301861-1 0339387401121986568134497304353564371909018441162307008000000/44382046 034353700870844645521467409531546774125745618706117*7^(1/2), d[11,4] = -11981006228626295462560796941505115508704941017998062630793220138212 16000/6500478393154211394061918311368508591815563525270376790171202592 17719-5246910725371604176437215470414409301918039054299216617679970826 112000/114043480581652831474770496690675589330097605706497838424056185 82767*7^(1/2), d[10,7] = -48353116460521333282762104805305279216991173 3168059717563493500416000/74515956604900292480552815395887119792926081 8114161914940179650083*7^(1/2)-482504110415092660293138266356471191063 590434946166327068843254107648000/573772865857732252100256678548330822 40553082994790467450393833056391, d[18,3] = -2526784166660232040185324 78629807794349483090715026166341375/1431678904333990350672407920047335 791340218520185342538907*7^(1/2)+2067355049882807580588801884762356866 603197474794104605922875/286335780866798070134481584009467158268043704 0370685077814, d[16,7] = -13661914698306229195729216969711744974728895 376228541750000000/128851101390059131560516712804260221220619666816680 82850163+232083093008364823648443315110295386092801194235238000000000/ 12885110139005913156051671280426022122061966681668082850163*7^(1/2), d [18,5] = 1151711925877643737505847632473788074536642325115699176157850 0/1431678904333990350672407920047335791340218520185342538907-200127195 0418236472823158285298015115392245938807932725791125/14316789043339903 50672407920047335791340218520185342538907*7^(1/2), d[16,8] = -58247520 424623867073113342801624660418662229380970000000000/128851101390059131 56051671280426022122061966681668082850163*7^(1/2)+34288264824201544624 10850018536495909254143411432276250000000/1288511013900591315605167128 0426022122061966681668082850163, d[20,2] = -18724388334826048809573294 764906626932882278291570734642700/111352803670421471718965060448126117 1042392182366377530261+11853580241930494292048949394379682023094861487 04187724700/159075433814887816741378657783037310148913168909482504323* 7^(1/2), d[8,7] = -375784335076447250223603077283108559615445652570024 21026817336669921875/1053345503259368006375251590008012968235193078331 13006585993490519333*7^(1/2)-14609849716667403498351011131564469186659 56503535718197130879330322265625/3160036509778104019125754770024038904 70557923499339019757980471557999, d[16,6] = -3778126173645638167812746 52988438993015682333907315400000000/1288511013900591315605167128042602 2122061966681668082850163*7^(1/2)+222404987949399822783380734599709307 66377295148098911525000000/1288511013900591315605167128042602212206196 6681668082850163, d[1,6] = -401822368168950462616648969731266769950301 63633026716312006139827990900/9826892724289944795893181969768579286131 7402784447583992645622313311+87593279292444328573400503228921644809254 715323653677309899691125/120560578141208990257553453193087710540200469 61654715248760351161*7^(1/2), d[12,2] = 220462599128642020502802009787 550092329168455670656206543685061/557175376818923737496397391357747001 1360871707104440929211789898+38194153832181999803960746931575834214126 19088899955423132510917/2228701507275694949985589565430988004544348682 8417763716847159592*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 "s ubs(dd,matrix([seq([seq(d[j,i],j=1..11)],i=1..8)])):\nevalf[8](%);\nsu bs(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-$!)d>=6!\"'F+F+F+F+$!)oJ\\9!\"($!)\\u?5F2$\")9XT@F/$!)D %)4@F/$\")G2&*QF/$!))z-7%F/7-$\")Jv.iF/F+F+F+F+$\")s)3%>F/$\")\\&pO\"F /$!)exnG!\"&$\")fWDGFF$!))zh@&FF$\")7xFFF+F+F+F+$!)a[w5F F$!))>;e(F/$\")4d!f\"!\"%$!)K4n:FV$\")'zI*GFV$!)INgIFV7-$\")R&Hh$FFF+F +F+F+$\")AW&4$FF$\")35!=#FF$!)\"*ptXFV$\")!*=1XFV$!)M2>$)FV$\")\"p+!)) FV7-$!)/y'*QFFF+F+F+F+$!)wp,[FF$!)f!=Q$FF$\")Dz%4(FV$!)(p+*pFV$\")uY!H \"!\"$$!).3l8Fcp7-$\")')f_AFFF+F+F+F+$\")M$yw$FF$\")nbFV$\" )A-&[&FV$!)Sh75Fcp$\")=;r5Fcp7-$!)[f!Q&F/F+F+F+F+$!)(\\6;\"FF$!)`!z<)F /$\")um:F9Q)pprint226\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can c heck which 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->`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 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 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 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#*$%\"uG\"\"*" }{TEXT -1 45 " in order to provide appropriate \+ weighting. 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{XPPMATH 20 "6#/&%\"aG6$\"#9\"#8,&#\"\"$\"$#R\"\"\"*( F+F-F,!\"\"\"\"(#F-\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#:\"\"\",&#\"4ZL:3v4cYF-\"7ON?\\P0ZXK.b!\"\"F(#F-\"\"#F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"),&#\"2vmsn/$4_:\"1[; xv4QDc\"\"\"*(\"3vAqni>BOAF-\"3Whg@tm5v:!\"\"\"\"(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"*,&#\"en2:!yQvWC29#)R>c\"[2 R`^9Rx.MDtbgC\"en+++74ZW$f&>QP\\@jKG!3%*f(3#o!=xO%f#!\"\"*(\"Wn<[%[R_I j7#fGRIXpNq90;<)f94*>\"\"\"\"W+++?Nkd%zD]*=6-IZWE:K?]AdnnTF-\"\"(#F0\" \"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"#5,&#\"2C-38_ \"fgU\"2v$4hH_u!p\"\"\"\"*(\"4\\jDj<_]m5\"F-\"3++v$)f-V2')!\"\"\"\"(#F -\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"#6,&#\"27:UF(\"U++?2^usu.F-_7'p1VG]`Il+2 \\_&!\"\"\"\"(#F(\"\"#F-#\"WTz]qC2xbZ*4%4A&Gf#3!z,VG?$[\\Y@\"X++!3?/*) o*3/?3hmzg-ar$oj'\\ibtI%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$\"#;\"\"',&*(\"U$G5vteI&*p?sn#=1([VV%*>J\"X+c=%QVZ$**zP!)oRW#)GIsVs1C&Rb%4O$F. 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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#=\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#=\" \"',&*(\"7Fc7.6Tp^Sl7\"\"\"\"8+P8TPpnR%e&o#!\"\"\"\"(#F,\"\"#F.#\"7$)* [_56K&)pf(RF-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#=\"\"(, &*(\";+++D'p[(Q@4zl7\"\"\"\"JZKD!Q\\\"pN<-UMAP>OQ(oRy^\"!\"\"\"\"(#F,\"\"#F,#\"6oJY6_3/)zKB\"6DMx&3:i!>N b%F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#=\"#7,&*(\"1h\\UW5 RaS\"\"\"\"4+DZTZ\"\\.m8!\"\"\"\"(#F,\"\"#F.#\"36ErVv/P`C\"5+v3A6sB0\\ ?F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#=\"#8,&#\"))*f.5\"+ D\"GgX$\"\"\"*(\"(cQX\"F-F,!\"\"\"\"(#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#=\"#9,&#\",%=>()QG\"-vV.(G'f\"\"\"*(\"+c;&f $RF-\",DTWa:(!\"\"\"\"(#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#=\"#:,&*(\")%yYI*\"\"\"\"+n_ufX!\"\"\"\"(#F,\"\"#F,#\"*'\\9 VD\"+04'pD$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#=\"#;,&*( \");C*f\"\"\"\"\"*B*yxQ!\"\"\"\"(#F,\"\"#F,#\"()3D@\")*)pRbF," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#=\"#<\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"\",&*(\"0hfqMbQ3%F(\"3++S7Peg_9! \"\"\"\"(#F(\"\"#F-#\"6pn13\"R!3#>\\P\"8++!R6$QkH)z>5F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"#\"\"!" }}{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$\"#>\" \"',&*(\"7tZZr/)[$4*G%=\"\"\"\"8+k?&*e*\\QTS:q!\"\"\"\"(#F,\"\"#F.#\"8 n6S?(pAHuDS\"*\"9+%yir*>mp+(Qf)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#>\"\"(,&*(\":Dc,Tv-fPUwi,#\"\"\"\";%p6Kl%*G)y$>WrW'!\"\"F(# F,\"\"#F.#\">D\"G8'G-mnr`>(=%Q$\"?*)***3g(zR`$*yLqmuDF," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"),&*(\"3DYJQ^XXM5\"\"\"\"4[)=V* e_Uww#!\"\"\"\"(#F,\"\"#F.#\"3v$fPKZbs]$\"4KqcWhxaV7\"F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"\"*,&*(\"Q(Qr/0Fl.B0AzXwK!zmAOKm3 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1 "" {XPPMATH 20 "6#/&%\"aG6$\"#>\"#=\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#?\"\"\",&*(\"3^f>ek#ysw#F( \"5++SgT6k2ov!\"\"\"\"(#F(\"\"#F(#\"5$>\")\\vG[G$*z#\"6++q#eqx:X3^F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#?\"\"#\"\"!" }}{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$\"#?\"\"',&*(\"7$oWSCeZ^poR#\"\"\"\"8+k?&*e*\\QTS:q!\"\"\"\"(#F,\" \"#F,#\"6\"4mJw%zd(Rf8\"6+o:k=9$z8SZF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#?\"\"(,&*(\":vV['R*3a^?vBi#\"\"\"\";%p6Kl%*G)y$>WrW' !\"\"F(#F,\"\"#F,#\"RVxMd')*f?P)R@%!\"\"\"\"(#F,\"\"#F,#\"fnH# =(zRnwUe$)*)*eS$e&e.*[cV8%eh`\\_s>\"fn++:$Qe@dE^y3\"\"!" }}}{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 "inte rpolation coefficients " 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FhpFgp/&F&6$FAFE,&*(\"io+++]7yvw63%Ra28#=Y(yXWjh@eh?P[^'=,x&f?[$\\]F(F _]lFgpFAFhpFgp#\"]p+++]P%['*)oo8H4ojiB%[B'>p_3(Gr%HDRx7>T\"yPu_#HeFdgl Fgp/&F&6$FEFE,&#\"covV[r#G_\\d'3[v6?.$z8Mev7A'\\1f0$=usdd$)Hn%FddnF(*( \"aoDJqq*p/\\;4k[i#H=d/hMO>%HSE!*H)GCxGGKJ%*F(F[[nFgpFAFhpF(/&F&6$FIFE ,&#\"iqD\"eyKk+O(*)o&*)GBbC<%pb6\"[JF7y`tA;n6Ni8W$o'*zwn\\Hl`s#e(3!fu, dV&\"fq6yE3egcn@f&GiS\">(Qdc]X$>mX#>W)3$RgY!**)=CQ6*G@zrL:['=9JNK)3t$F gp*(\"hqvV#*f#z!y`\\$=WO$pcT6Dl\"\\La8(fX+y2v='))ovh6!ef*Q'*)4f/rn9D43 pq4\"F(Fe[nFgpFAFhpFgp/&F&6$FMFE,&#\"bo++w@CGu51\\a_LZM.MMqyrZ46I(38vk 2e+r/AFailF(*(\"`o++/B_:R+'oQ6R:'4t+G8[]cI-m>Y+7KQCE)fdP.2uZmp-]nk>F(FcuFgpFAFhpFgp/&F&6$FUFE,&#\"^o D\"3jo4m,xTb+i-N@Z\\,7(*f&GJefL>Q))GK)R\"]oeyeZM^fK>\\eIy&Q-(o7&=_MCu[ )R(>8Ho+<\"F(*(\"\\ovVFfy%\\'e8&Q[-k>$FealF(*(\"hn++y(>mF#yV\")[I4?V^P%3v'zN$ R3))y[yO8F(FealFgpFAFhpFgp/&F&6$F[oFE,&*(\"hn+++++obj)3')zqZ\\%)>iV4+: WazlzI_rQF(F^blFgpFAFhpFgp#\"jn+++++CY@ZX>+fbyhlY'eacFwp:7T(z6R\"FablF (/&F&6$F_oFE,&*(\"fn+++++(4QHAm=/mC;!GM8J2nQiC/_Z#eF(FhblFgpFAFhpFgp# \"hn+++]iFK9TVTD4f\\O&=+&3TiW:?C[E)GMFhblF(/&F&6$FcoFE,&*(\"jn+++#>?tZ !R:n!zReA]1(y1LZ75]jw)oZNCF(FaclFgpFAFhpFgp#\"[o+++SY]&f\\vqp77t-wp(=z%G0.OF(FaxFgpFAFhpF(/&F&6$F[ pFE,&*(\"hn++D;Az!)eXK])3$\\QhZ[wL^eK4&fuUUM6F(FfdlFgpFAFhpF(#\"in+++& [L-)y)4>Q))[)*)F(FcdlFgpFAFhpF(#\"gn+++5DcCC&)o&=iE[E] #o@#\\VFcdlFgp" }}}{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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Checking t he interpolation scheme .. A" }}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "The coefficients have be en extracted from Maple code (with indices modified to allow for the r emoval of the 13th stage)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13627 "ee := \{c[2]=1/16,\nc[3]=112/1065,\nc[4]=56/355,\nc[5]=39/100, \nc[6]=7/15,\nc[7]=39/250,\nc[8]=24/25,\nc[9]=14435868/16178861,\nc[10 ]=11/12,\nc[11]=19/20,\nc[12]=1,\nc[13]=1,\nc[14]=1/2-7^(1/2)/14,\nc[1 5]=9/20,\nc[16]=1/10,\nc[17]=13/40,\nc[18]=2/5,\nc[19]=3/10,\nc[20]=7/ 10,\na[2,1]=1/16,\na[3,1]=18928/1134225,\na[3,2]=100352/1134225,\na[4, 1]=14/355,\na[4,2]=0,\na[4,3]=42/355,\na[5,1]=94495479/250880000,\na[5 ,2]=0,\na[5,3]=-352806597/250880000,\na[5,4]=178077159/125440000,\na[6 ,1]=12089/252720,\na[6,2]=0,\na[6,3]=0,\na[6,4]=2505377/10685520,\na[6 ,5]=960400/5209191,\na[7,1]=21400899/350000000,\na[7,2]=0,\na[7,3]=0, \na[7,4]=3064329829899/27126050000000,\na[7,5]=-21643947/592609375,\na [7,6]=124391943/6756250000,\na[8,1]=-15365458811/13609565775,\na[8,2]= 0,\na[8,3]=0,\na[8,4]=-7/5,\na[8,5]=-8339128164608/939060038475,\na[8, 6]=341936800488/47951126225,\na[8,7]=1993321838240/380523459069,\na[9, 1]=\n-1840911252282376584438157336464708426954728061551/29919236151711 51921596253813483118262195533733898,\na[9,2]=0,\na[9,3]=0,\na[9,4]=\n- 14764960804048657303638372252908780219281424435/2981692102565021975611 711269209606363661854518,\na[9,5]=-87532504850213044111861342178526674 2862694404520560000/\n170212030428894418395571677575961339495435011888 324169,\na[9,6]=\n7632051964154290925661849798370645637589377834346780 /1734087257418811583049800347581865260479233950396659,\na[9,7]=\n75198 34791971137517048532179652347729899303513750000/1045677303502317596597 890707812349832637339039997351,\na[9,8]=\n1366042683489166351293315549 358278750/144631418224267718165055326464180836641,\na[10,1]=-630777367 05254280154824845013881/78369357853786633855112190394368,\na[10,2]=0, \na[10,3]=0,\na[10,4]=-31948346510820970247215/6956009216960026632192, \na[10,5]=-3378604805394255292453489375/517042670569824692230499952,\n a[10,6]=1001587844183325981198091450220795/184232684207722503701669953 872896,\na[10,7]=187023075231349900768014890274453125/2522469884980817 8010752575653374848,\na[10,8]=1908158550070998850625/11708706703918992 9394176,\na[10,9]=\n-52956818288156668227044990077324877908565/2912779 959477433986349822224412353951940608,\na[11,1]=\n-10116106591826909534 781157993685116703/9562819945036894030442231411871744000,\na[11,2]=0, \na[11,3]=0,\na[11,4]=-9623541317323077848129/3864449564977792573440, \na[11,5]=-4823348333146829406881375/576413233634141239944816,\na[11,6 ]=6566119246514996884067001154977284529/970305487021846325473990863582 315520,\na[11,7]=\n2226455130519213549256016892506730559375/3648804431 59675255577435648380047355776,\na[11,8]=39747262782380466933662225/175 6032802431424164410720256,\na[11,9]=4817577141926095533524468380517154 8038966866545122229/\n198978642051381514652888016595206411890385284361 2160000,\na[11,10]=-2378292068163246/47768728487211875,\na[12,1]=-3218 022174758599831659045535578571/1453396753634469525663775847094384,\na[ 12,2]=0,\na[12,3]=0,\na[12,4]=26290092604284231996745/5760876126062860 430544,\na[12,5]=-697069297560926452045586710000/411079677552454305940 36502319,\na[12,6]=1827357820434213461438077550902273440/1393810139142 45317709567680839641697,\na[12,7]=\n6435048028142415509419492271941075 00000/242124609118836550860494007545333945331,\na[12,8]=16225993815138 0266113750/59091082835244183497007,\na[12,9]=-230282516328735238185454 14856857015616678575554130463402/\n20013169183191444503443905240405603 349978424504151629055,\na[12,10]=7958341351371843889152/32844679884432 03581305,\na[12,11]=-507974327957860843878400/121555654819179042718967 ,\n\na[13,1]=4631674879841/103782082379976,\na[13,2]=0,\na[13,3]=0,\na [13,4]=0,\na[13,5]=0,\na[13,6]=14327219974204125/40489566827933216,\na [13,7]=2720762324010009765625000/10917367480696813922225349,\na[13,8]= -498533005859375/95352091037424,\na[13,9]= 405932030463777247926705030 596175437402459637909765779/788039194363218410832018860412015372297691 15088303952,\na[13,10]=-10290327637248/1082076946951,\na[13,11]=863264 105888000/85814662253313,\na[13,12]=-29746300739/247142463456,\na[14,1 ]=\n-(121914470767797245 + 18481672943302477*7^(1/2))*(-7 + 7^(1/2))/1 5947569906836632064,\na[14,2]=0,\na[14,3]=0,\na[14,4]=0,\na[14,5]=0,\n a[14,6]=\n-1125*(-14397833825653754 + 1311411061499725*7^(1/2))*(-7 + \+ 7^(1/2))^2/5444065197416588490496,\na[14,7]=\n12207031250*(11317817629 4624287 + 27447016019592206*7^(1/2))*(-7 + 7^(1/2))^2/1834881952480713 51590841440643\n,\na[14,8]=\n390625*(82208106769201 + 40422290280248*7 ^(1/2))*(-7 + 7^(1/2))^2/358978501070780677632,\na[14,9]=-179344334901 09959879230535901290222453687161*\n(31825092488959 + 16394910776018*7^ (1/2))*(-7 + 7^(1/2))^2/\n15136656845328699235261418270793991271094051 626161423100160,\na[14,10]=\n144*(56617048920691 + 27819317025697*7^(1 /2))*(-7 + 7^(1/2))^2/90932336237027285,\na[14,11]=\n-2000*(1015750445 62621 + 49167448518551*7^(1/2))*(-7 + 7^(1/2))^2/1442287028491431591, \na[14,12]=-29746300739*(10 + 11*7^(1/2))*(-7 + 7^(1/2))^2/67815891972 3264,\na[14,13]=-(1 + 7^(1/2))*(-7 + 7^(1/2))^2/5488,\na[15,1]=\n37917 56097508153347/82000657682944000000 + 821104665636321*7^(1/2)/64224291 3320960000,\na[15,2]=0,\na[15,3]=0,\na[15,4]=0,\na[15,5]=0,\na[15,6]= \n768457351210057567857/5182664553975451648000 - 62532827932224226599* 7^(1/2)/2267415742364260096000,\na[15,7]=\n253006492725129434765625/10 78258516612031004664232 - 46193539913722265625*7^(1/2)/550332454705374 9203536,\na[15,8]=\n15520930467726675/5625380975771648 - 2236231962677 02275*7^(1/2)/157510667321606144,\na[15,9]=-24605573253403773914515339 07481561939821407244475387801507/\n25943677180682087599408028326321493 73819559344470912000000 +\n1990914598171605147035694530392859212633052 39484481767*7^(1/2)/\n416767572250203215264447300211189502579457643520 000000,\na[15,10]=\n42605915213080224/16907452296109375 - 110665052176 3256349*7^(1/2)/860743025983750000,\na[15,11]=\n-61485359208535917/143 02443708885500 + 887648262380253891*7^(1/2)/400468423848794000,\na[15, 12]=-1539638779949901/263618627686400000,\na[15,13]=101277*7^(1/2)/640 00000 - 79893/16000000,\na[15,14]=708939*7^(1/2)/16000000,\na[16,1]=-4 215121951102178392285859687667763860057798131121*7^(1/2)/\n55249070065 30535028430669612520227037472745107200000 +\n2146494832028430179008259 28522094099475577072470507941/\n43073556249663683715402607966610820040 89688904200800000,\na[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[ 16,6]=1185913430248674317434870618267722069953058737510283*7^(1/2)/\n3 0008442317179171818955614503925786051785131637358800 -\n31199443191133 7356840024036100680775990247322848978459/\n336094553952406724372302882 4439688037799934743384185600,\na[16,7]=3993218188734046844685987846507 8752005970844232914062500*7^(1/2)/\n7991406437185379813952210414813050 100306894094759398308201 +\n354364517241886228469307408685840332599095 120809109472656250/\n4531127449884110354510903305198999406874008951728 578840749967,\na[16,8]=\n448242553947529695555930499668879693603303962 9875*7^(1/2)/20845977587384504244419538387493205979265982789632\n - 46 323379005207505717635146937831732889161714866875/487074284012926397249 41806040008163970784940556544,\na[16,9]=\n2257427684900935524736890230 539751689489149601754038401722717509558293702749265180053129*7^(1/2)/1 5382319963668643757680340043617345728033389807810527409513111150183672 0940560681990528000 +\n42492774700711579417354994420731305451245624654 71930167206290592193363892663828255136238663/\n51104152848047584046512 223465220908365985929810714271725745707161059426234354242200368775000, \na[16,10]=\n82674886809376207867730668398503287366098540852*7^(1/2)/7 29064214207911394139200255820908786902329141125\n - 728800494986325248 78182125169775730823179168857424/1079575855654022641321508071119422626 75921815128125,\na[16,11]=-1445233402113318024032534072777477015793202 1057436*7^(1/2)/\n42400459244893414400850901677665011873556514674975\n + 1379482776995886509031984497778742508172264966358576/89040964414276 1702417868935230965249344686808174475,\na[16,12]=\n2400779388266927753 1292945295458719370616706273*7^(1/2)/305278657349843870837391340007961 8470033568180000\n - 8388482751130705054783902671749922482556395577590 3/2564340721738688515034087256066879514828197271200000,\na[16,13]=\n-4 114436729416819286145536389728774047607/456542312392180862139685668388 139491300000\n + 1673093837200937887039405670571696062733*7^(1/2)/9130 84624784361724279371336776278982600000,\na[16,14]=\n100137330804174498 9114984611143231300353/40206851375447746381620044659182739290625\n - 8 92131449472993412180461043706539941657*7^(1/2)/25732384880286557684236 828581876953146000,\na[16,15]=\n-2975918446967643161447796202231979904 *7^(1/2)/141528116841576067263302557200323242303\n + 53303208736809836 098042063038756642432/707640584207880336316512786001616211515,\na[17,1 ]=-674107528569903747657*7^(1/2)/194736308727800627200000\n + 66762908 17642855930112293/173510051076470358835200000,\na[17,2]=0,\na[17,3]=0, \na[17,4]=0,\na[17,5]=0,\na[17,6]=-227711699316765384611310459*7^(1/2) /7040098361067108075752652800\n + 860764742109302032039902111/70400983 61067108075752652800,\na[17,7]=-304194866461696889173828125*7^(1/2)/78 99680270269138271354212352\n + 54888690084350082907540673828125/365048 175129272014088413829892096,\na[17,8]=\n-9661338110897182125*7^(1/2)/2 09930810853020925952 + 17032572234499325505625/51013187037284085006336 ,\na[17,9]=-399866634050793590138577034430419123328121213083504583*7^( 1/2)/\n15662284975171767637632388396363220624764129071529984000 -\n355 3613878989287978546743631019160506304244781237959476464169103/\n901446 23003970691762221902742145266933120080348307809053900800000,\na[17,10] =17668130783823819/66589350581600000,\na[17,11]=\n66167102143387747604 7*7^(1/2)/9714173999255163166400 - 65155243311676871271269/11657008799 1061957996800,\na[17,12]=\n-729594201713541940737*7^(1/2)/358097856689 988567040000 + 62747303504529774530387/5371467850349828505600000,\na[1 7,13]=3949144173/1143910400000 - 908415729*7^(1/2)/3145753600000,\na[1 7,14]=319287235181733/9769566771200000 + 29511701788023*7^(1/2)/781565 341696000\n,\na[17,15]=908415729*7^(1/2)/64849710464 - 2317999437/2358 1712896,\na[17,16]=9992573019*7^(1/2)/352964993024 + 112099727493/1764 824965120,\na[18,1]=\n-146095905498393119*7^(1/2)/28971542512452028125 + 362058929348034963652/8604548126198252353125,\na[18,2]=0,\na[18,3]= 0,\na[18,4]=0,\na[18,5]=0,\na[18,6]=\n-1265405169411103125627*7^(1/2)/ 26855843967693741133700 + 3975969853211105248983/268558439676937411337 00,\na[18,7]=-12657909213874869625000000*7^(1/2)/225650046782590131286 240929\n + 415816207712929843732812500000/2317200330410418058178408099 901,\na[18,8]=\n-1449590403533375*7^(1/2)/21622207233549366 + 40101229 2446311250/1264899123162637911,\na[18,9]=-5999612359694084022437474903 7212780460893575326973*7^(1/2)/\n1613165643042136101593301722342708423 113370837903250 +\n454533103388283279317267935846326248185987493846411 2848008/\n171937223442021735691493802532473119608154450127235048396875 ,\na[18,10]=596839002960384/3381490459221875,\na[18,11]=\n150607403991 58849536*7^(1/2)/151783968738361924475 - 233279804085211463168/4553519 06215085773425,\na[18,12]=\n-4054391044424961*7^(1/2)/1366034914741472 500 + 245337047543712611/20490523721122087500,\na[18,13]=10035998/3456 028125 - 1453856*7^(1/2)/3456028125,\na[18,14]=28388719184/59628703437 5 + 3935951656*7^(1/2)/71554444125,\na[18,15]=93046784*7^(1/2)/4559745 267 - 254314496/3256960905,\na[18,16]=15992416*7^(1/2)/387778923 + 212 5088/55396989,\na[18,17]=0,\na[19,1]=\n-408385534705961*7^(1/2)/145260 583712400000 + 374919208039108066769/10197982964383113900000,\na[19,2] =0,\na[19,3]=0,\na[19,4]=0,\na[19,5]=0,\na[19,6]=\n-184289093488047147 4773*7^(1/2)/70154041384995895206400 + 91402574292269720401167/8593870 06966199716278400,\na[19,7]=-2016276423759027541015625*7^(1/2)/6447144 1937882894653211694\n + 33841871953716766022861328125/2574667033789353 39797600899989,\na[19,8]=\n-103445455138314625*7^(1/2)/276764252589431 8848 + 350725547323759375/1124354776144567032,\na[19,9]=\n-16770866323 6226679032764579220523036527050471387*7^(1/2)/808826050019168087288740 7280342617343355065504000\n - 1606556638244155264171556950161091515234 7902393579453431911/\n305666175008038641229322315613285545970052355781 751197150000,\na[19,10]=901046813736528/3381490459221875,\na[19,11]=\n 1199511517980117216*7^(1/2)/21683424105480274925 - 7994843106761284891 2/151783968738361924475,\na[19,12]=\n-5904670442992239*7^(1/2)/3568417 736467520000 + 2337899411211529353/218565586358635600000,\na[19,13]=36 7198237/110592900000 - 3242183*7^(1/2)/13824112500,\na[19,14]=12661697 2699/4770296275000 + 35109599707*7^(1/2)/1144871106000,\na[19,15]=7410 704*7^(1/2)/651392181 - 298713632/3256960905,\na[19,16]=5094859*7^(1/2 )/221587956 + 42920003/553969890,\na[19,17]=0,\na[19,18]=0,\na[20,1]= \n276727826458195951*7^(1/2)/75680764114160400000 + 279932848287549811 93/510845157770582700000,\na[20,2]=0,\na[20,3]=0,\na[20,4]=0,\na[20,5] =0,\na[20,6]=\n2396869514758244044683*7^(1/2)/70154041384995895206400 \+ + 135939757794763166091/474013793141864156800,\na[20,7]=26223752051540 89396484375*7^(1/2)/64471441937882894653211694\n + 2553815259503825962 40234375/705817950170703033255683247,\na[20,8]=\n134541471320113375*7^ (1/2)/2767642525894318848 + 555006824763125/510245713256409,\na[20,9]= 113641767182046923760020640897484653184317971780717*7^(1/2)/\n42139837 20599865734774339193058503635887989127584000 -\n1972524953615841343564 8903585583405898983584276673979718229/\n561427668382111790013040987861 13671708785126572158383150000,\na[20,10]=7074600250608/6273637215625, \na[20,11]=\n-1560088302368475936*7^(1/2)/21683424105480274925 - 12263 0185337518052432/65050272316440824775,\na[20,12]=\n7679632208088369*7^ (1/2)/3568417736467520000 + 468087818175204739/13381566511753200000,\n a[20,13]=1321004573/110592900000 + 4216793*7^(1/2)/13824112500,\na[20, 14]=-164678417029/4770296275000 - 45663651397*7^(1/2)/1144871106000,\n a[20,15]=-9638384*7^(1/2)/651392181 + 57814624/651392181,\na[20,16]=-6 626389*7^(1/2)/221587956 - 46233929/553969890,\na[20,17]=0,\na[20,18]= 0,\na[20,19]=0\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "i nterpolation coefficients: dd " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 120 "The coefficients have been extracted fro m Maple code (with indices modified to allow for the removal of the 13 th stage)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33339 "dd := \{d[1 ,1]=1,seq(d[j,1]=0,j=2..20),\nd[1,2]=356436803021983973929777916110069 02844604115504781863583325615*7^(1/2)/\n382731994099076159547788740295 516541397461808306498896786042894 -\n499134300946037516415666665532904 357414491300955138537955224539656077/\n4367507877462197687063636431008 2574605029956793087815107842498805916,\nd[2,2]=0,\nd[3,2]=0,\nd[4,2]=0 ,\nd[5,2]=0,\nd[6,2]=-236859708878371301763021532740510680335334135437 52892588902929763625*\n7^(1/2)/470124624692006623419839923593907861216 49003097783705034965814997192 -\n3185555254263547852672636017785134690 51754900650813057635058169612625/\n27384759388309385814205675549345132 91586054430445900818286758723586434,\nd[7,2]=-512593083799952678590586 10262718392599578938940008688345921713867187500000\n*7^(1/2)/144457692 387411698446617427382693665797531195246660764237040764885295387027 -\n 9191902943338743687894134552913884099193965310325712891127074414062500 00000/\n11219547442088975246020620193389208043608256164157319355743499 406091275059097,\nd[8,2]=268082252135945924435008113195522200373278388 69838401763840251953125*\n7^(1/2)/360118120772433506453077466669406142 9863907960106427575589521043396 +\n17193471936928197999297729709829937 3880896289047165344616861328125/\n100032811325675974014743740741501706 385108554447400765988597806761,\nd[9,2]=-\n579128236369464738349493062 3934076601319818366128184086760571640512757656160930124011293784825753 160970893*7^(1/2)/ 78960491752622616892134852743469023966537545959871 5531094020245912074193807136326057747285010721816047996 -\n37142425507 7197382133419500692053803036941586595728362939460454134605398282516230 48052653226120876208741/ 219334699312840602478152368731858399907048738 77742098085945006830890949827976009057159646805853383779111,\nd[10,2]= 110876201679254623037871993864066005939382896536227671113630596096*7^( 1/2)\n/818856665987915301984096872482276041680506393532046060373823791 3 +\n28159766316316226442327715443324263660962796329673063061358459699 2/\n90074233258670683218250655973050364584855703288525066641120617043, \nd[11,2]=-16334733089154587640390384290059499224472629993502097884922 2045568000*\n7^(1/2)/1140434805816528314747704966906755893300976057064 9783842405618582767 -\n71657832081718310340617577294538331105648420163 7487951993884383744000/\n216682613105140379802063943712283619727185450 842345893005706753072573,\nd[12,2]=38194153832181999803960746931575834 21412619088899955423132510917*7^(1/2)/\n222870150727569494998558956543 09880045443486828417763716847159592 +\n2204625991286420205028020097875 50092329168455670656206543685061/\n55717537681892373749639739135774700 11360871707104440929211789898,\nd[13,2]=402585847555372253030773794622 450357040040289344403163858*7^(1/2)/\n42950367130019710520172237601420 07374020655560556027616721 +\n2721489311917062733130875756847927895477 861273414985588679/\n4724540384302168157218946136156208111422721116611 6303783931,\nd[14,2]=-123492430408563200278801156813453826186179158427 94662375528*7^(1/2)/\n164377948275384077299424613042471887153876941206 4652544671 +\n29553726321805022820241612807131610602365379038713689058 228/\n1643779482753840772994246130424718871538769412064652544671,\nd[1 5,2]=-35765200672383149604976200334469073612125825077957568000000*7^(1 /2)/\n4931338448261522318982738391274156614616308236193957634013 +\n12 85174376823368046557017759143675528393594918717610624000000/\n16273416 8792630236526430366912047168282338171794400601922429,\nd[16,2]=-538091 653799326086073602371256585993376203834697972000000*7^(1/2)/\n14316789 04333990350672407920047335791340218520185342538907 +\n3167556144993275 9585814449048894682735398944250970524500000/\n143167890433399035067240 7920047335791340218520185342538907,\nd[17,2]=-173069075495440698843906 977750181786857481132028743640678400*7^(1/2)/\n42950367130019710520172 23760142007374020655560556027616721 +\n1393815010347086709454061781438 8965150457511215120796621209600/\n558354772690256236762239088818460958 62268522287228359017373,\nd[18,2]=332850677853640526165412989983121657 1480624909539443236125*7^(1/2)/\n1590754338148878167413786577830373101 48913168909482504323 -\n4151925955283244785540616549867793550067856286 5236277287875/\n636301735259551266965514631132149240595652675637930017 292,\nd[19,2]=14971188707800967856538513527912635104056060277205358955 00*7^(1/2)/\n53025144604962605580459552594345770049637722969827501441 \+ -\n10810719405586300269221788426869054893442683433666843966500/\n53025 144604962605580459552594345770049637722969827501441,\nd[20,2]=11853580 24193049429204894939437968202309486148704187724700*7^(1/2)/\n159075433 814887816741378657783037310148913168909482504323 -\n187243883348260488 09573294764906626932882278291570734642700/\n11135280367042147171896506 04481261171042392182366377530261,\nd[1,3]=-246230945735113892339372585 690030669766812810495003823744255113735*7^(1/2)\n/31345750316714337466 9638978302028047404521221003022596467769130186 +\n32310749520312616452 3432378313687072540527571660364270765036005492591/\n503943216630253579 2765734343471066300580379629971670973981826785298,\nd[2,3]=0,\nd[3,3]= 0,\nd[4,3]=0,\nd[5,3]=0,\nd[6,3]=4822446573891340110911165752064519388 4053404474223761391463039244625*\n7^(1/2)/1134783576842774608254786022 4680534581087690402913308111888300171736 +\n44721084360784434677784729 801158958699023144255148014082283092570062625/\n5476951877661877162841 135109869026583172108860891801636573517447172868,\nd[7,3]=\n3541056331 19922767953259870997321382253869744249799701441590241518554687500000*7 ^(1/2)/\n1183108500652901810277796730264261122881780489070151659101363 86441056921975113 +\n6794085196529111781136735778012722437427090588701 8262802541051081181640625000000/\n118141834565196909340597130636388360 69919493740857657281597904874614112637229141,\nd[8,3]=-264563615359334 05820655294979290299863007510302862110709158453404296875*\n7^(1/2)/421 338201303747202550100636003205187294077231332452026343973962077332 -\n 7625009568771142617570744472154927592785083145755636228628007352929687 5/\n632007301955620803825150954004807780941115846998678039515960943115 998,\nd[9,3]=307745337126029212059142399973448631566530326017344081741 1443098308014249943618454887532557145872390493628329*7^(1/2)/ 497451 0980415224864204495722838548509891865395471907845892327549246067420984 9588541638078955675474411023748\n + 1267079071292303233179256524471818 1749790218683950038864605441245430452908531872827577086602928164586660 30863/ 106596663866040532804382051203683182354825687045826596697692733 19813001616396340401779588347644744516647946,\nd[10,3]=-85105166119311 678537257515346907811541623568055391798062460169174016*\n7^(1/2)/74515 9566049002924805528153958871197929260818114161914940179650083 -\n25696 2403700955597405150437397475457337271082383744977126104318773248/\n117 0965032362718881837258527649654739603124142750825866334568021559,\nd[1 1,3]=47510569498982918125134697696975104131967515527218789537205684352 000*\n7^(1/2)/39325338131604424646472585065750203217275036450516496008 2952364923 +\n15089752168079918698697352256226095367728491644635906439 8733016553216000/\n650047839315421139406191831136850859181556352527037 679017120259217719,\nd[12,3]=-2255127622897123456876081777974926631981 73982033850914510537775289*\n7^(1/2)/156009105509298646498991269580169 160318104407798924346017930117144 -\n928503106091229406723421830741263 80973594493023226754287693593983/\n33430522609135424249783843481464820 068165230242626645575270739388,\nd[13,3]=-2139317505131212408973113339 48469797620487410133479950210474*7^(1/2)/\n270587312919124176277085096 888946464563301300315029739853423 -\n602655454921832537181960288056651 4575441175091155923561220047/\n297646044211036593904793606577841111019 6314303465327138387653,\nd[14,3]=9374735838632006782900231997021291271 11060390461868310702712*7^(1/2)/\n147940153447845669569482151738224698 43848924708581872902039 -\n2243525181240063021054547606023596942176560 700713986922286012/\n1479401534478456695694821517382246984384892470858 1872902039,\nd[15,3]=2715059598470792261411620674740552998114611488544 663872000000*7^(1/2)/\n44382046034353700870844645521467409531546774125 745618706117 -\n975620145254031770622577302338700408050112332982923272 96000000/\n14646075191336721287378733022084245145410435461496054173018 61,\nd[16,3]=408483912305568790490948742180409553333366096463537880000 00*7^(1/2)/\n128851101390059131560516712804260221220619666816680828501 63 -\n2404600995794184108553369816093489658400897946481484185500000/\n 12885110139005913156051671280426022122061966681668082850163,\nd[17,3]= 170797506941086482591181116160029856187748421507562731687116800*7^(1/2 )/\n502519295421230613086015179936614862760416700585055231156357 -\n14 69604360050531892734122746115860001716171503446532235375411200/\n50251 9295421230613086015179936614862760416700585055231156357,\nd[18,3]=-252 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5208907139108930574693744269511602453920849054481209066699823988304843 75*7^(1/2)/\n291226707853021984068380733603510430247822889617268100701 874182008755500246432 +\n198558424207784729337652027809140421811808471 629632719255620101996220416489390625/\n2035674687892623668637981327888 53790743228199842470402390610053224120094672255968,\nd[8,10]=-17220145 818752448585882335349654548932718460749261190595807234473819511*\n7^(1 /2)/165942430051937359773578096641262350688128878801704182683165129679 687680 -\n275326450733850708599990832157965144231292906079013890219305 269105385061/\n1345479162583275890056038621415640681255099017311114994 7283659163217920,\nd[9,10]=2991731810660521340697977984382054012136005 4382105043848318440878012665027684872455588432977526033043794799024326 37*7^(1/2)/29261829296560146260026445428462050058187443502775928505248 985583800396594029169730375340562162043771190440000000000 + 1769846066 1668508536765691798853715651824843464782976814220306208341107790335716 58092337624515695561338372711474996019 / 877854878896804387800793362 8538615017456233050832778551574695675140118978208750919112602168648613 1313571320000000000,\nd[10,10]=-10565564706577006714302229306481376308 9349431414614572661549378773572343\n*7^(1/2)/5597652990151764759656912 21423430887867533667453547111583668607333984375 -\n2291801398858447505 41453333118293522016656389442391856325876820302141271217/\n61574182891 66941235622603435657739766542870341989018227420354680673828125,\nd[11, 10]=177892797093142412257316821457180742473932854488415637220647129912 851361*\n7^(1/2)/89096469204416274589664450539590304164138754458201436 2687938951778671875 +\n19995156164211388134790788863663720062236612236 14802558012444400980074115933/\n50784987446517276516108736807566473373 559090041174818673212520251384296875,\nd[12,10]=-219231170471239840655 164853036290144997241456374763617246805840968109517\n*7^(1/2)/91770062 064293321469994864458923035481237886940543732951723598320000000000 -\n 1296925824892082782003428494309379004662579485507049921827455196040336 68579/\n27531018619287996440998459337676910644371366082163119885517079 4960000000000,\nd[13,10]=-17677683272242250326798012111044889846169058 43180510475361582909737*\n7^(1/2)/135293656459562088138542548444473232 2816506501575148699267115000000000 -\n26588531625001786960277154122158 332644613904116765362996783730510181/\n1352936564595620881385425484444 73232281650650157514869926711500000000,\nd[14,10]=19366411310941722108 39367126017060646687896130434817268490574099339*\n7^(1/2)/184925191809 80708696185268967278087304811155885727341127548750000000 -\n9269387894 067927883683399568166931603178032678022732194932880857003/\n3698503836 1961417392370537934556174609622311771454682255097500000000,\nd[15,10]= 2804396935699600127491897980287593626603967194833957908363492*7^(1/2)/ \n27738778771471063044277903450917130957216733828591011691323125 -\n91 61111960650286998565238931184900112340007538498176522151496/\n83216336 314413189132833710352751392871650201485773035073969375,\nd[16,10]=1687 69927937866556774048075154855360475964617996887862377947*7^(1/2)/\n322 12775347514782890129178201065055305154916704170207125407500 -\n7947913 2382756329649073748419693764677462714391216826926191871/\n257702202780 118263121033425608520442441239333633361657003260000,\nd[17,10]=8481610 194042494659726503523227707518701575387141587131434342656*7^(1/2)/\n15 099738444147554479748052281749244674291367205079784590034765625 +\n471 17667764615581929392219462770092611530419464898143535626026624/\n30199 47688829510895949610456349848934858273441015956918006953125,\nd[18,10] =-33407058158239317897612851438336426258720662498149916004293814231*\n 7^(1/2)/11453431234671922805379263360378686330721748161482740311256000 0000 -\n29719767644019792182873122228978382593230595512292483793922808 7333/\n572671561733596140268963168018934316536087408074137015562800000 00,\nd[19,10]=-2629560217206334579988022605476192091185408071560023403 7708637423*\n7^(1/2)/6681168220225288303137903626887567026254353094198 2651815660000000 -\n12356380684601353053292666137812785102426329689771 66544306108743161/\n10021752330337932454706855440331350539381529641297 3977723490000000,\nd[20,10]=-10409895848618224279376577757783711716939 8651627706117231606646591*\n7^(1/2)/1002175233033793245470685544033135 053938152964129739777234900000000 +\n302379860731446351726387098064390 123896211318014197571588306389/\n6681168220225288303137903626887567026 254353094198265181566000000\}:" }}}{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,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*,$*(\"$7\"F*\"%l5F.F-F.F* ,$*(\"&G*=F*\"(DU8\"F.F-F.F*,$*(\"'_.5F*F8F.F-F.F*F/F/F/F/F/7*,$*(\"#c F*\"$b$F.F-F.F*,$*(\"#9F*F@F.F-F.F*\"\"!,$*(\"#UF*F@F.F-F.F*F/F/F/F/7* ,$*(\"#RF*\"$+\"F.F-F.F*,$*(\")za\\%*F*\"*++)3DF.F-F.F*FD,$*(\"*(f1GNF *FPF.F-F.F.,$*(\"*fr2y\"F*\"*++WD\"F.F-F.F*F/F/F/7*,$*(\"\"(F*\"#:F.F- F.F*,$*(\"&*37F*\"'?FDF.F-F.F*FDFD,$*(\"(x`]#F*\")?bo5F.F-F.F*,$*(\"'+ /'*F*\"(\">4_F.F-F.F*F/F/7*,$*(FKF*\"$]#F.F-F.F*,$*(\")**3S@F*\"*+++]$ F.F-F.F*FDFD,$*(\".**)H)HV1$F*\"/+++]g7FF.F-F.F*,$*(\")ZRk@F*\"*v$4EfF .F-F.F.,$*(\"*V>RC\"F*\"+++DcnF.F-F.F*F/7*,$*(\"#CF*\"#DF.F-F.F*,$*(\" ,6)eaO:F*\",vdc4O\"F.F-F.F.FDFD,$*(FenF*\"\"&F.F-F.F.,$*(\".3Y;G\"R$)F *\"-v%Q+1R*F.F-F.F.,$*(\"-)[+o$>MF*\",Di7^z%F.F-F.F*,$*(\".S#Q=K$*>F* \"-p!fM_!QF.F-F.F*Q(pprint36\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The new polynomials (of degree " }{XPPEDIT 18 0 "`` <= 7;" "6#1%!G \"\"(" }{TEXT -1 55 " ) are obtained as follows (re-using the weight s ymbol " }{XPPEDIT 18 0 "b[j]" "6#&%\"bG6#%\"jG" }{TEXT -1 3 "). " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "[seq(b[j]=add(d[j,i]*u^(i-1) ,i=1..8),j=1..20)]:\npols := eval(subs(dd,%)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The first few non-zero po lynomials with rough approximations for the coefficients are . . . " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct to 12 do\n if rhs( pols[ct])<>0 then print(evalf[6](pols[ct])) end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\",2$F'\"\"!F'*&$\"'?=6!\"%F'% \"uGF'!\"\"*&$\"'w.iF.F')F/\"\"#F'F'*&$\"'#)[>!\"$F')F/\"\"$F'F0*&$\"' &Hh$F9F')F/\"\"%F'F'*&$\"'y'*QF9F')F/\"\"&F'F0*&$\"'g_AF9F')F/\"\"'F'F '*&$\"'f!Q&F.F')F/\"\"(F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG 6#\"\"',0*&$\"'K\\9!\"&\"\"\"%\"uGF-!\"\"*&$\"')3%>!\"%F-)F.\"\"#F-F-* &$\"'[w5!\"$F-)F.\"\"$F-F/*&$\"'W&4$F9F-)F.\"\"%F-F-*&$\"'q,[F9F-)F.\" \"&F-F/*&$\"'$yw$F9F-)F.F'F-F-*&$\"':h6F9F-)F.\"\"(F-F/" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(,0*&$\"'v?5!\"&\"\"\"%\"uGF-!\"\"* &$\"''pO\"!\"%F-)F.\"\"#F-F-*&$\"'j\"e(F3F-)F.\"\"$F-F/*&$\"'5!=#!\"$F -)F.\"\"%F-F-*&$\"'\"=Q$F>F-)F.\"\"&F-F/*&$\"'m`EF>F-)F.\"\"'F-F-*&$\" '!z<)F3F-)F.F'F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"),0 *&$\"'XT@!\"%\"\"\"%\"uGF-F-*&$\"'xnG!\"$F-)F.\"\"#F-!\"\"*&$\"'d!f\"! \"#F-)F.\"\"$F-F-*&$\"'qtXF9F-)F.\"\"%F-F5*&$\"'z%4(F9F-)F.\"\"&F-F-*& $\"'?nbF9F-)F.\"\"'F-F5*&$\"'n:1XF9F-)F.\"\"%F- F-*&$\"'2!*pF9F-)F.\"\"&F-F/*&$\"'-&[&F9F-)F.\"\"'F-F-*&$\"'M!p\"F9F-) F.\"\"(F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5,0*&$\"'3& *Q!\"%\"\"\"%\"uGF-F-*&$\"'>;_!\"$F-)F.\"\"#F-!\"\"*&$\"'3$*G!\"#F-)F. \"\"$F-F-*&$\"'2>$)F9F-)F.\"\"%F-F5*&$\"'Y!H\"F5F-)F.\"\"&F-F-*&$\"'h7 5F5F-)F.\"\"'F-F5*&$\"'i?JF9F-)F.\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6,0*&$\"'H?T!\"%\"\"\"%\"uGF-!\"\"*&$\"'x \+ " 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 satisf ies the order conditions (and row sum conditions) for a 20 stage, orde r 8 Runge-Kutta scheme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 " RK8_20eqs := [op(RowSumConditions(20,'expanded')),op(OrderConditions(8 ,20,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "simplify(subs(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 an interpolation scheme B .. [7 stage scheme] .. (shorter method) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "The \+ interpolation scheme constructed in this section is a minor variation \+ of the interpolation scheme for Verner's Maple dverk78 Runge Kutta met hod." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 172 " If the only criterion in the selection of the nodes is to obtain a rea sonable principal error curve for an order 8 interpolant (without cons idering an order 7 interpolant) " }}{PARA 0 "" 0 "" {TEXT -1 71 "then \+ one can improve slightly on the situation given by Verner's nodes." }} {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 3978 "e1 := \{c[2]=1/16,c[3]=112/1065,c[4]=56/355,c[5]=39/100,c[6]=7 /15,c[7]=39/250,c[8]=24/25,\n c[9]=14435868/16178861,c[10]=11/12,c[11 ]=19/20,c[12]=1,c[13]=1,\n a[2,1]=1/16,a[3,1]=18928/1134225,a[3,2]=10 0352/1134225,a[4,1]=14/355,a[4,2]=0,\n a[4,3]=42/355,\n a[5,1]=94495 479/250880000,a[5,2]=0,a[5,3]=-352806597/250880000,a[5,4]=178077159/12 5440000,\n a[6,1]=12089/252720,a[6,2]=0,a[6,3]=0,a[6,4]=2505377/10685 520,a[6,5]=960400/5209191,\n a[7,1]=21400899/350000000,a[7,2]=0,a[7,3 ]=0,a[7,4]=3064329829899/27126050000000,\n a[7,5]=-21643947/592609375 ,a[7,6]=124391943/6756250000,\n a[8,1]=-15365458811/13609565775,a[8,2 ]=0,a[8,3]=0,a[8,4]=-7/5,\n a[8,5]=-8339128164608/939060038475,a[8,6] =341936800488/47951126225,\n a[8,7]=1993321838240/380523459069,\n a[ 9,1]=-1840911252282376584438157336464708426954728061551/\n \+ 2991923615171151921596253813483118262195533733898,a[9,2]=0,a [9,3]=0,\n a[9,4]=-14764960804048657303638372252908780219281424435/\n 2981692102565021975611711269209606363661854518,\n a[ 9,5]=-875325048502130441118613421785266742862694404520560000/\n \+ 170212030428894418395571677575961339495435011888324169,\n a[9 ,6]=7632051964154290925661849798370645637589377834346780/\n \+ 1734087257418811583049800347581865260479233950396659,\n a[9,7]=75 19834791971137517048532179652347729899303513750000/\n 10 45677303502317596597890707812349832637339039997351,\n a[9,8]=13660426 83489166351293315549358278750/\n 1446314182242677181650 55326464180836641,\n a[10,1]=-63077736705254280154824845013881/783693 57853786633855112190394368,\n a[10,2]=0,a[10,3]=0,a[10,4]=-3194834651 0820970247215/6956009216960026632192,\n a[10,5]=-33786048053942552924 53489375/517042670569824692230499952,\n a[10,6]=100158784418332598119 8091450220795/184232684207722503701669953872896,\n a[10,7]=1870230752 31349900768014890274453125/25224698849808178010752575653374848,\n a[1 0,8]=1908158550070998850625/117087067039189929394176,\n a[10,9]=-5295 6818288156668227044990077324877908565/\n 29127799594 77433986349822224412353951940608,\n a[11,1]=-101161065918269095347811 57993685116703/9562819945036894030442231411871744000,\n a[11,2]=0,a[1 1,3]=0,a[11,4]=-9623541317323077848129/3864449564977792573440,\n a[11 ,5]=-4823348333146829406881375/576413233634141239944816,\n a[11,6]=65 66119246514996884067001154977284529/9703054870218463254739908635823155 20,\n a[11,7]=2226455130519213549256016892506730559375/36488044315967 5255577435648380047355776,\n a[11,8]=39747262782380466933662225/17560 32802431424164410720256,\n a[11,9]=4817577141926095533524468380517154 8038966866545122229/\n 198978642051381514652888016595206411 8903852843612160000,\n a[11,10]=-2378292068163246/47768728487211875, \n a[12,1]=-3218022174758599831659045535578571/1453396753634469525663 775847094384,\n a[12,2]=0,a[12,3]=0,a[12,4]=26290092604284231996745/5 760876126062860430544,\n a[12,5]=-697069297560926452045586710000/4110 7967755245430594036502319,\n a[12,6]=18273578204342134614380775509022 73440/139381013914245317709567680839641697,\n a[12,7]=643504802814241 550941949227194107500000/242124609118836550860494007545333945331,\n a [12,8]=162259938151380266113750/59091082835244183497007,\n a[12,9]=-2 3028251632873523818545414856857015616678575554130463402/\n \+ 20013169183191444503443905240405603349978424504151629055,\n a[12,10 ]=7958341351371843889152/3284467988443203581305,\n a[12,11]=-50797432 7957860843878400/121555654819179042718967,\n a[13,1]=4631674879841/10 3782082379976,a[13,2]=0,a[13,3]=0,a[13,4]=0,a[13,5]=0,\n a[13,6]=1432 7219974204125/40489566827933216,\n a[13,7]=2720762324010009765625000/ 10917367480696813922225349,\n a[13,8]=-498533005859375/95352091037424 ,\n a[13,9]=405932030463777247926705030596175437402459637909765779/\n 78803919436321841083201886041201537229769115088303952, \n a[13,10]=-10290327637248/1082076946951,\n a[13,11]=86326410588800 0/85814662253313,\n a[13,12]=-29746300739/247142463456\}:" }}}{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),``$(13-i)],i=2. .13)])):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7 /$\"++++]i!#6F(%!GF+F+F+F+F+F+F+F+F+F+7/$\"+#>V;0\"!#5$\"+!p/)o;F*$\"+ MsiZ))F*F+F+F+F+F+F+F+F+F+F+7/$\"+*ykud\"F/$\"+s>mVRF*$\"\"!F:$\"+#f)4 $=\"F/F+F+F+F+F+F+F+F+F+7/$\"+++++RF/$\"+m3cmPF/F9$!+&HwiS\"!\"*$\"+4- i>9FDF+F+F+F+F+F+F+F+7/$\"+nmmmYF/$\"+A\\b$y%F*F9F9$\"+unkWBF/$\"++WmV =F/F+F+F+F+F+F+F+7/$\"++++g:F/$\"+rDa9hF*F9F9$\"+@JmH6F/$!+AEJ_OF*$\"+ U)Q6%=F*F+F+F+F+F+F+7/$\"+++++'*F/$!+M!>!H6FDF9F9$!+++++9FD$!+qGH!)))F D$\"+(4W48(FD$\"+3yOQ_FDF+F+F+F+F+7/$\"+;EnA*)F/$!+K`$H:'F/F9F9$!+jI(= &\\FD$!+#)obU^FD$\"+wD>,WFD$\"+rJN\">(FD$\"+gE*\\W*!#7F+F+F+F+7/$\"+nm mm\"*F/$!+&\\v([!)F/F9F9$!+%>8Hf%FD$!+V'zW`'FD$\"+;s`OaFD$\"+oPG9uFD$ \"+))>pH;F*$!+&4&3==F*F+F+F+7/$\"+++++&*F/$!+1\"ey0\"FDF9F9$!+(yu-\\#F D$!+k`'yO)FD$\"+9R1nnFD$\"+ed(=5'FD$\"+m#pME#F*$\"+!*G:@CF*$!+/Owy\\F* F+F+7/$\"\"\"F:$!+@)QT@#FDF9F9$\"+Z!eNc%FD$!+@Oq&p\"!\")$\"+z@068F_s$ \"+ZAudEFD$\"+T$Hfu#FD$!+A\\l]6FD$\"+hH-BCFD$!+TW%*yTFDF+7/Fgr$\"+(*[) GY%F*F9F9F9F9$\"+Gn\\QNF/$\"+y89#\\#F/$!+r%Q$G_FD$\"+#\\l6:&FD$!+sGz4& *FD$\"+9M'f+\"F_s$!+i%4O?\"F/Q)pprint206\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "convert(ListTools [Enumerate](SimpleOrderConditions(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(F2F(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(#F(FTQ)pprint236\"" }}}{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 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_13 := SimpleOrderConditions(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 whc h do\n temp_eqn := convert(SO7_13[ct],'interpolation_order_condition '):\n interp_order_eqns14 := [op(interp_order_eqns14),temp_eqn];\nen d do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, the order conditions can be specified explicitly as fo llows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 347 "interp_order_eqns 14 := [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,inter p_order_eqns14)):\nnops(eqs_14);\nindets(eqs_14);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<,& %\"aG6$\"#9\"\"(&F%6$F'\"\"'&F%6$F'\"#8&F%6$F'\"#6&%\"cG6#F'&F%6$F'\" \"\"&F%6$F'\"#7&F%6$F'\"\")&F%6$F'\"#5&F%6$F'\"\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We solve for the linking coefficients in terms of \+ " }{XPPEDIT 18 0 "c[14];" "6#&%\"cG6#\"#9" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "sol_14 := solve(\{op(eqs_14)\},inde ts(eqs_14) minus \{c[14]\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "We choose the node " } {XPPEDIT 18 0 "c[14];" "6#&%\"cG6#\"#9" }{TEXT -1 63 " so that an add itional (adapted) order condition is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 6 "EITHER" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "extra_ eqn := add(a[14,i]*add(a[i,j]*add(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i= 2..13)=c[14]^6/120:" }}}{PARA 0 "" 0 "" {TEXT -1 6 "OR ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "extra_eqn := convert(SO7_13[27],'in terpolation_order_condition'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "expand(subs(e2,extra_eqn)): \neq_14 := subs(sol_14,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eq_14 G/,,*&#\"?,#pC'y1TRIf4-a]L\"@?hDAsv^`H=7a)3!)p\"\"\"*$)&%\"cG6#\"#9\" \"'F+F+F+*&#\"M;lQSW\\\">,)=Nkk7Y_,^/u;eF+*$)F.\"\"#F+F+F+*&# \"=+on+WcQoKIx!))))*\"?.kbIRz$Qd/`8A]u\"F+*$)F.\"\"$F+F+!\"\"*&#\"=+#p ,5T'4?sC\">ni]9#)3([F+*$)F.\"\"%F+F+F+*&#\"=gPZ!3&*pyG7Tl@ #pF6F+*$)F.\"\"&F+F+FA,$*&#F+\"$?\"F+F,F+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#,,*&#\"=gPZ!3&*pyG 7Tl@#p\"?.kbIRz$Qd/`8A]u\"\"\"\"*$)&%\"cG6#\"#9\"\"'F(F(F(*&#\"M;lQSW\\\">,)=Nkk7Y_,^/u;eF(*$)F+\"\"#F(F(F(*&#\"=+on+WcQoKIx!))))*F' F(*$)F+\"\"$F(F(!\"\"*&#\"=+#p,5T'4?sC\">ni]9#)3([F(*$)F+ \"\"%F(F(F(*&#\"=gPZ!3&*pyG7Tl@#pF3F(*$)F+\"\"&F(F(F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"M;lQSW\\\"?.kbIRz$Qd/`8A]u\"\"\"\"* ()&%\"cG6#\"#9\"\"#F(,(*&F.F(F*F(F(*&F.F(F+F(!\"\"\"\"$F(F(),&F+F(F(F3 F/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$F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 69 "We can now obtain values for the linking coefficie nts for this stage." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "e3 := \{c[14]=1/2-1/14*7^(1/2)\}:\ne4 := solve(\{op(subs(e3,eqs_14))\}):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "e5 := `union`(e2,e3,e4):\n[c[14]=subs(e5,c[14]),seq(a[14,i]=sub s(e5,a[14,i]),i=1..13)]:\nevalf[20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&%\"cG6#\"#9$\"5R'Q'Q&\\jx,6$!#?/&%\"aG6$F(\"\"\"$\"5s%4+P#p+ yjY!#@/&F.6$F(\"\"#$\"\"!F9/&F.6$F(\"\"$F8/&F.6$F(\"\"%F8/&F.6$F(\"\"& F8/&F.6$F(\"\"'$\"41!e([t6p:G%F+/&F.6$F(\"\"($\"5;/C/@93]VBF+/&F.6$F( \"\")$\"5D:;Y(RxXC!R!#>/&F.6$F(\"\"*$!5d\"3Ml^\"Qr`!4'FX/&F.6$F(\"#7$!5n!>dR2HG>D$ F3/&F.6$F(\"#8$!5$oi;'QaN]f7F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "These linking coefficients can be compare d with those of the published scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 70 "#------------------------------------ ---------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We c an check which of the (adapted) simple order conditions are satisfied \+ at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "recd := []:\nfor ct to nops(SO7_13) do\n tt : = convert(SO7_13[ct],'interpolation_order_condition'):\n if expand(s ubs(e5,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do: \nop(recd);\nnops(recd);\nop(\{seq(i,i=1..nops(SO7_13))\} minus \{op(r ecd)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6J\"\"\"\"\"#\"\"$\"\"%\"\"& \"\"'\"\"(\"\")\"\"*\"#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 4966 "e5 := \{a[14,2] = 0, a[9,3] = 0, \+ a[14,12] = 1100613127343/48439922837376-29746300739/1424703612864*7^(1 /2), a[7,6] = 124391943/6756250000, a[9,8] = 1366042683489166351293315 549358278750/144631418224267718165055326464180836641, a[9,2] = 0, a[8, 2] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[10,7] = 187023075231349900768014890274453125/25 224698849808178010752575653374848, a[12,5] = -697069297560926452045586 710000/41107967755245430594036502319, a[7,2] = 0, a[11,4] = -962354131 7323077848129/3864449564977792573440, a[11,5] = -482334833314682940688 1375/576413233634141239944816, a[12,1] = -3218022174758599831659045535 578571/1453396753634469525663775847094384, a[12,3] = 0, a[12,4] = 2629 0092604284231996745/5760876126062860430544, a[12,7] = 6435048028142415 50941949227194107500000/242124609118836550860494007545333945331, a[11, 1] = -10116106591826909534781157993685116703/9562819945036894030442231 411871744000, a[11,2] = 0, a[11,3] = 0, a[12,6] = 18273578204342134614 38077550902273440/139381013914245317709567680839641697, a[10,8] = 1908 158550070998850625/117087067039189929394176, a[10,9] = -52956818288156 668227044990077324877908565/291277995947743398634982222441235395194060 8, a[12,8] = 162259938151380266113750/59091082835244183497007, a[12,9] = -23028251632873523818545414856857015616678575554130463402/200131691 83191444503443905240405603349978424504151629055, a[12,2] = 0, a[12,10] = 7958341351371843889152/3284467988443203581305, a[12,11] = -50797432 7957860843878400/121555654819179042718967, a[11,8] = 39747262782380466 933662225/1756032802431424164410720256, a[11,9] = 48175771419260955335 244683805171548038966866545122229/198978642051381514652888016595206411 8903852843612160000, a[11,10] = -2378292068163246/47768728487211875, a [11,6] = 6566119246514996884067001154977284529/97030548702184632547399 0863582315520, a[11,7] = 2226455130519213549256016892506730559375/3648 80443159675255577435648380047355776, a[3,2] = 100352/1134225, a[13,3] \+ = 0, a[3,1] = 18928/1134225, a[13,6] = 14327219974204125/4048956682793 3216, a[13,7] = 2720762324010009765625000/10917367480696813922225349, \+ a[13,8] = -498533005859375/95352091037424, a[10,1] = -6307773670525428 0154824845013881/78369357853786633855112190394368, a[10,2] = 0, a[13,1 1] = 863264105888000/85814662253313, a[6,1] = 12089/252720, a[6,2] = 0 , a[5,4] = 178077159/125440000, a[13,12] = -29746300739/247142463456, \+ c[11] = 19/20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[10,6] = 10015878 44183325981198091450220795/184232684207722503701669953872896, a[10,3] \+ = 0, a[7,5] = -21643947/592609375, a[7,4] = 3064329829899/271260500000 00, a[9,4] = -14764960804048657303638372252908780219281424435/29816921 02565021975611711269209606363661854518, a[8,7] = 1993321838240/3805234 59069, a[9,1] = -1840911252282376584438157336464708426954728061551/299 1923615171151921596253813483118262195533733898, a[13,4] = 0, a[13,5] = 0, a[9,5] = -875325048502130441118613421785266742862694404520560000/1 70212030428894418395571677575961339495435011888324169, a[8,1] = -15365 458811/13609565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -8339128164608 /939060038475, a[8,6] = 341936800488/47951126225, a[10,4] = -319483465 10820970247215/6956009216960026632192, a[10,5] = -33786048053942552924 53489375/517042670569824692230499952, a[13,9] = 4059320304637772479267 05030596175437402459637909765779/7880391943632184108320188604120153722 9769115088303952, a[13,10] = -10290327637248/1082076946951, a[4,3] = 4 2/355, a[5,1] = 94495479/250880000, a[4,1] = 14/355, a[9,6] = 76320519 64154290925661849798370645637589377834346780/1734087257418811583049800 347581865260479233950396659, a[9,7] = 75198347919711375170485321796523 47729899303513750000/1045677303502317596597890707812349832637339039997 351, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a [7,1] = 21400899/350000000, a[7,3] = 0, c[2] = 1/16, a[5,2] = 0, a[5,3 ] = -352806597/250880000, a[13,1] = 4631674879841/103782082379976, a[1 3,2] = 0, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, a[14,3] = 0, a[14,5] = 0, c[3] = 112/1065, c[4] = 56/355, c[ 5] = 39/100, c[6] = 7/15, a[14,7] = 129834778101757666015625000/534951 006554143882189042101-82766411529674389648437500/262125993211530502272 63062949*7^(1/2), a[14,1] = 923507123432989/20341288146475296+37286199 17660047/7973784953418316032*7^(1/2), a[14,11] = -5071676622092000/420 4918450412337-380378998046332000/206041004070204513*7^(1/2), a[14,6] = 1533012537239841375/7935955098274910336-11049456415304617875/19443089 9907735303232*7^(1/2), a[14,8] = 13061564753515625/18689009843335104+3 1047286856168359375/25641321505055762688*7^(1/2), a[14,9] = -321179942 8826566849882311586102611111468427562764815533/15445568209519080852307 569664075501297034746557307574592-356099260585467255084405198117663828 33790503793377686129/6359939850978445056832528685207559357602542700067 8248320*7^(1/2), a[14,10] = 37302437685024/53021770400599+157421431244 43936/12990333748146755*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 1 5" }}{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 scheme that avoids d ealing 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 := SimpleOr derConditions(7,14,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "whch := [1,2,4,8,16,17,25,2 7,32,64]:\ninterp_order_eqns15 := []:\nfor ct in whch do\n temp_eqn \+ := convert(SO7_14[ct],'interpolation_order_condition'):\n interp_ord er_eqns15 := [op(interp_order_eqns15),temp_eqn];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Alternatively, th e order conditions can be specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 439 "interp_order_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]*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..14)=c[15]^6/72 0, ##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[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" }}}{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] = 3/4 0;" "6#/&%\"cG6#\"#:*&\"\"$\"\"\"\"#S!\"\"" }{TEXT -1 80 ", and have e nough equations to determine the corresponding linking coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "e6 := `union`(e5,\{c[15]= 3/40,seq(a[15,i]=0,i=2..5)\}):\neqs_15 := expand(subs(e6,interp_order_ eqns15)):\nnops(eqs_15);\nindets(eqs_15);\nnops(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<,&%\"aG6$\"# :\"\"\"&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_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[20](%);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#70/&%\"aG6$\"#:\"\"\"$\"5(*o;!HI-@-M%!#@/&F&6$F(\"\"# $\"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F&6$F(\"\"'$\"5 a$Gu#=VKAP:F,/&F&6$F(\"\"($\"5k`&)[;3*ehq&F,/&F&6$F(\"\")$!5-Ts1C3OtX8 !#>/&F&6$F(\"\"*$\"5q>XAb#[H#*G&!#?/&F&6$F(\"#5$!56*Q1mj0M!>8FQ/&F&6$F (\"#6$\"5z37(e[%ygP@FQ/&F&6$F(\"#7$!5diXD$o4\"\\o@!#A/&F&6$F(\"#8$!4[c b%\\QP(>X#Feo/&F&6$F(\"#9$!59ev%G!yra=SF," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#------------------------------ ---------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check which of the (adapted) simple order conditions are sa tisfied at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "recd := []:\nfor ct to nops(SO7_14 ) do\n tt := convert(SO7_14[ct],'interpolation_order_condition'):\n \+ if expand(subs(e8,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end i f: \nend do:\nop(recd);\nnops(recd);\nop(\{seq(i,i=1..nops(SO7_14))\} \+ minus \{op(recd)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6J\"\"\"\"\"#\" \"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#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 6079 "e8 := \{c[8] = 24/25, c[9] = 14435868/16178861, a[14,10] = 37302437685024/53021770400599+1574214 3124443936/12990333748146755*7^(1/2), a[9,3] = 0, a[11,4] = -962354131 7323077848129/3864449564977792573440, a[14,9] = -321179942882656684988 2311586102611111468427562764815533/15445568209519080852307569664075501 297034746557307574592-356099260585467255084405198117663828337905037933 77686129/63599398509784450568325286852075593576025427000678248320*7^(1 /2), a[11,9] = 48175771419260955335244683805171548038966866545122229/1 989786420513815146528880165952064118903852843612160000, a[9,4] = -1476 4960804048657303638372252908780219281424435/29816921025650219756117112 69209606363661854518, a[6,5] = 960400/5209191, a[14,12] = 110061312734 3/48439922837376-29746300739/1424703612864*7^(1/2), a[14,11] = -507167 6622092000/4204918450412337-380378998046332000/206041004070204513*7^(1 /2), a[11,10] = -2378292068163246/47768728487211875, a[14,2] = 0, a[4, 1] = 14/355, a[9,6] = 763205196415429092566184979837064563758937783434 6780/1734087257418811583049800347581865260479233950396659, a[12,4] = 2 6290092604284231996745/5760876126062860430544, a[9,2] = 0, a[8,2] = 0, a[10,9] = -52956818288156668227044990077324877908565/2912779959477433 986349822224412353951940608, a[7,6] = 124391943/6756250000, a[9,8] = 1 366042683489166351293315549358278750/144631418224267718165055326464180 836641, a[11,2] = 0, a[11,3] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392* 7^(1/2), a[12,9] = -23028251632873523818545414856857015616678575554130 463402/20013169183191444503443905240405603349978424504151629055, a[12, 2] = 0, a[7,2] = 0, a[10,7] = 187023075231349900768014890274453125/252 24698849808178010752575653374848, a[12,5] = -6970692975609264520455867 10000/41107967755245430594036502319, a[4,2] = 0, c[14] = 1/2-1/14*7^(1 /2), a[12,8] = 162259938151380266113750/59091082835244183497007, a[12, 10] = 7958341351371843889152/3284467988443203581305, a[12,11] = -50797 4327957860843878400/121555654819179042718967, a[11,8] = 39747262782380 466933662225/1756032802431424164410720256, a[11,5] = -4823348333146829 406881375/576413233634141239944816, a[12,1] = -32180221747585998316590 45535578571/1453396753634469525663775847094384, a[12,6] = 182735782043 4213461438077550902273440/139381013914245317709567680839641697, a[10,8 ] = 1908158550070998850625/117087067039189929394176, a[11,6] = 6566119 246514996884067001154977284529/970305487021846325473990863582315520, a [12,7] = 643504802814241550941949227194107500000/242124609118836550860 494007545333945331, a[11,1] = -10116106591826909534781157993685116703/ 9562819945036894030442231411871744000, a[12,3] = 0, a[2,1] = 1/16, a[1 0,6] = 1001587844183325981198091450220795/1842326842077225037016699538 72896, a[10,3] = 0, a[7,4] = 3064329829899/27126050000000, a[7,5] = -2 1643947/592609375, a[14,3] = 0, a[14,5] = 0, c[10] = 11/12, c[4] = 56/ 355, c[5] = 39/100, c[6] = 7/15, c[3] = 112/1065, a[13,5] = 0, a[8,3] \+ = 0, a[8,7] = 1993321838240/380523459069, a[14,6] = 153301253723984137 5/7935955098274910336-11049456415304617875/194430899907735303232*7^(1/ 2), a[14,8] = 13061564753515625/18689009843335104+31047286856168359375 /25641321505055762688*7^(1/2), c[11] = 19/20, c[12] = 1, c[13] = 1, c[ 2] = 1/16, a[10,5] = -3378604805394255292453489375/5170426705698246922 30499952, a[13,9] = 40593203046377724792670503059617543740245963790976 5779/78803919436321841083201886041201537229769115088303952, a[13,10] = -10290327637248/1082076946951, a[7,1] = 21400899/350000000, a[14,1] = 923507123432989/20341288146475296+3728619917660047/797378495341831603 2*7^(1/2), a[14,7] = 129834778101757666015625000/534951006554143882189 042101-82766411529674389648437500/26212599321153050227263062949*7^(1/2 ), a[6,3] = 0, a[6,4] = 2505377/10685520, a[9,7] = 7519834791971137517 048532179652347729899303513750000/104567730350231759659789070781234983 2637339039997351, a[5,3] = -352806597/250880000, a[5,2] = 0, a[5,1] = \+ 94495479/250880000, a[4,3] = 42/355, a[9,1] = -18409112522823765844381 57336464708426954728061551/2991923615171151921596253813483118262195533 733898, a[13,4] = 0, a[9,5] = -875325048502130441118613421785266742862 694404520560000/170212030428894418395571677575961339495435011888324169 , a[8,1] = -15365458811/13609565775, a[7,3] = 0, a[13,1] = 46316748798 41/103782082379976, a[13,2] = 0, c[7] = 39/250, a[10,4] = -31948346510 820970247215/6956009216960026632192, a[8,6] = 341936800488/47951126225 , c[15] = 3/40, a[8,5] = -8339128164608/939060038475, a[8,4] = -7/5, a [11,7] = 2226455130519213549256016892506730559375/36488044315967525557 7435648380047355776, a[6,2] = 0, a[5,4] = 178077159/125440000, a[13,12 ] = -29746300739/247142463456, a[13,11] = 863264105888000/858146622533 13, a[6,1] = 12089/252720, a[13,7] = 2720762324010009765625000/1091736 7480696813922225349, a[13,8] = -498533005859375/95352091037424, a[13,6 ] = 14327219974204125/40489566827933216, a[3,2] = 100352/1134225, a[13 ,3] = 0, a[3,1] = 18928/1134225, a[10,1] = -63077736705254280154824845 013881/78369357853786633855112190394368, a[10,2] = 0, a[15,1] = 539681 19671846752033/1211086636548096000000-18013979306423851/41103546452541 440000*7^(1/2), a[15,9] = 14378292160028055491458882417317120757649007 49803936348249893/1494355805607288245725902431596118039320066182415245 312000000-4367810325925642786942443213218202333579800829827668077/2667 3124624013005776924627213516128165085289185280000000*7^(1/2), a[15,10] = -43024834368326969649/17313231151216000000+24278487789419081919/550 87553662960000000*7^(1/2), a[15,13] = 609279/512000000-2221887/4096000 000*7^(1/2), a[15,6] = -3197585392137639804267/33169053145442890547200 0+37078101149895800937/3922016419224666112000*7^(1/2), a[15,14] = -155 53209/1024000000*7^(1/2), a[15,7] = 12599626362153730936328125/2548007 81771704557409886208+1013426797972694921875/352212771011439949026304*7 ^(1/2), a[15,8] = -8532637324759244825/3240219442044469248+13259473087 7413325/272450884015751168*7^(1/2), a[15,11] = 15187119353760445983/36 61425589474688000-19473860153393911521/25629979126322816000*7^(1/2), a [15,2] = 0, a[15,4] = 0, a[15,5] = 0, a[15,3] = 0, a[15,12] = -3658589 7414219009/16871592171929600000\}: " }{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 1 6" }}{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 scheme that avoids d ealing 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_15 := SimpleOr derConditions(7,15,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "whch := [1,2,4,8,16,17,25,2 7,32,63,64]:\ninterp_order_eqns16 := []:\nfor ct in whch do\n temp_e qn := convert(SO7_15[ct],'interpolation_order_condition'):\n interp_ order_eqns16 := [op(interp_order_eqns16),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 0 509 "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]*a dd(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 spec ify the node " }{XPPEDIT 18 0 "c[16] = 3/25;" "6#/&%\"cG6#\"#;*&\"\"$ \"\"\"\"#D!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "e9 := `union`(e8,\{c[16]=3/25,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'\"\"(&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$F'\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "info level[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e10 := \+ solve(\{op(eqs_16)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "e11 := `unio n`(e9,e10):\n[seq(a[16,i]=subs(e11,a[16,i]),i=1..15)]:\nevalf[20](%); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#71/&%\"aG6$\"#;\"\"\"$\"5T;r/-,l&Q F#!#@/&F&6$F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\"\"&F 1/&F&6$F(\"\"'$!5\"[\"o-Q@(oa[$!#A/&F&6$F(\"\"($!5>f@+iWqU=8F,/&F&6$F( \"\")$\"5\"o(3Y`sUr`GF,/&F&6$F(\"\"*$!5x^QD!)olQ>eFE/&F&6$F(\"#5$\"5\\ \"*H5%)>RJRBF,/&F&6$F(\"#6$!5!QS+$yoo!eo%F,/&F&6$F(\"#7$\"5X8YL*odBA>* !#B/&F&6$F(\"#8$\"5')HU,0c!Q6u#Fdo/&F&6$F(\"#9$\"5=hs1m*yiBn)FE/&F&6$F (\"#:$\"5/VpHP&[E\"[5!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 70 "#------------------------------------------------- --------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check whic h of the (adapted) simple order conditions are satisfied at this stage ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "recd := []:\nfor ct to nops(SO7_15) do\n tt := con vert(SO7_15[ct],'interpolation_order_condition'):\n if expand(subs(e 11,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop( recd);\nnops(recd);\nop(\{seq(i,i=1..nops(SO7_15))\} minus \{op(recd) \});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6R\"\"\"\"\"#\"\"$\"\"%\"\"&\"\" '\"\"(\"\")\"\"*\"#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 8850 "e11 := \{c[8] = 24/25, c[9] = 14435868/1617 8861, a[14,10] = 37302437685024/53021770400599+15742143124443936/12990 333748146755*7^(1/2), a[9,3] = 0, a[11,4] = -9623541317323077848129/38 64449564977792573440, a[14,9] = -3211799428826566849882311586102611111 468427562764815533/154455682095190808523075696640755012970347465573075 74592-35609926058546725508440519811766382833790503793377686129/6359939 8509784450568325286852075593576025427000678248320*7^(1/2), a[11,9] = 4 8175771419260955335244683805171548038966866545122229/19897864205138151 46528880165952064118903852843612160000, a[9,4] = -14764960804048657303 638372252908780219281424435/298169210256502197561171126920960636366185 4518, a[6,5] = 960400/5209191, a[14,12] = 1100613127343/48439922837376 -29746300739/1424703612864*7^(1/2), a[14,11] = -5071676622092000/42049 18450412337-380378998046332000/206041004070204513*7^(1/2), a[11,10] = \+ -2378292068163246/47768728487211875, a[14,2] = 0, a[4,1] = 14/355, a[9 ,6] = 7632051964154290925661849798370645637589377834346780/17340872574 18811583049800347581865260479233950396659, a[12,4] = 26290092604284231 996745/5760876126062860430544, a[9,2] = 0, a[8,2] = 0, a[10,9] = -5295 6818288156668227044990077324877908565/29127799594774339863498222244123 53951940608, a[7,6] = 124391943/6756250000, a[9,8] = 13660426834891663 51293315549358278750/144631418224267718165055326464180836641, a[11,2] \+ = 0, a[11,3] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12,9] = -23028251632873523818545414856857015616678575554130463402/200131691 83191444503443905240405603349978424504151629055, a[12,2] = 0, a[7,2] = 0, a[10,7] = 187023075231349900768014890274453125/2522469884980817801 0752575653374848, a[12,5] = -697069297560926452045586710000/4110796775 5245430594036502319, a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,8] = 1 62259938151380266113750/59091082835244183497007, a[12,10] = 7958341351 371843889152/3284467988443203581305, a[12,11] = -507974327957860843878 400/121555654819179042718967, a[11,8] = 39747262782380466933662225/175 6032802431424164410720256, a[11,5] = -4823348333146829406881375/576413 233634141239944816, a[12,1] = -3218022174758599831659045535578571/1453 396753634469525663775847094384, a[12,6] = 1827357820434213461438077550 902273440/139381013914245317709567680839641697, a[10,8] = 190815855007 0998850625/117087067039189929394176, a[11,6] = 65661192465149968840670 01154977284529/970305487021846325473990863582315520, a[12,7] = 6435048 02814241550941949227194107500000/2421246091188365508604940075453339453 31, a[11,1] = -10116106591826909534781157993685116703/9562819945036894 030442231411871744000, a[12,3] = 0, a[2,1] = 1/16, a[10,6] = 100158784 4183325981198091450220795/184232684207722503701669953872896, a[10,3] = 0, a[7,4] = 3064329829899/27126050000000, a[7,5] = -21643947/59260937 5, a[14,3] = 0, a[14,5] = 0, c[10] = 11/12, c[4] = 56/355, c[5] = 39/1 00, c[6] = 7/15, c[3] = 112/1065, a[13,5] = 0, a[8,3] = 0, a[8,7] = 19 93321838240/380523459069, a[14,6] = 1533012537239841375/79359550982749 10336-11049456415304617875/194430899907735303232*7^(1/2), a[14,8] = 13 061564753515625/18689009843335104+31047286856168359375/256413215050557 62688*7^(1/2), c[11] = 19/20, c[12] = 1, c[13] = 1, c[2] = 1/16, a[10, 5] = -3378604805394255292453489375/517042670569824692230499952, a[13,9 ] = 405932030463777247926705030596175437402459637909765779/78803919436 321841083201886041201537229769115088303952, a[13,10] = -10290327637248 /1082076946951, a[7,1] = 21400899/350000000, a[14,1] = 923507123432989 /20341288146475296+3728619917660047/7973784953418316032*7^(1/2), a[14, 7] = 129834778101757666015625000/534951006554143882189042101-827664115 29674389648437500/26212599321153050227263062949*7^(1/2), a[6,3] = 0, a [6,4] = 2505377/10685520, a[9,7] = 75198347919711375170485321796523477 29899303513750000/1045677303502317596597890707812349832637339039997351 , a[5,3] = -352806597/250880000, a[5,2] = 0, a[5,1] = 94495479/2508800 00, a[4,3] = 42/355, a[9,1] = -184091125228237658443815733646470842695 4728061551/2991923615171151921596253813483118262195533733898, a[13,4] \+ = 0, a[9,5] = -875325048502130441118613421785266742862694404520560000/ 170212030428894418395571677575961339495435011888324169, a[8,1] = -1536 5458811/13609565775, a[7,3] = 0, a[13,1] = 4631674879841/1037820823799 76, a[13,2] = 0, c[7] = 39/250, a[10,4] = -31948346510820970247215/695 6009216960026632192, a[8,6] = 341936800488/47951126225, c[15] = 3/40, \+ a[8,5] = -8339128164608/939060038475, a[8,4] = -7/5, a[11,7] = 2226455 130519213549256016892506730559375/364880443159675255577435648380047355 776, a[6,2] = 0, a[5,4] = 178077159/125440000, a[13,12] = -29746300739 /247142463456, a[13,11] = 863264105888000/85814662253313, a[6,1] = 120 89/252720, a[13,7] = 2720762324010009765625000/10917367480696813922225 349, a[13,8] = -498533005859375/95352091037424, a[13,6] = 143272199742 04125/40489566827933216, a[3,2] = 100352/1134225, a[13,3] = 0, a[3,1] \+ = 18928/1134225, a[10,1] = -63077736705254280154824845013881/783693578 53786633855112190394368, a[10,2] = 0, c[16] = 3/25, a[16,3] = 0, a[16, 4] = 0, a[16,2] = 0, a[15,1] = 53968119671846752033/121108663654809600 0000-18013979306423851/41103546452541440000*7^(1/2), a[15,9] = 1437829 216002805549145888241731712075764900749803936348249893/149435580560728 8245725902431596118039320066182415245312000000-43678103259256427869424 43213218202333579800829827668077/2667312462401300577692462721351612816 5085289185280000000*7^(1/2), a[15,10] = -43024834368326969649/17313231 151216000000+24278487789419081919/55087553662960000000*7^(1/2), a[15,1 3] = 609279/512000000-2221887/4096000000*7^(1/2), a[15,6] = -319758539 2137639804267/331690531454428905472000+37078101149895800937/3922016419 224666112000*7^(1/2), a[15,14] = -15553209/1024000000*7^(1/2), a[15,7] = 12599626362153730936328125/254800781771704557409886208+101342679797 2694921875/352212771011439949026304*7^(1/2), a[15,8] = -85326373247592 44825/3240219442044469248+132594730877413325/272450884015751168*7^(1/2 ), a[15,11] = 15187119353760445983/3661425589474688000-194738601533939 11521/25629979126322816000*7^(1/2), a[16,5] = 0, a[15,2] = 0, a[15,4] \+ = 0, a[15,5] = 0, a[15,3] = 0, a[15,12] = -36585897414219009/168715921 71929600000, a[16,1] = 33317884221304756710581664536827945414542207027 177538799/1391748247282161490991675299723566089376318996169218750000-3 022858984983534314773428506933040222615091059789769/665908252288115545 9290312438868737269743153091718750000*7^(1/2), a[16,8] = 1287168286319 1528707812574253462271793937198554473/81836773382225296828453851756295 0447076898208684200+7923793831170279449496891966550656386059347318121/ 1636735467644505936569077035125900894153796417368400*7^(1/2), a[16,12] = 8314643765317918980786740197327855795129766290647999/14914139040691 643843786103420465233322847554416562500000+127439374782036697408691950 12551757724982855520824/9321336900432277402366314637790770826779721510 3515625*7^(1/2), a[16,11] = -20330802395832958719982915003380332526117 8399821537536/8285758985671105201066755800819103710987390140968171875- 18395594530987044403702119081901330600389569409698304/2180462890966080 316070198894952395713417734247623203125*7^(1/2), a[16,13] = 8124463314 10101756322947639951373288835154/6698441904183349335450501610911809049 423828125+386918100083180096375404793490759651703962/66984419041833493 35450501610911809049423828125*7^(1/2), a[16,14] = 34809815267436573518 7679001267334640046338304/97942082977383567310235712743332126911845703 125+7578809409397518388644113904993904548128984/3917683319095342692409 428509733285076473828125*7^(1/2), a[16,6] = -6167678449409636588173213 3250119640765816127842489932383/58641515918369718256965079710079269237 067699352911722500000-276340795345731296984607784346910271936235143373 49372/30041760204082847467707520343278314158333862373417890625*7^(1/2) , a[16,10] = 1568221980335495859574819360318483217200155100501135104/1 95898096351550311454966852305863720094941335219755859375+5918726234654 06532819676818747091838359436881921536/1017652448579482137428399232757 73361088281213101171875*7^(1/2), a[16,9] = 406115460287238735025830165 985301412067123513597095836215277820544199798339360779698993414799/528 3918680341811073090017427537789807044572034179162813474927957845326668 01613048775622343750000-4313791558950843964899327540844721567810470317 95210079948966119713148914776761016489118643/1732432354210429860029513 91066812780558838427350136485687702555994928743213643622549384375000*7 ^(1/2), a[16,15] = 2901044122821490120249433136363475030176169984/2899 0856561305535923829770972026309565906328125+20797772815384263809952654 53726087303397376/1159634262452221436953190838881052382636253125*7^(1/ 2), a[16,7] = -5362312067621069347891531839642343206187360246368132602 5000/7495941567532910607209275449283868805039087051861676693501311-244 0862180486837508553671181096852807790831050835529920000/10708487953618 44372458467921326266972148441007408810956214473*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 17" }}{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_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],'interpolation_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 specified expl icitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 585 "inter p_order_eqns17 := [add(a[17,i],i=1..16)=c[17],seq(op(StageOrderConditi ons(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 ad d(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] = \+ 13/40;" "6#/&%\"cG6#\"#<*&\"#8\"\"\"\"#S!\"\"" }{TEXT -1 2 ". " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "e12 := `union`(e11,\{c[17]= 13/40,seq(a[17,i]=0,i=2..5)\}):\neqs_17 := expand(subs(e12,interp_orde r_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$F'\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e13 := solve(\{op(eqs_17)\} ):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "e14 := `union`(e12,e13):\n[seq(a[17 ,i]=subs(e14,a[17,i]),i=1..16)]:\nevalf[20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#72/&%\"aG6$\"#<\"\"\"$\"5D\"z=soK)f&R%!#@/&F&6$F(\"\"#$ \"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F&6$F(\"\"'$!4ce dDH!HAy!)F,/&F&6$F(\"\"($!5ooOT6%H?16\"F,/&F&6$F(\"\")$!5aaU#R+Ig(4hF, /&F&6$F(\"\"*$\"5(f2_8)G%G&yNF,/&F&6$F(\"#5$!4!*)ofK#\\n)H#)!#?/&F&6$F (\"#6$\"5g>qd6pL<06Fgn/&F&6$F(\"#7$!5sm*)4RKgNV " 0 " " {MPLTEXT 1 0 205 "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);" }}{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 10610 "e14 := \{c[8] = 24/25, c[9] = 14435868/16178861, a[14,10] = 3 7302437685024/53021770400599+15742143124443936/12990333748146755*7^(1/ 2), a[9,3] = 0, a[17,4] = 0, a[17,3] = 0, a[11,4] = -96235413173230778 48129/3864449564977792573440, a[14,9] = -32117994288265668498823115861 02611111468427562764815533/1544556820951908085230756966407550129703474 6557307574592-35609926058546725508440519811766382833790503793377686129 /63599398509784450568325286852075593576025427000678248320*7^(1/2), a[1 1,9] = 48175771419260955335244683805171548038966866545122229/198978642 0513815146528880165952064118903852843612160000, a[9,4] = -147649608040 48657303638372252908780219281424435/2981692102565021975611711269209606 363661854518, a[6,5] = 960400/5209191, a[14,12] = 1100613127343/484399 22837376-29746300739/1424703612864*7^(1/2), a[14,11] = -50716766220920 00/4204918450412337-380378998046332000/206041004070204513*7^(1/2), a[1 1,10] = -2378292068163246/47768728487211875, a[14,2] = 0, a[4,1] = 14/ 355, a[9,6] = 7632051964154290925661849798370645637589377834346780/173 4087257418811583049800347581865260479233950396659, a[12,4] = 262900926 04284231996745/5760876126062860430544, a[9,2] = 0, a[8,2] = 0, a[10,9] = -52956818288156668227044990077324877908565/291277995947743398634982 2224412353951940608, a[7,6] = 124391943/6756250000, a[9,8] = 136604268 3489166351293315549358278750/144631418224267718165055326464180836641, \+ a[11,2] = 0, a[11,3] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12,9] = -23028251632873523818545414856857015616678575554130463402/2 0013169183191444503443905240405603349978424504151629055, a[12,2] = 0, \+ a[7,2] = 0, a[10,7] = 187023075231349900768014890274453125/25224698849 808178010752575653374848, a[12,5] = -697069297560926452045586710000/41 107967755245430594036502319, a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[1 2,8] = 162259938151380266113750/59091082835244183497007, a[12,10] = 79 58341351371843889152/3284467988443203581305, a[12,11] = -5079743279578 60843878400/121555654819179042718967, a[11,8] = 3974726278238046693366 2225/1756032802431424164410720256, a[11,5] = -482334833314682940688137 5/576413233634141239944816, a[12,1] = -3218022174758599831659045535578 571/1453396753634469525663775847094384, a[12,6] = 18273578204342134614 38077550902273440/139381013914245317709567680839641697, a[10,8] = 1908 158550070998850625/117087067039189929394176, a[11,6] = 656611924651499 6884067001154977284529/970305487021846325473990863582315520, a[12,7] = 643504802814241550941949227194107500000/24212460911883655086049400754 5333945331, a[11,1] = -10116106591826909534781157993685116703/95628199 45036894030442231411871744000, a[12,3] = 0, a[2,1] = 1/16, a[10,6] = 1 001587844183325981198091450220795/184232684207722503701669953872896, a [10,3] = 0, a[7,4] = 3064329829899/27126050000000, a[7,5] = -21643947/ 592609375, a[14,3] = 0, a[14,5] = 0, c[10] = 11/12, c[4] = 56/355, c[5 ] = 39/100, c[6] = 7/15, c[3] = 112/1065, a[13,5] = 0, a[8,3] = 0, a[8 ,7] = 1993321838240/380523459069, a[14,6] = 1533012537239841375/793595 5098274910336-11049456415304617875/194430899907735303232*7^(1/2), a[14 ,8] = 13061564753515625/18689009843335104+31047286856168359375/2564132 1505055762688*7^(1/2), c[11] = 19/20, c[12] = 1, c[13] = 1, c[2] = 1/1 6, a[10,5] = -3378604805394255292453489375/517042670569824692230499952 , a[13,9] = 405932030463777247926705030596175437402459637909765779/788 03919436321841083201886041201537229769115088303952, a[13,10] = -102903 27637248/1082076946951, a[7,1] = 21400899/350000000, a[14,1] = 9235071 23432989/20341288146475296+3728619917660047/7973784953418316032*7^(1/2 ), a[14,7] = 129834778101757666015625000/534951006554143882189042101-8 2766411529674389648437500/26212599321153050227263062949*7^(1/2), a[6,3 ] = 0, a[6,4] = 2505377/10685520, a[9,7] = 751983479197113751704853217 9652347729899303513750000/10456773035023175965978907078123498326373390 39997351, a[5,3] = -352806597/250880000, a[5,2] = 0, a[5,1] = 94495479 /250880000, a[4,3] = 42/355, a[9,1] = -1840911252282376584438157336464 708426954728061551/2991923615171151921596253813483118262195533733898, \+ a[13,4] = 0, a[9,5] = -87532504850213044111861342178526674286269440452 0560000/170212030428894418395571677575961339495435011888324169, a[8,1] = -15365458811/13609565775, a[7,3] = 0, a[13,1] = 4631674879841/10378 2082379976, a[13,2] = 0, c[7] = 39/250, a[10,4] = -3194834651082097024 7215/6956009216960026632192, a[8,6] = 341936800488/47951126225, c[15] \+ = 3/40, a[8,5] = -8339128164608/939060038475, a[8,4] = -7/5, a[11,7] = 2226455130519213549256016892506730559375/3648804431596752555774356483 80047355776, a[6,2] = 0, a[5,4] = 178077159/125440000, a[13,12] = -297 46300739/247142463456, a[13,11] = 863264105888000/85814662253313, a[6, 1] = 12089/252720, a[13,7] = 2720762324010009765625000/109173674806968 13922225349, a[13,8] = -498533005859375/95352091037424, a[13,6] = 1432 7219974204125/40489566827933216, a[3,2] = 100352/1134225, a[13,3] = 0, a[3,1] = 18928/1134225, a[10,1] = -63077736705254280154824845013881/7 8369357853786633855112190394368, a[10,2] = 0, c[16] = 3/25, a[17,5] = \+ 0, a[16,3] = 0, a[16,4] = 0, a[16,2] = 0, a[15,1] = 539681196718467520 33/1211086636548096000000-18013979306423851/41103546452541440000*7^(1/ 2), a[15,9] = 14378292160028055491458882417317120757649007498039363482 49893/1494355805607288245725902431596118039320066182415245312000000-43 67810325925642786942443213218202333579800829827668077/2667312462401300 5776924627213516128165085289185280000000*7^(1/2), a[15,10] = -43024834 368326969649/17313231151216000000+24278487789419081919/550875536629600 00000*7^(1/2), a[15,13] = 609279/512000000-2221887/4096000000*7^(1/2), a[15,6] = -3197585392137639804267/331690531454428905472000+3707810114 9895800937/3922016419224666112000*7^(1/2), a[15,14] = -15553209/102400 0000*7^(1/2), a[15,7] = 12599626362153730936328125/2548007817717045574 09886208+1013426797972694921875/352212771011439949026304*7^(1/2), a[15 ,8] = -8532637324759244825/3240219442044469248+132594730877413325/2724 50884015751168*7^(1/2), a[15,11] = 15187119353760445983/36614255894746 88000-19473860153393911521/25629979126322816000*7^(1/2), a[16,5] = 0, \+ a[15,2] = 0, a[15,4] = 0, a[15,5] = 0, a[15,3] = 0, c[17] = 13/40, a[1 7,2] = 0, a[15,12] = -36585897414219009/16871592171929600000, a[16,1] \+ = 33317884221304756710581664536827945414542207027177538799/13917482472 82161490991675299723566089376318996169218750000-3022858984983534314773 428506933040222615091059789769/665908252288115545929031243886873726974 3153091718750000*7^(1/2), a[16,8] = 1287168286319152870781257425346227 1793937198554473/818367733822252968284538517562950447076898208684200+7 923793831170279449496891966550656386059347318121/163673546764450593656 9077035125900894153796417368400*7^(1/2), a[16,12] = 831464376531791898 0786740197327855795129766290647999/14914139040691643843786103420465233 322847554416562500000+127439374782036697408691950125517577249828555208 24/93213369004322774023663146377907708267797215103515625*7^(1/2), a[16 ,11] = -203308023958329587199829150033803325261178399821537536/8285758 985671105201066755800819103710987390140968171875-183955945309870444037 02119081901330600389569409698304/2180462890966080316070198894952395713 417734247623203125*7^(1/2), a[16,13] = 8124463314101017563229476399513 73288835154/6698441904183349335450501610911809049423828125+38691810008 3180096375404793490759651703962/66984419041833493354505016109118090494 23828125*7^(1/2), a[16,14] = 34809815267436573518767900126733464004633 8304/97942082977383567310235712743332126911845703125+75788094093975183 88644113904993904548128984/3917683319095342692409428509733285076473828 125*7^(1/2), a[16,6] = -6167678449409636588173213325011964076581612784 2489932383/58641515918369718256965079710079269237067699352911722500000 -27634079534573129698460778434691027193623514337349372/300417602040828 47467707520343278314158333862373417890625*7^(1/2), a[16,10] = 15682219 80335495859574819360318483217200155100501135104/1958980963515503114549 66852305863720094941335219755859375+5918726234654065328196768187470918 38359436881921536/1017652448579482137428399232757733610882812131011718 75*7^(1/2), a[16,9] = 406115460287238735025830165985301412067123513597 095836215277820544199798339360779698993414799/528391868034181107309001 7427537789807044572034179162813474927957845326668016130487756223437500 00-4313791558950843964899327540844721567810470317952100799489661197131 48914776761016489118643/1732432354210429860029513910668127805588384273 50136485687702555994928743213643622549384375000*7^(1/2), a[16,15] = 29 01044122821490120249433136363475030176169984/2899085656130553592382977 0972026309565906328125+2079777281538426380995265453726087303397376/115 9634262452221436953190838881052382636253125*7^(1/2), a[16,7] = -536231 20676210693478915318396423432061873602463681326025000/7495941567532910 607209275449283868805039087051861676693501311-244086218048683750855367 1181096852807790831050835529920000/10708487953618443724584679213262669 72148441007408810956214473*7^(1/2), a[17,10] = -1489758732528250543113 521169/6777268656058713348166400000+182993414470798047123969/352065904 2108422518528000*7^(1/2), a[17,1] = 1204472757949318149125704321217/20 800267292079442987248844800000-2198100637293808474641843/4168724405178 66021068800000*7^(1/2), a[17,12] = -1906612263858924745067238619/12878 55848637222919589068800000-1218769836385461326457543/12265293796544980 186562560000*7^(1/2), a[17,14] = 27217572159567342015099/8936684185518 73945600000+989931556574911765269/35746736742074957824000*7^(1/2), a[1 7,11] = 10565149828794636491259146771/55897372479169766976172492800-78 967956837666628354617891/2661779641865226998865356800*7^(1/2), a[17,7] = 10629566348189721232485330989365234375/3500938220035776504779508667 58750994432-9677609936618471677294132306640625/61744942152306463929091 8636258820096*7^(1/2), a[17,6] = 48686024023005238414132359876273/1687 924437393983211523871237734400-90947086755739272427273888113/651708276 9860939040632707481600*7^(1/2), a[17,8] = -139813563097406086392326875 /1528856569422457169284694016+7802957598378772386375/68017198061281600 2351104*7^(1/2), a[17,9] = 1389822806559455341499614630827323136554807 9803650544776770188499895421/10266152477644947347145999385280743005156 8429917145049778991367782400000-18175102671333836006788877922318371017 2814810723200929646376184583/48282904068876883466883007103025246349943 99996103235733285896192000*7^(1/2), a[17,16] = 13295441661397228125/46 517010473975545856-682342228168453125/46517010473975545856*7^(1/2), a[ 17,13] = -30165509761493343939/16966969514364108800000+159134879536428 159/458566743631462400000*7^(1/2), a[17,15] = -1204548165456597/101863 57540916840+350890378839/13765348028266*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 "calculatio n for stage 18" }}{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_17 := SimpleOrderConditions(7,17,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1, 2,4,8,16,17,25,27,32,61,63,64]:\ninterp_order_eqns18 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_17[ct],'interpolation_order_condi tion'):\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 explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 590 "interp_orde r_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]*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..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 ad d(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] = 2/5;" "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]=2/5,seq(a[18,i]=0,i=2..5),a[18,17]=0\}):\neqs_18 := expand(subs(e1 5,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'\"#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[so lve]:=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 80 "e17 := `union`(e15,e16): \n[seq(a[18,i]=subs(e17,a[18,i]),i=1..17)]:\nevalf[20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#73/&%\"aG6$\"#=\"\"\"$\"5)*3NVTyNm@K!#@/&F&6 $F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F&6$F( \"\"'$\"4fDdPM_S\"pgF,/&F&6$F(\"\"($\"4Z&4h0y!>E+)F,/&F&6$F(\"\")$\"5$ *R&=7z'yw_MF,/&F&6$F(\"\"*$!4_v6VKY<C:ORLN+H%Fdo/&F&6$F(\"#9$\"5nvu*G.>:U!=FW/&F&6$F(\"#:$\"5HzufZP IG0CF,/&F&6$F(\"#;$\"5%G#)QEHA<5\\\"FW/&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 12217 "e17 := \{c[8] = 24/25, c[ 9] = 14435868/16178861, a[14,10] = 37302437685024/53021770400599+15742 143124443936/12990333748146755*7^(1/2), a[9,3] = 0, a[17,4] = 0, a[17, 3] = 0, a[11,4] = -9623541317323077848129/3864449564977792573440, a[14 ,9] = -3211799428826566849882311586102611111468427562764815533/1544556 8209519080852307569664075501297034746557307574592-35609926058546725508 440519811766382833790503793377686129/635993985097844505683252868520755 93576025427000678248320*7^(1/2), a[11,9] = 481757714192609553352446838 05171548038966866545122229/1989786420513815146528880165952064118903852 843612160000, a[9,4] = -1476496080404865730363837225290878021928142443 5/2981692102565021975611711269209606363661854518, a[6,5] = 960400/5209 191, a[14,12] = 1100613127343/48439922837376-29746300739/1424703612864 *7^(1/2), a[14,11] = -5071676622092000/4204918450412337-38037899804633 2000/206041004070204513*7^(1/2), a[11,10] = -2378292068163246/47768728 487211875, a[14,2] = 0, a[4,1] = 14/355, a[9,6] = 76320519641542909256 61849798370645637589377834346780/1734087257418811583049800347581865260 479233950396659, a[12,4] = 26290092604284231996745/5760876126062860430 544, a[9,2] = 0, a[8,2] = 0, a[10,9] = -529568182881566682270449900773 24877908565/2912779959477433986349822224412353951940608, a[7,6] = 1243 91943/6756250000, a[9,8] = 1366042683489166351293315549358278750/14463 1418224267718165055326464180836641, a[11,2] = 0, a[11,3] = 0, a[14,4] \+ = 0, a[14,13] = 3/392-3/392*7^(1/2), a[12,9] = -2302825163287352381854 5414856857015616678575554130463402/20013169183191444503443905240405603 349978424504151629055, a[12,2] = 0, a[7,2] = 0, a[10,7] = 187023075231 349900768014890274453125/25224698849808178010752575653374848, a[12,5] \+ = -697069297560926452045586710000/41107967755245430594036502319, a[4,2 ] = 0, c[14] = 1/2-1/14*7^(1/2), a[12,8] = 162259938151380266113750/59 091082835244183497007, a[12,10] = 7958341351371843889152/3284467988443 203581305, a[12,11] = -507974327957860843878400/1215556548191790427189 67, a[11,8] = 39747262782380466933662225/1756032802431424164410720256, a[11,5] = -4823348333146829406881375/576413233634141239944816, a[12,1 ] = -3218022174758599831659045535578571/145339675363446952566377584709 4384, a[12,6] = 1827357820434213461438077550902273440/1393810139142453 17709567680839641697, a[10,8] = 1908158550070998850625/117087067039189 929394176, a[11,6] = 6566119246514996884067001154977284529/97030548702 1846325473990863582315520, a[12,7] = 643504802814241550941949227194107 500000/242124609118836550860494007545333945331, a[11,1] = -10116106591 826909534781157993685116703/9562819945036894030442231411871744000, a[1 2,3] = 0, a[2,1] = 1/16, a[10,6] = 1001587844183325981198091450220795/ 184232684207722503701669953872896, a[10,3] = 0, a[7,4] = 3064329829899 /27126050000000, a[7,5] = -21643947/592609375, a[14,3] = 0, a[14,5] = \+ 0, c[10] = 11/12, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, c[3] = 11 2/1065, a[13,5] = 0, a[8,3] = 0, a[8,7] = 1993321838240/380523459069, \+ a[14,6] = 1533012537239841375/7935955098274910336-11049456415304617875 /194430899907735303232*7^(1/2), a[14,8] = 13061564753515625/1868900984 3335104+31047286856168359375/25641321505055762688*7^(1/2), c[11] = 19/ 20, c[12] = 1, c[13] = 1, c[2] = 1/16, a[10,5] = -33786048053942552924 53489375/517042670569824692230499952, a[13,9] = 4059320304637772479267 05030596175437402459637909765779/7880391943632184108320188604120153722 9769115088303952, a[13,10] = -10290327637248/1082076946951, a[7,1] = 2 1400899/350000000, a[14,1] = 923507123432989/20341288146475296+3728619 917660047/7973784953418316032*7^(1/2), a[14,7] = 129834778101757666015 625000/534951006554143882189042101-82766411529674389648437500/26212599 321153050227263062949*7^(1/2), a[6,3] = 0, a[6,4] = 2505377/10685520, \+ a[9,7] = 7519834791971137517048532179652347729899303513750000/10456773 03502317596597890707812349832637339039997351, a[5,3] = -352806597/2508 80000, a[5,2] = 0, a[5,1] = 94495479/250880000, a[4,3] = 42/355, a[9,1 ] = -1840911252282376584438157336464708426954728061551/299192361517115 1921596253813483118262195533733898, a[13,4] = 0, a[9,5] = -87532504850 2130441118613421785266742862694404520560000/17021203042889441839557167 7575961339495435011888324169, a[8,1] = -15365458811/13609565775, a[7,3 ] = 0, a[13,1] = 4631674879841/103782082379976, a[13,2] = 0, c[7] = 39 /250, a[10,4] = -31948346510820970247215/6956009216960026632192, a[8,6 ] = 341936800488/47951126225, c[15] = 3/40, a[8,5] = -8339128164608/93 9060038475, a[8,4] = -7/5, a[11,7] = 222645513051921354925601689250673 0559375/364880443159675255577435648380047355776, a[6,2] = 0, a[5,4] = \+ 178077159/125440000, a[13,12] = -29746300739/247142463456, a[13,11] = \+ 863264105888000/85814662253313, a[6,1] = 12089/252720, a[13,7] = 27207 62324010009765625000/10917367480696813922225349, a[13,8] = -4985330058 59375/95352091037424, a[13,6] = 14327219974204125/40489566827933216, a [3,2] = 100352/1134225, a[13,3] = 0, a[3,1] = 18928/1134225, a[10,1] = -63077736705254280154824845013881/78369357853786633855112190394368, a [10,2] = 0, c[16] = 3/25, a[17,5] = 0, a[16,3] = 0, a[16,4] = 0, c[18] = 2/5, a[18,2] = 0, a[16,2] = 0, a[15,1] = 53968119671846752033/12110 86636548096000000-18013979306423851/41103546452541440000*7^(1/2), a[15 ,9] = 1437829216002805549145888241731712075764900749803936348249893/14 94355805607288245725902431596118039320066182415245312000000-4367810325 925642786942443213218202333579800829827668077/266731246240130057769246 27213516128165085289185280000000*7^(1/2), a[15,10] = -4302483436832696 9649/17313231151216000000+24278487789419081919/55087553662960000000*7^ (1/2), a[15,13] = 609279/512000000-2221887/4096000000*7^(1/2), a[15,6] = -3197585392137639804267/331690531454428905472000+370781011498958009 37/3922016419224666112000*7^(1/2), a[15,14] = -15553209/1024000000*7^( 1/2), a[15,7] = 12599626362153730936328125/254800781771704557409886208 +1013426797972694921875/352212771011439949026304*7^(1/2), a[18,17] = 0 , a[15,8] = -8532637324759244825/3240219442044469248+13259473087741332 5/272450884015751168*7^(1/2), a[15,11] = 15187119353760445983/36614255 89474688000-19473860153393911521/25629979126322816000*7^(1/2), a[18,14 ] = 1083595819903600472/20454446835751509375+39411512915496232/8181778 73430060375*7^(1/2), a[18,16] = 259448245437500/1197776110455837-30561 362328125/1197776110455837*7^(1/2), a[18,13] = -56700339581774/4854288 9159290625+29301850944203/48542889159290625*7^(1/2), a[18,10] = -84424 01866475056490423808/43019771742560192151446875+1087953059968188774912 /12033502585331522280125*7^(1/2), a[18,6] = 56575761137587272324976173 /804865091988555532228408450-39074664028872736786931649/16097301839771 11064456816900*7^(1/2), a[18,12] = 911312938075044590941/6140975230394 47269243750-14152280688871027947/81879669738592969232500*7^(1/2), a[18 ,9] = 6110065569363880271648643567888751040237399254671572996643015547 3/391622637849614995847549415026883812147401542347507666698422881250-1 70949024966397582880137105076434230151670370842813631056736667/2610817 585664099972316996100179225414316010282316717777989485875*7^(1/2), a[1 8,8] = -2770031482617170508125/151635249347625299077614+38792939239451 158625/1944041658302888449713*7^(1/2), a[18,7] = 111299441065094395309 72892187500000/138892202332962324713542519160881437-126220514080179501 0221127437500000/46297400777654108237847506386960479*7^(1/2), a[18,11] = 2045460261009505062889984/27293638905844612781334225-46948919188939 0453370368/9097879635281537593778075*7^(1/2), a[18,1] = 29131180830546 450644731399/515753697961869733494253125-143343963156330219505552/1562 8899938238476772553125*7^(1/2), a[18,15] = -3206129319411712/343789567 00594335+304869207965696/6875791340118867*7^(1/2), 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, c[17] = 13/40, a[17,2] = 0, a[15,12] = -365858974142190 09/16871592171929600000, a[16,1] = 33317884221304756710581664536827945 414542207027177538799/139174824728216149099167529972356608937631899616 9218750000-3022858984983534314773428506933040222615091059789769/665908 2522881155459290312438868737269743153091718750000*7^(1/2), a[16,8] = 1 2871682863191528707812574253462271793937198554473/81836773382225296828 4538517562950447076898208684200+79237938311702794494968919665506563860 59347318121/1636735467644505936569077035125900894153796417368400*7^(1/ 2), a[16,12] = 8314643765317918980786740197327855795129766290647999/14 914139040691643843786103420465233322847554416562500000+127439374782036 69740869195012551757724982855520824/9321336900432277402366314637790770 8267797215103515625*7^(1/2), a[16,11] = -20330802395832958719982915003 3803325261178399821537536/82857589856711052010667558008191037109873901 40968171875-18395594530987044403702119081901330600389569409698304/2180 462890966080316070198894952395713417734247623203125*7^(1/2), a[16,13] \+ = 812446331410101756322947639951373288835154/6698441904183349335450501 610911809049423828125+386918100083180096375404793490759651703962/66984 41904183349335450501610911809049423828125*7^(1/2), a[16,14] = 34809815 2674365735187679001267334640046338304/97942082977383567310235712743332 126911845703125+7578809409397518388644113904993904548128984/3917683319 095342692409428509733285076473828125*7^(1/2), a[16,6] = -6167678449409 6365881732133250119640765816127842489932383/58641515918369718256965079 710079269237067699352911722500000-276340795345731296984607784346910271 93623514337349372/3004176020408284746770752034327831415833386237341789 0625*7^(1/2), a[16,10] = 156822198033549585957481936031848321720015510 0501135104/195898096351550311454966852305863720094941335219755859375+5 91872623465406532819676818747091838359436881921536/1017652448579482137 42839923275773361088281213101171875*7^(1/2), a[16,9] = 406115460287238 7350258301659853014120671235135970958362152778205441997983393607796989 93414799/5283918680341811073090017427537789807044572034179162813474927 95784532666801613048775622343750000-4313791558950843964899327540844721 56781047031795210079948966119713148914776761016489118643/1732432354210 4298600295139106681278055883842735013648568770255599492874321364362254 9384375000*7^(1/2), a[16,15] = 290104412282149012024943313636347503017 6169984/28990856561305535923829770972026309565906328125+20797772815384 26380995265453726087303397376/1159634262452221436953190838881052382636 253125*7^(1/2), a[16,7] = -5362312067621069347891531839642343206187360 2463681326025000/74959415675329106072092754492838688050390870518616766 93501311-2440862180486837508553671181096852807790831050835529920000/10 70848795361844372458467921326266972148441007408810956214473*7^(1/2), a [17,10] = -1489758732528250543113521169/6777268656058713348166400000+1 82993414470798047123969/3520659042108422518528000*7^(1/2), a[17,1] = 1 204472757949318149125704321217/20800267292079442987248844800000-219810 0637293808474641843/416872440517866021068800000*7^(1/2), a[17,12] = -1 906612263858924745067238619/1287855848637222919589068800000-1218769836 385461326457543/12265293796544980186562560000*7^(1/2), a[17,14] = 2721 7572159567342015099/893668418551873945600000+989931556574911765269/357 46736742074957824000*7^(1/2), a[17,11] = 10565149828794636491259146771 /55897372479169766976172492800-78967956837666628354617891/266177964186 5226998865356800*7^(1/2), a[17,7] = 1062956634818972123248533098936523 4375/350093822003577650477950866758750994432-9677609936618471677294132 306640625/617449421523064639290918636258820096*7^(1/2), a[17,6] = 4868 6024023005238414132359876273/1687924437393983211523871237734400-909470 86755739272427273888113/6517082769860939040632707481600*7^(1/2), a[17, 8] = -139813563097406086392326875/1528856569422457169284694016+7802957 598378772386375/680171980612816002351104*7^(1/2), a[17,9] = 1389822806 5594553414996146308273231365548079803650544776770188499895421/10266152 4776449473471459993852807430051568429917145049778991367782400000-18175 1026713338360067888779223183710172814810723200929646376184583/48282904 06887688346688300710302524634994399996103235733285896192000*7^(1/2), a [17,16] = 13295441661397228125/46517010473975545856-682342228168453125 /46517010473975545856*7^(1/2), a[17,13] = -30165509761493343939/169669 69514364108800000+159134879536428159/458566743631462400000*7^(1/2), a[ 17,15] = -1204548165456597/10186357540916840+350890378839/137653480282 66*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 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] = 3/10;" "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$\" #>\"#=\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "e18 := `union`(e17,\{c[19]=3/10,seq(a[19,i]=0,i=2..5),a[19,17]=0, a[19,18]=0\}):\neqs_19 := expand(subs(e18,interp_order_eqns19)):\nnops (eqs_19);\nindets(eqs_19);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#>\"#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$F'\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "e19 := solve(\{op(eqs_19)\}):\ninfolevel[sol ve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "e20 := `union`(e18,e19):\n[seq(a[19,i]=subs(e20,a[19, i]),i=1..18)]:\nevalf[20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#74/&% \"aG6$\"#>\"\"\"$\"5E%y*yw>0u2W!#@/&F&6$F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1 /&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F&6$F(\"\"'$!4\"ocDQ*)4*G>)F,/&F&6$F( \"\"($!5o\\MJPx.>E6F,/&F&6$F(\"\")$!5uzA$Q5*\\***='F,/&F&6$F(\"\"*$\"5 X]$\\0#Rp!Qi$F,/&F&6$F(\"#5$!4vs^EPq6 Fgn/&F&6$F(\"#7$!5vLw:@yo[m`Urey&\\e^F,/&F&6$F(\"#;$\"5c8m >RcY-![#Fgn/&F&6$F(\"# " 0 "" {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_18) do\n t t := convert(SO7_18[ct],'interpolation_order_condition'):\n if expan d(subs(e20,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend \+ do:\nop(recd);\nnops(recd);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6\\o\"\" \"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#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 13869 "e20 := \{c[8] = 24/25, c[9] = 14435868/16178861, a [14,10] = 37302437685024/53021770400599+15742143124443936/129903337481 46755*7^(1/2), a[9,3] = 0, a[17,4] = 0, a[17,3] = 0, a[11,4] = -962354 1317323077848129/3864449564977792573440, a[14,9] = -321179942882656684 9882311586102611111468427562764815533/15445568209519080852307569664075 501297034746557307574592-356099260585467255084405198117663828337905037 93377686129/63599398509784450568325286852075593576025427000678248320*7 ^(1/2), a[11,9] = 4817577141926095533524468380517154803896686654512222 9/1989786420513815146528880165952064118903852843612160000, a[9,4] = -1 4764960804048657303638372252908780219281424435/29816921025650219756117 11269209606363661854518, a[6,5] = 960400/5209191, a[14,12] = 110061312 7343/48439922837376-29746300739/1424703612864*7^(1/2), a[14,11] = -507 1676622092000/4204918450412337-380378998046332000/206041004070204513*7 ^(1/2), a[11,10] = -2378292068163246/47768728487211875, a[14,2] = 0, a [4,1] = 14/355, a[9,6] = 763205196415429092566184979837064563758937783 4346780/1734087257418811583049800347581865260479233950396659, a[12,4] \+ = 26290092604284231996745/5760876126062860430544, a[9,2] = 0, a[8,2] = 0, a[10,9] = -52956818288156668227044990077324877908565/2912779959477 433986349822224412353951940608, a[7,6] = 124391943/6756250000, a[9,8] \+ = 1366042683489166351293315549358278750/144631418224267718165055326464 180836641, a[11,2] = 0, a[11,3] = 0, a[14,4] = 0, a[14,13] = 3/392-3/3 92*7^(1/2), a[12,9] = -23028251632873523818545414856857015616678575554 130463402/20013169183191444503443905240405603349978424504151629055, a[ 12,2] = 0, a[7,2] = 0, a[10,7] = 187023075231349900768014890274453125/ 25224698849808178010752575653374848, a[12,5] = -6970692975609264520455 86710000/41107967755245430594036502319, a[4,2] = 0, c[14] = 1/2-1/14*7 ^(1/2), a[12,8] = 162259938151380266113750/59091082835244183497007, a[ 12,10] = 7958341351371843889152/3284467988443203581305, a[12,11] = -50 7974327957860843878400/121555654819179042718967, a[11,8] = 39747262782 380466933662225/1756032802431424164410720256, a[11,5] = -4823348333146 829406881375/576413233634141239944816, a[12,1] = -32180221747585998316 59045535578571/1453396753634469525663775847094384, a[12,6] = 182735782 0434213461438077550902273440/139381013914245317709567680839641697, a[1 0,8] = 1908158550070998850625/117087067039189929394176, a[11,6] = 6566 119246514996884067001154977284529/970305487021846325473990863582315520 , a[12,7] = 643504802814241550941949227194107500000/242124609118836550 860494007545333945331, a[11,1] = -101161065918269095347811579936851167 03/9562819945036894030442231411871744000, a[12,3] = 0, a[2,1] = 1/16, \+ a[10,6] = 1001587844183325981198091450220795/1842326842077225037016699 53872896, a[10,3] = 0, a[7,4] = 3064329829899/27126050000000, a[7,5] = -21643947/592609375, a[14,3] = 0, a[14,5] = 0, c[10] = 11/12, c[4] = \+ 56/355, c[5] = 39/100, c[6] = 7/15, c[3] = 112/1065, a[13,5] = 0, a[8, 3] = 0, a[8,7] = 1993321838240/380523459069, a[14,6] = 153301253723984 1375/7935955098274910336-11049456415304617875/194430899907735303232*7^ (1/2), a[14,8] = 13061564753515625/18689009843335104+31047286856168359 375/25641321505055762688*7^(1/2), c[11] = 19/20, c[12] = 1, c[13] = 1, c[2] = 1/16, a[10,5] = -3378604805394255292453489375/5170426705698246 92230499952, a[13,9] = 40593203046377724792670503059617543740245963790 9765779/78803919436321841083201886041201537229769115088303952, a[13,10 ] = -10290327637248/1082076946951, a[7,1] = 21400899/350000000, a[14,1 ] = 923507123432989/20341288146475296+3728619917660047/797378495341831 6032*7^(1/2), a[14,7] = 129834778101757666015625000/534951006554143882 189042101-82766411529674389648437500/26212599321153050227263062949*7^( 1/2), a[6,3] = 0, a[6,4] = 2505377/10685520, a[9,7] = 7519834791971137 517048532179652347729899303513750000/104567730350231759659789070781234 9832637339039997351, a[5,3] = -352806597/250880000, a[5,2] = 0, a[5,1] = 94495479/250880000, a[4,3] = 42/355, a[9,1] = -18409112522823765844 38157336464708426954728061551/2991923615171151921596253813483118262195 533733898, a[13,4] = 0, a[9,5] = -875325048502130441118613421785266742 862694404520560000/170212030428894418395571677575961339495435011888324 169, a[8,1] = -15365458811/13609565775, a[7,3] = 0, a[13,1] = 46316748 79841/103782082379976, a[13,2] = 0, c[7] = 39/250, a[10,4] = -31948346 510820970247215/6956009216960026632192, a[8,6] = 341936800488/47951126 225, c[15] = 3/40, a[8,5] = -8339128164608/939060038475, a[8,4] = -7/5 , a[11,7] = 2226455130519213549256016892506730559375/36488044315967525 5577435648380047355776, a[6,2] = 0, a[5,4] = 178077159/125440000, a[13 ,12] = -29746300739/247142463456, a[13,11] = 863264105888000/858146622 53313, a[6,1] = 12089/252720, a[13,7] = 2720762324010009765625000/1091 7367480696813922225349, a[13,8] = -498533005859375/95352091037424, a[1 3,6] = 14327219974204125/40489566827933216, a[3,2] = 100352/1134225, a [13,3] = 0, a[3,1] = 18928/1134225, a[10,1] = -63077736705254280154824 845013881/78369357853786633855112190394368, a[10,2] = 0, c[16] = 3/25, a[17,5] = 0, a[16,3] = 0, a[16,4] = 0, c[18] = 2/5, a[18,2] = 0, a[19 ,2] = 0, a[16,2] = 0, a[19,17] = 0, a[19,18] = 0, a[15,1] = 5396811967 1846752033/1211086636548096000000-18013979306423851/411035464525414400 00*7^(1/2), a[15,9] = 143782921600280554914588824173171207576490074980 3936348249893/14943558056072882457259024315961180393200661824152453120 00000-4367810325925642786942443213218202333579800829827668077/26673124 624013005776924627213516128165085289185280000000*7^(1/2), a[15,10] = - 43024834368326969649/17313231151216000000+24278487789419081919/5508755 3662960000000*7^(1/2), a[15,13] = 609279/512000000-2221887/4096000000* 7^(1/2), a[15,6] = -3197585392137639804267/331690531454428905472000+37 078101149895800937/3922016419224666112000*7^(1/2), a[15,14] = -1555320 9/1024000000*7^(1/2), a[15,7] = 12599626362153730936328125/25480078177 1704557409886208+1013426797972694921875/352212771011439949026304*7^(1/ 2), a[18,17] = 0, a[15,8] = -8532637324759244825/3240219442044469248+1 32594730877413325/272450884015751168*7^(1/2), a[15,11] = 1518711935376 0445983/3661425589474688000-19473860153393911521/25629979126322816000* 7^(1/2), a[18,14] = 1083595819903600472/20454446835751509375+394115129 15496232/818177873430060375*7^(1/2), a[18,16] = 259448245437500/119777 6110455837-30561362328125/1197776110455837*7^(1/2), a[18,13] = -567003 39581774/48542889159290625+29301850944203/48542889159290625*7^(1/2), a [18,10] = -8442401866475056490423808/43019771742560192151446875+108795 3059968188774912/12033502585331522280125*7^(1/2), a[18,6] = 5657576113 7587272324976173/804865091988555532228408450-3907466402887273678693164 9/1609730183977111064456816900*7^(1/2), a[18,12] = 9113129380750445909 41/614097523039447269243750-14152280688871027947/818796697385929692325 00*7^(1/2), a[18,9] = 611006556936388027164864356788875104023739925467 15729966430155473/3916226378496149958475494150268838121474015423475076 66698422881250-1709490249663975828801371050764342301516703708428136310 56736667/2610817585664099972316996100179225414316010282316717777989485 875*7^(1/2), a[18,8] = -2770031482617170508125/15163524934762529907761 4+38792939239451158625/1944041658302888449713*7^(1/2), a[18,7] = 11129 944106509439530972892187500000/138892202332962324713542519160881437-12 62205140801795010221127437500000/46297400777654108237847506386960479*7 ^(1/2), a[18,11] = 2045460261009505062889984/2729363890584461278133422 5-469489191889390453370368/9097879635281537593778075*7^(1/2), a[18,1] \+ = 29131180830546450644731399/515753697961869733494253125-1433439631563 30219505552/15628899938238476772553125*7^(1/2), a[18,15] = -3206129319 411712/34378956700594335+304869207965696/6875791340118867*7^(1/2), a[1 8,5] = 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, c[17] = 13/40, a[17,2] = 0, a[15,12] = \+ -36585897414219009/16871592171929600000, a[16,1] = 3331788422130475671 0581664536827945414542207027177538799/13917482472821614909916752997235 66089376318996169218750000-3022858984983534314773428506933040222615091 059789769/6659082522881155459290312438868737269743153091718750000*7^(1 /2), a[16,8] = 12871682863191528707812574253462271793937198554473/8183 67733822252968284538517562950447076898208684200+7923793831170279449496 891966550656386059347318121/163673546764450593656907703512590089415379 6417368400*7^(1/2), a[16,12] = 831464376531791898078674019732785579512 9766290647999/14914139040691643843786103420465233322847554416562500000 +12743937478203669740869195012551757724982855520824/932133690043227740 23663146377907708267797215103515625*7^(1/2), a[16,11] = -2033080239583 29587199829150033803325261178399821537536/8285758985671105201066755800 819103710987390140968171875-183955945309870444037021190819013306003895 69409698304/2180462890966080316070198894952395713417734247623203125*7^ (1/2), a[16,13] = 812446331410101756322947639951373288835154/669844190 4183349335450501610911809049423828125+38691810008318009637540479349075 9651703962/6698441904183349335450501610911809049423828125*7^(1/2), a[1 6,14] = 348098152674365735187679001267334640046338304/9794208297738356 7310235712743332126911845703125+75788094093975183886441139049939045481 28984/3917683319095342692409428509733285076473828125*7^(1/2), a[19,3] \+ = 0, a[19,4] = 0, a[19,5] = 0, c[19] = 3/10, a[16,6] = -61676784494096 365881732133250119640765816127842489932383/586415159183697182569650797 10079269237067699352911722500000-2763407953457312969846077843469102719 3623514337349372/30041760204082847467707520343278314158333862373417890 625*7^(1/2), a[16,10] = 1568221980335495859574819360318483217200155100 501135104/195898096351550311454966852305863720094941335219755859375+59 1872623465406532819676818747091838359436881921536/10176524485794821374 2839923275773361088281213101171875*7^(1/2), a[16,9] = 4061154602872387 3502583016598530141206712351359709583621527782054419979833936077969899 3414799/52839186803418110730900174275377898070445720341791628134749279 5784532666801613048775622343750000-43137915589508439648993275408447215 6781047031795210079948966119713148914776761016489118643/17324323542104 2986002951391066812780558838427350136485687702555994928743213643622549 384375000*7^(1/2), a[16,15] = 2901044122821490120249433136363475030176 169984/28990856561305535923829770972026309565906328125+207977728153842 6380995265453726087303397376/11596342624522214369531908388810523826362 53125*7^(1/2), a[16,7] = -53623120676210693478915318396423432061873602 463681326025000/749594156753291060720927544928386880503908705186167669 3501311-2440862180486837508553671181096852807790831050835529920000/107 0848795361844372458467921326266972148441007408810956214473*7^(1/2), a[ 17,10] = -1489758732528250543113521169/6777268656058713348166400000+18 2993414470798047123969/3520659042108422518528000*7^(1/2), a[17,1] = 12 04472757949318149125704321217/20800267292079442987248844800000-2198100 637293808474641843/416872440517866021068800000*7^(1/2), a[17,12] = -19 06612263858924745067238619/1287855848637222919589068800000-12187698363 85461326457543/12265293796544980186562560000*7^(1/2), a[17,14] = 27217 572159567342015099/893668418551873945600000+989931556574911765269/3574 6736742074957824000*7^(1/2), a[17,11] = 10565149828794636491259146771/ 55897372479169766976172492800-78967956837666628354617891/2661779641865 226998865356800*7^(1/2), a[17,7] = 10629566348189721232485330989365234 375/350093822003577650477950866758750994432-96776099366184716772941323 06640625/617449421523064639290918636258820096*7^(1/2), a[17,6] = 48686 024023005238414132359876273/1687924437393983211523871237734400-9094708 6755739272427273888113/6517082769860939040632707481600*7^(1/2), a[17,8 ] = -139813563097406086392326875/1528856569422457169284694016+78029575 98378772386375/680171980612816002351104*7^(1/2), a[17,9] = 13898228065 594553414996146308273231365548079803650544776770188499895421/102661524 776449473471459993852807430051568429917145049778991367782400000-181751 026713338360067888779223183710172814810723200929646376184583/482829040 6887688346688300710302524634994399996103235733285896192000*7^(1/2), a[ 17,16] = 13295441661397228125/46517010473975545856-682342228168453125/ 46517010473975545856*7^(1/2), a[17,13] = -30165509761493343939/1696696 9514364108800000+159134879536428159/458566743631462400000*7^(1/2), a[1 7,15] = -1204548165456597/10186357540916840+350890378839/1376534802826 6*7^(1/2), a[19,7] = 7475544817818255952599589560546875/37037920622123 2865902780051095683832-89837679811524715021428060546875/75587593106374 05426587347981544568*7^(1/2), a[19,9] = 186742589470370201052578395915 7970802614636176859438775066185829501/16709232548250239822828775041147 042651622465806826993779132709600000-389354491375761447942130557969761 865140472157707449139336703651/136401898353063182227173673805281980829 57114944348566350312416000*7^(1/2), a[19,10] = -7338365977866748561867 86/3910888340232744741040625+9679407044454704106/245581685414929026125 *7^(1/2), a[19,14] = 15116525284746085123/654542298744048300000+549803 829577523213/26181691949761932000*7^(1/2), a[19,1] = 54669805406555815 509959179/1000249596047262513443400000-20405065149351317359441/5103314 265547257721650000*7^(1/2), a[19,8] = -549387050628922094174375/646977 0638832012760644864+4329396153890301811625/497674664525539443126528*7^ (1/2), a[19,6] = 8161475216083083924801944517/412090927098140432500945 126400-88996681564173938312881497/8410018920370212908182553600*7^(1/2) , a[19,13] = -19472213594618603/12426979624778400000+297287747500681/1 129725420434400000*7^(1/2), a[19,12] = -164244455362347392642157/10480 5977265399000617600000-32233316630530693491/427779499042444900480000*7 ^(1/2), a[19,15] = 132907117156352/6875791340118867*7^(1/2)-3531632929 957888/34378956700594335, a[19,11] = 1560146031091654580841646/9097879 635281537593778075-29238981082342572586738/1299697090754505370539725*7 ^(1/2), a[19,16] = 85069109306103125/306630684276694272-31006637035937 5/27875516752426752*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 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_19 := SimpleOrderCon ditions(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_condition'):\n interp_orde r_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_order_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]*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..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 add(a[20,i]*add(a[i,j]*ad d(a[j,k]*c[k]^3,k=2..j-1),j=2..i-1),i=2..19)=c[20]^6/120, ##27\n ad d(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 "W e specify " }{XPPEDIT 18 0 "c[20] = 7/10;" "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$\"#?\"#>\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "e21 := `union`(e20,\{c[20]=7/10,seq(a[20 ,i]=0,i=2..5),seq(a[20,i]=0,i=17..19)\}):\neqs_20 := expand(subs(e21,i nterp_order_eqns20)):\nnops(eqs_20);\nindets(eqs_20);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<.&%\"aG6$\"#?\"\"'&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$F '\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[so lve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "e22 := solve(\{ op(eqs_20)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "e23 := `union`(e21,e22): \n[seq(a[20,i]=subs(e23,a[20,i]),i=1..19)]:\nevalf[20](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#75/&%\"aG6$\"#?\"\"\"$\"5ehW?qV'>]Q&!#@/&F&6 $F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F&6$F( \"\"'$\"5?\\ma09g$G8%!#?/&F&6$F(\"\"($\"5!f*z?\"z$H'[<&FE/&F&6$F(\"\") $\"5],%RO9!#>/&F&6$F(\"\"*$!5'z#*['zt,n[RFE/&F&6$F(\"#5$\"5uv\"e .*=e`29FR/&F&6$F(\"#6$!5ZTC*)*4:D&pCFR/&F&6$F(\"#7$\"59`1\"y>#=j9ZF,/& F&6$F(\"#8$\"5A**p')o*fh2c\"F,/&F&6$F(\"#9$!5:]!*)Gd`z#f6FE/&F&6$F(\"# :$\"5C8t6#R^Ivs\"F,/&F&6$F(\"#;$!5VFTvn)R!f#G#FE/&F&6$F(\"#F1" }}}{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 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],'interpolation_order_condition'):\n if expand(subs(e23,lhs(tt)= rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop(recd);\nnop s(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 15535 "e23 := \{c[8] = 24/25, c[9] = 14435868/16178861, a[14,10] = 373 02437685024/53021770400599+15742143124443936/12990333748146755*7^(1/2) , a[9,3] = 0, a[17,4] = 0, a[17,3] = 0, a[11,4] = -9623541317323077848 129/3864449564977792573440, a[14,9] = -3211799428826566849882311586102 611111468427562764815533/154455682095190808523075696640755012970347465 57307574592-35609926058546725508440519811766382833790503793377686129/6 3599398509784450568325286852075593576025427000678248320*7^(1/2), a[11, 9] = 48175771419260955335244683805171548038966866545122229/19897864205 13815146528880165952064118903852843612160000, a[9,4] = -14764960804048 657303638372252908780219281424435/298169210256502197561171126920960636 3661854518, a[6,5] = 960400/5209191, a[14,12] = 1100613127343/48439922 837376-29746300739/1424703612864*7^(1/2), a[14,11] = -5071676622092000 /4204918450412337-380378998046332000/206041004070204513*7^(1/2), a[11, 10] = -2378292068163246/47768728487211875, a[14,2] = 0, a[4,1] = 14/35 5, a[9,6] = 7632051964154290925661849798370645637589377834346780/17340 87257418811583049800347581865260479233950396659, a[12,4] = 26290092604 284231996745/5760876126062860430544, a[9,2] = 0, a[8,2] = 0, a[10,9] = -52956818288156668227044990077324877908565/29127799594774339863498222 24412353951940608, a[7,6] = 124391943/6756250000, a[9,8] = 13660426834 89166351293315549358278750/144631418224267718165055326464180836641, a[ 11,2] = 0, a[11,3] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a [12,9] = -23028251632873523818545414856857015616678575554130463402/200 13169183191444503443905240405603349978424504151629055, a[12,2] = 0, a[ 7,2] = 0, a[10,7] = 187023075231349900768014890274453125/2522469884980 8178010752575653374848, a[12,5] = -697069297560926452045586710000/4110 7967755245430594036502319, a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[12, 8] = 162259938151380266113750/59091082835244183497007, a[12,10] = 7958 341351371843889152/3284467988443203581305, a[12,11] = -507974327957860 843878400/121555654819179042718967, a[11,8] = 397472627823804669336622 25/1756032802431424164410720256, a[11,5] = -4823348333146829406881375/ 576413233634141239944816, a[12,1] = -321802217475859983165904553557857 1/1453396753634469525663775847094384, a[12,6] = 1827357820434213461438 077550902273440/139381013914245317709567680839641697, a[10,8] = 190815 8550070998850625/117087067039189929394176, a[11,6] = 65661192465149968 84067001154977284529/970305487021846325473990863582315520, a[12,7] = 6 43504802814241550941949227194107500000/2421246091188365508604940075453 33945331, a[11,1] = -10116106591826909534781157993685116703/9562819945 036894030442231411871744000, a[12,3] = 0, a[2,1] = 1/16, a[10,6] = 100 1587844183325981198091450220795/184232684207722503701669953872896, a[1 0,3] = 0, a[7,4] = 3064329829899/27126050000000, a[7,5] = -21643947/59 2609375, a[14,3] = 0, a[14,5] = 0, c[10] = 11/12, c[4] = 56/355, c[5] \+ = 39/100, c[6] = 7/15, c[3] = 112/1065, a[13,5] = 0, a[8,3] = 0, a[8,7 ] = 1993321838240/380523459069, a[14,6] = 1533012537239841375/79359550 98274910336-11049456415304617875/194430899907735303232*7^(1/2), a[14,8 ] = 13061564753515625/18689009843335104+31047286856168359375/256413215 05055762688*7^(1/2), c[11] = 19/20, c[12] = 1, c[13] = 1, c[2] = 1/16, a[10,5] = -3378604805394255292453489375/517042670569824692230499952, \+ a[13,9] = 405932030463777247926705030596175437402459637909765779/78803 919436321841083201886041201537229769115088303952, a[13,10] = -10290327 637248/1082076946951, a[7,1] = 21400899/350000000, a[14,1] = 923507123 432989/20341288146475296+3728619917660047/7973784953418316032*7^(1/2), a[14,7] = 129834778101757666015625000/534951006554143882189042101-827 66411529674389648437500/26212599321153050227263062949*7^(1/2), a[6,3] \+ = 0, a[6,4] = 2505377/10685520, a[9,7] = 75198347919711375170485321796 52347729899303513750000/1045677303502317596597890707812349832637339039 997351, a[5,3] = -352806597/250880000, a[5,2] = 0, a[5,1] = 94495479/2 50880000, a[4,3] = 42/355, a[9,1] = -184091125228237658443815733646470 8426954728061551/2991923615171151921596253813483118262195533733898, a[ 13,4] = 0, a[9,5] = -8753250485021304411186134217852667428626944045205 60000/170212030428894418395571677575961339495435011888324169, a[8,1] = -15365458811/13609565775, a[7,3] = 0, a[13,1] = 4631674879841/1037820 82379976, a[13,2] = 0, c[7] = 39/250, a[10,4] = -319483465108209702472 15/6956009216960026632192, a[8,6] = 341936800488/47951126225, c[15] = \+ 3/40, a[8,5] = -8339128164608/939060038475, a[8,4] = -7/5, a[11,7] = 2 226455130519213549256016892506730559375/364880443159675255577435648380 047355776, a[6,2] = 0, a[5,4] = 178077159/125440000, a[13,12] = -29746 300739/247142463456, a[13,11] = 863264105888000/85814662253313, a[6,1] = 12089/252720, a[13,7] = 2720762324010009765625000/10917367480696813 922225349, a[13,8] = -498533005859375/95352091037424, a[13,6] = 143272 19974204125/40489566827933216, a[3,2] = 100352/1134225, a[13,3] = 0, a [3,1] = 18928/1134225, a[10,1] = -63077736705254280154824845013881/783 69357853786633855112190394368, a[10,2] = 0, c[16] = 3/25, a[17,5] = 0, a[16,3] = 0, a[16,4] = 0, a[20,5] = 0, a[20,17] = 0, a[20,8] = 285378 44110255116480186875/19409311916496038281934592-6381054972281168645375 /497674664525539443126528*7^(1/2), a[20,1] = 2577220290983004472718148 7/673637483052238019257800000+2313448267872880980787/39256263581132751 7050000*7^(1/2), a[20,9] = -517827546394048821394457686498439852369824 124297313387912381872377/102301423764797386670380255353961485622178362 0826142476273431200000+57386580596021382751753630948493217104662642122 4269648745064341/13640189835306318222717367380528198082957114944348566 350312416000*7^(1/2), a[20,11] = -9970960321488888197319914/3899091272 263516111619175+43095050438446097412958/1299697090754505370539725*7^(1 /2), a[20,10] = 1370691905165773958915514/877954525358371268396875-142 66384099373886246/245581685414929026125*7^(1/2), a[20,12] = 3006379957 62234907715489/6416692485636673507200000+47508372541408403781/42777949 9042444900480000*7^(1/2), a[20,14] = -22280099903809149493/65454229874 4048300000-810350528295354683/26181691949761932000*7^(1/2), a[20,7] = \+ 10683684211320620397871859541015625/2267627793191221627976204394463370 4+10185452642688448704857556640625/581443023895185032814411383195736*7 ^(1/2), a[20,6] = 3128676148761100227755815437/84100189203702129081825 53600+131171345200484783698014927/8410018920370212908182553600*7^(1/2) , a[20,13] = 206707689553129373/12426979624778400000-4819864782877781/ 12426979624778400000*7^(1/2), a[20,15] = 3185294836283392/343789567005 94335-195890510047232/6875791340118867*7^(1/2), c[18] = 2/5, a[18,2] = 0, a[19,2] = 0, a[20,16] = -83291528425371875/306630684276694272+5027 041953171875/306630684276694272*7^(1/2), c[20] = 7/10, a[16,2] = 0, a[ 19,17] = 0, a[19,18] = 0, a[15,1] = 53968119671846752033/1211086636548 096000000-18013979306423851/41103546452541440000*7^(1/2), a[15,9] = 14 37829216002805549145888241731712075764900749803936348249893/1494355805 607288245725902431596118039320066182415245312000000-436781032592564278 6942443213218202333579800829827668077/26673124624013005776924627213516 128165085289185280000000*7^(1/2), a[15,10] = -43024834368326969649/173 13231151216000000+24278487789419081919/55087553662960000000*7^(1/2), a [15,13] = 609279/512000000-2221887/4096000000*7^(1/2), a[15,6] = -3197 585392137639804267/331690531454428905472000+37078101149895800937/39220 16419224666112000*7^(1/2), a[15,14] = -15553209/1024000000*7^(1/2), a[ 15,7] = 12599626362153730936328125/254800781771704557409886208+1013426 797972694921875/352212771011439949026304*7^(1/2), a[18,17] = 0, a[15,8 ] = -8532637324759244825/3240219442044469248+132594730877413325/272450 884015751168*7^(1/2), a[15,11] = 15187119353760445983/3661425589474688 000-19473860153393911521/25629979126322816000*7^(1/2), a[18,14] = 1083 595819903600472/20454446835751509375+39411512915496232/818177873430060 375*7^(1/2), a[18,16] = 259448245437500/1197776110455837-3056136232812 5/1197776110455837*7^(1/2), a[18,13] = -56700339581774/485428891592906 25+29301850944203/48542889159290625*7^(1/2), a[18,10] = -8442401866475 056490423808/43019771742560192151446875+1087953059968188774912/1203350 2585331522280125*7^(1/2), a[18,6] = 56575761137587272324976173/8048650 91988555532228408450-39074664028872736786931649/1609730183977111064456 816900*7^(1/2), a[18,12] = 911312938075044590941/614097523039447269243 750-14152280688871027947/81879669738592969232500*7^(1/2), a[18,9] = 61 100655693638802716486435678887510402373992546715729966430155473/391622 637849614995847549415026883812147401542347507666698422881250-170949024 966397582880137105076434230151670370842813631056736667/261081758566409 9972316996100179225414316010282316717777989485875*7^(1/2), a[18,8] = - 2770031482617170508125/151635249347625299077614+38792939239451158625/1 944041658302888449713*7^(1/2), a[18,7] = 11129944106509439530972892187 500000/138892202332962324713542519160881437-12622051408017950102211274 37500000/46297400777654108237847506386960479*7^(1/2), a[18,11] = 20454 60261009505062889984/27293638905844612781334225-4694891918893904533703 68/9097879635281537593778075*7^(1/2), a[18,1] = 2913118083054645064473 1399/515753697961869733494253125-143343963156330219505552/156288999382 38476772553125*7^(1/2), a[18,15] = -3206129319411712/34378956700594335 +304869207965696/6875791340118867*7^(1/2), 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, c[17] = 13/40, a[17,2] = 0, a[20,2] = 0, a[15,12] = -36585897414219009/16871592171929600000, a[20,18] = 0, \+ a[20,19] = 0, a[16,1] = 3331788422130475671058166453682794541454220702 7177538799/1391748247282161490991675299723566089376318996169218750000- 3022858984983534314773428506933040222615091059789769/66590825228811554 59290312438868737269743153091718750000*7^(1/2), a[16,8] = 128716828631 91528707812574253462271793937198554473/8183677338222529682845385175629 50447076898208684200+7923793831170279449496891966550656386059347318121 /1636735467644505936569077035125900894153796417368400*7^(1/2), a[16,12 ] = 8314643765317918980786740197327855795129766290647999/1491413904069 1643843786103420465233322847554416562500000+12743937478203669740869195 012551757724982855520824/932133690043227740236631463779077082677972151 03515625*7^(1/2), a[16,11] = -2033080239583295871998291500338033252611 78399821537536/8285758985671105201066755800819103710987390140968171875 -18395594530987044403702119081901330600389569409698304/218046289096608 0316070198894952395713417734247623203125*7^(1/2), a[16,13] = 812446331 410101756322947639951373288835154/669844190418334933545050161091180904 9423828125+386918100083180096375404793490759651703962/6698441904183349 335450501610911809049423828125*7^(1/2), a[16,14] = 3480981526743657351 87679001267334640046338304/9794208297738356731023571274333212691184570 3125+7578809409397518388644113904993904548128984/391768331909534269240 9428509733285076473828125*7^(1/2), a[19,3] = 0, a[19,4] = 0, a[19,5] = 0, c[19] = 3/10, a[16,6] = -61676784494096365881732133250119640765816 127842489932383/586415159183697182569650797100792692370676993529117225 00000-27634079534573129698460778434691027193623514337349372/3004176020 4082847467707520343278314158333862373417890625*7^(1/2), a[16,10] = 156 8221980335495859574819360318483217200155100501135104/19589809635155031 1454966852305863720094941335219755859375+59187262346540653281967681874 7091838359436881921536/10176524485794821374283992327577336108828121310 1171875*7^(1/2), a[16,9] = 4061154602872387350258301659853014120671235 13597095836215277820544199798339360779698993414799/5283918680341811073 0900174275377898070445720341791628134749279578453266680161304877562234 3750000-43137915589508439648993275408447215678104703179521007994896611 9713148914776761016489118643/17324323542104298600295139106681278055883 8427350136485687702555994928743213643622549384375000*7^(1/2), a[16,15] = 2901044122821490120249433136363475030176169984/28990856561305535923 829770972026309565906328125+207977728153842638099526545372608730339737 6/1159634262452221436953190838881052382636253125*7^(1/2), a[16,7] = -5 3623120676210693478915318396423432061873602463681326025000/74959415675 32910607209275449283868805039087051861676693501311-2440862180486837508 553671181096852807790831050835529920000/107084879536184437245846792132 6266972148441007408810956214473*7^(1/2), a[17,10] = -14897587325282505 43113521169/6777268656058713348166400000+182993414470798047123969/3520 659042108422518528000*7^(1/2), a[17,1] = 12044727579493181491257043212 17/20800267292079442987248844800000-2198100637293808474641843/41687244 0517866021068800000*7^(1/2), a[17,12] = -1906612263858924745067238619/ 1287855848637222919589068800000-1218769836385461326457543/122652937965 44980186562560000*7^(1/2), a[17,14] = 27217572159567342015099/89366841 8551873945600000+989931556574911765269/35746736742074957824000*7^(1/2) , a[17,11] = 10565149828794636491259146771/558973724791697669761724928 00-78967956837666628354617891/2661779641865226998865356800*7^(1/2), a[ 17,7] = 10629566348189721232485330989365234375/35009382200357765047795 0866758750994432-9677609936618471677294132306640625/617449421523064639 290918636258820096*7^(1/2), a[17,6] = 48686024023005238414132359876273 /1687924437393983211523871237734400-90947086755739272427273888113/6517 082769860939040632707481600*7^(1/2), a[17,8] = -1398135630974060863923 26875/1528856569422457169284694016+7802957598378772386375/680171980612 816002351104*7^(1/2), a[17,9] = 13898228065594553414996146308273231365 548079803650544776770188499895421/102661524776449473471459993852807430 051568429917145049778991367782400000-181751026713338360067888779223183 710172814810723200929646376184583/482829040688768834668830071030252463 4994399996103235733285896192000*7^(1/2), a[17,16] = 132954416613972281 25/46517010473975545856-682342228168453125/46517010473975545856*7^(1/2 ), a[17,13] = -30165509761493343939/16966969514364108800000+1591348795 36428159/458566743631462400000*7^(1/2), a[17,15] = -1204548165456597/1 0186357540916840+350890378839/13765348028266*7^(1/2), a[19,7] = 747554 4817818255952599589560546875/370379206221232865902780051095683832-8983 7679811524715021428060546875/7558759310637405426587347981544568*7^(1/2 ), a[19,9] = 186742589470370201052578395915797080261463617685943877506 6185829501/16709232548250239822828775041147042651622465806826993779132 709600000-389354491375761447942130557969761865140472157707449139336703 651/13640189835306318222717367380528198082957114944348566350312416000* 7^(1/2), a[19,10] = -733836597786674856186786/391088834023274474104062 5+9679407044454704106/245581685414929026125*7^(1/2), a[19,14] = 151165 25284746085123/654542298744048300000+549803829577523213/26181691949761 932000*7^(1/2), a[19,1] = 54669805406555815509959179/10002495960472625 13443400000-20405065149351317359441/5103314265547257721650000*7^(1/2), a[19,8] = -549387050628922094174375/6469770638832012760644864+4329396 153890301811625/497674664525539443126528*7^(1/2), a[19,6] = 8161475216 083083924801944517/412090927098140432500945126400-88996681564173938312 881497/8410018920370212908182553600*7^(1/2), a[19,13] = -1947221359461 8603/12426979624778400000+297287747500681/1129725420434400000*7^(1/2), a[19,12] = -164244455362347392642157/104805977265399000617600000-3223 3316630530693491/427779499042444900480000*7^(1/2), a[19,15] = 13290711 7156352/6875791340118867*7^(1/2)-3531632929957888/34378956700594335, a [19,11] = 1560146031091654580841646/9097879635281537593778075-29238981 082342572586738/1299697090754505370539725*7^(1/2), a[19,16] = 85069109 306103125/306630684276694272-310066370359375/27875516752426752*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 interpolation coefficients" }}{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 t o be satisfied by the \"d\" coefficients of the weight polynomials for a given stage (corresponding to an \"approximate\" interpolation sche me)." }}{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]:\nordeqns := []:\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(ordeqn s) do\n eqns := [op(eqns),expand(subs(e23,ordeqns[ct]))];\nend do:\n nops(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 equat ions. " }}{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 3 "dd " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37233 "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[19,6] = -5099139251805566854 03243049723802960846039798301015253619510340070087500/1576428865474945 3392868088032626473294716734693025959018163727466171+36416064331916719 5012810735331892208467000617042115329794290741575000/22520412363927790 56124012576089496184959533527575137002594818209453*7^(1/2), d[14,4] = \+ 7408796283045694306410397470048681014672681412521125135646448795379876 /121835430888849346936309080366441743606310763841814911840379665131407 3+60762219836580780920113047520614263606058041362497437616647661031265 24/1218354308888493469363090803664417436063107638418149118403796651314 073*7^(1/2), d[20,7] = -3469104744662012692142612460845000024443119693 119638369589638456150000/525476288491648446428936267754215776490557823 1008653006054575822057+16912163618512752757691188131716413718508067126 312923646892737450000/525476288491648446428936267754215776490557823100 8653006054575822057*7^(1/2), d[12,2] = 6613050907348868412707820628904 03445628082324744813509222072805967406213/1298430939238840493314882354 661316709771126535458938871060868108639878984-133094086031670304464782 90092404437718941128474129897768024725744010607/2596861878477680986629 764709322633419542253070917877742121736217279757968*7^(1/2), d[9,2] = \+ -902448074446078309442732091516821092268874228983186950696478185131860 9947081887822066370347859939623672563906675893/41401807564171213800910 9431860455802636898242931929601393311863989528535407956588523392454658 793583381057703478028+181626458562365911529539700610044543295598811095 695543607663001501051132398414948188212013405950150907492266416127/828 0361512834242760182188637209116052737964858638592027866237279790570708 15913177046784909317587166762115406956056*7^(1/2), d[1,6] = -686328002 16021672512749379562594026363792946379911701317408599854591400125/1715 8145287870448265001275177683070531639133765759214421558739165487148441 *7^(1/2)-5903328117765176767010978029089036533740897946503245658325642 265376667526822275/171753034331583187132662764528607536021707728995249 73635980297904652635589441, d[8,8] = 142132355004477425720175789100828 9382287782069757441748781634735576950683593750/82433834297861453355596 3691714326173991107646431503959925911604098826826107-79942046411322154 78393839496334125742651723598062930975797065315272216796875/2985441566 463090472878354991614046143643470935724906233245193377006562018874*7^( 1/2), d[10,7] = -34056085927085457843504690933339069833806307467101479 164777918027929994730496000/334277353289938770001778739091145488938377 7839505661753118450278433144976307+84247367677949476088464358666617265 6540299028482715740618317834465216768000/43412643284407632467763472609 239673888101010902670931858681172447183700991*7^(1/2), d[15,4] = 18801 71234541052193892003737727781655491481836831940832586545946624000000/4 5079109428874258366434359735583445134334982621471517380940476098620701 +150884742814730063539208210877344376980643719842352058981002362880000 00/1218354308888493469363090803664417436063107638418149118403796651314 073*7^(1/2), d[13,3] = -8117813850409698519451278851663930768957649704 6518215860691542694024109/19248196447449082592691935487836923892849133 060184695961177911236194791+709777584484724637375146468338854710232746 252951093504726875327007/270244948367133486734881509130739542195144023 30901644031137818513436*7^(1/2), d[14,8] = 277418039285217992712835456 8195155564887907520164855664145158728210000/12183543088884934693630908 03664417436063107638418149118403796651314073+2275205748099209159198801 748365943419270652578815001489555641065790000/121835430888849346936309 0803664417436063107638418149118403796651314073*7^(1/2), d[12,7] = -922 9321422855407030562516932164067886881661000752317997779792964310093271 375/715760055255410821939828898007050836261333502671740052672303544887 73328993+5909287684289989383798781112893327963627324510234658103409690 90457718875/2405916152119027972906987892460675079870028580409210261083 373260126834588*7^(1/2), d[19,2] = -5141727587949870460515375074094020 1155476439440081271114580632173000/25022680404364211734711250845438846 4995503725286126333621646467717+51864811762799559164877271671716824891 7051095645439658289278833500/25022680404364211734711250845438846499550 3725286126333621646467717*7^(1/2), d[20,2] = 1752946602184140642563771 862095248662860356224274118000141723011400/583862542768498273809929186 393573084989508692334294778450508424673-560162309163143808032749803317 1665021924765064524089718297822900/83408934681214039115704169484796154 998501241762042111207215489239*7^(1/2), d[15,8] = 56497908581323904574 92635700515467944395409342191600999444480000000000/1218354308888493469 363090803664417436063107638418149118403796651314073*7^(1/2)+7040191111 75268717534932230006377309033514430761903910070743040000000000/4507910 9428874258366434359735583445134334982621471517380940476098620701, d[12 ,4] = 1215266613496360933964347156790227268842134821527084481321950369 0759850401013/32720459668818780431535035337465181086232388693565259550 7338763377249503968-38476264368964968805560252770565072980662958360503 07984759738850291890167/2337175690629912887966788238390370077588027763 8260899679095625955517821712*7^(1/2), d[17,4] = 1164814440604816072895 331764271503128104872521702292846814934538466099200/878296082193183831 88836490467490351213421807575430343101197910168667-6868325160008912261 494693911012589565049932528664537266243896062771200/878296082193183831 88836490467490351213421807575430343101197910168667*7^(1/2), d[1,3] = 9 8015304108956167635553076862919642320828872767933808463142750861855889 7811172/17175303433158318713266276452860753602170772899524973635980297 904652635589441+192931688848163422064741452517950348120145548012868666 916234183477113213215/446111777484631654890033154619759833822617477909 739574960527218302665859466*7^(1/2), d[19,7] = 32163916820959686777539 9139665346050163858290439599839355336233198250000/15764288654749453392 868088032626473294716734693025959018163727466171-223696945440751218034 581866763240795245669141839282377901003915250000/225204123639277905612 4012576089496184959533527575137002594818209453*7^(1/2), d[12,3] = -551 3080653680945400671338331885079078677288124022747569246801137789088443 47/8180114917204695107883758834366295271558097173391314887683469084431 2375992+70725414170906642982948574454592884872641928793454859963196521 39700891171/1636022983440939021576751766873259054311619434678262977536 69381688624751984*7^(1/2), d[13,7] = -16158197214559374534407637428478 88476743783406455179606817380293723710000/1924819644744908259269193548 7836923892849133060184695961177911236194791+10081631870485733300164041 86196894351222548733108214542411355342375/6756123709178337168372037728 268488554878600582725411007784454628359*7^(1/2), d[16,4] = 18989053096 013528681288086717742824162415325204157313237136237792968750/247724536 00320569617364138336984458034554868803326507028543000303983-1492941024 48920629310267126763454066240717845071784859435802001953125/9008164945 571116224496050304357984739838134110300548010379272837812*7^(1/2), d[2 0,3] = -20416682045296556746036536726304333232635367045131497507634411 6875800/52547628849164844642893626775421577649055782310086530060545758 22057+2976669550070586611573699544020730516840700340550363347243520913 700/525476288491648446428936267754215776490557823100865300605457582205 7*7^(1/2), d[8,7] = -3093575494546088301222579677654348820472745492324 98955047310280884427935058593750/5523066897956717374824956734485985365 7404212310910765315036077474621397349169+31852338532705367355064473123 614010011621215379703524573534798363358642578125/298544156646309047287 8354991614046143643470935724906233245193377006562018874*7^(1/2), d[17, 5] = 14284773814649096068127661436676990475766429004191858948112254743 347200/878296082193183831888364904674903512134218075754303431011979101 68667*7^(1/2)-27673307529644483916839577694240990113991139390040877161 89060644628070400/8782960821931838318883649046749035121342180757543034 3101197910168667, d[1,2] = -808593402365133546450572161198817864985831 92039803960911417945806609253663737/7633468192514808317007233979049223 823187010177566654949324576846512282484196-195497637060025256117862699 941600460943029557474980519944371434271498545/381292117508232183666695 0039485126784808696392390936538124164258997144098*7^(1/2), d[15,2] = 6 503741362327325693514817822275293878921942613966692501486690304000000/ 5008789936541584262937151081731493903814998069052390820104497344291189 +52192870774763167012978480095237129754473425595550397746536448000000/ 135372700987610385484787867073824159562567515379794346489310739034897* 7^(1/2), d[6,8] = -930868719446585279356580156097361846744890987978857 8029468256078519053597890625/79771155395449081884291378407137874634221 221696659311898736845403069318893109+109401676964166530867536242641451 12997525774967515052378815053831415345078125/6036736083979930520973401 6091888121344816059662336776572017072196917322405596*7^(1/2), d[6,3] = 885119799756668662383781400633181468864222723561192725455206063838958 27826000375/4467184702145148585520317190799720979516388415012921466329 263342571881858014104-105823832839061371843896614499898960766036580154 9350854247491823877298115375/83265325296274903737564160126742236337677 32367218865734071320303023078952496*7^(1/2), d[10,5] = -40138444971875 12011429456845977150770131602282196775219367078287009030286702592/4775 3907612848395714539819870163641276911111992938025044549289691902071090 1+11744262640635267958864891832764014609929341885253880415952111965279 95674112/4341264328440763246776347260923967388810101090267093185868117 2447183700991*7^(1/2), d[18,6] = -195332347043361536836388414355422420 43378577022005640232490897982450000/2252041236392779056124012576089496 184959533527575137002594818209453+102777220898214297960112538383738662 8856684797851179251962327532584375/31528577309498906785736176065252946 589433469386051918036327454932342*7^(1/2), d[14,6] = 17994248469705639 220353743162669928256465752258551209088349757672692200/121835430888849 3469363090803664417436063107638418149118403796651314073+14757734448879 135259936429969395914778314308271027965880151310742387800/121835430888 8493469363090803664417436063107638418149118403796651314073*7^(1/2), d[ 14,7] = -1105352402029656240732254197313430740040170715264067768466169 1086334000/12183543088884934693630908036644174360631076384181491184037 96651314073-9065395117249487980524568422255648774497405007157123225220 383401266000/121835430888849346936309080366441743606310763841814911840 3796651314073*7^(1/2), d[11,8] = -376911565850605033329955286270660060 364701562054326801090127358264206943360000000/113614437437730602972560 382330870896266637507611512273928282261689568245136489+585354147023909 171436178362375654058239461909330256807418482131904456160000000/113614 4374377306029725603823308708962666375076115122739282822616895682451364 89*7^(1/2), d[8,3] = -646775169039721176744965483880298308680241503447 78011253622168763840726931640625/2209226759182686949929982693794394146 29616849243643061260144309898485589396676+2242501630129852792621352584 4180542214095509749486981554285278278176408203125/11941766265852361891 513419966456184574573883742899624932980773508026248075496*7^(1/2), d[1 0,4] = -80668725242564862856089625867427362882041688521020271661420196 742074036736/620180618348680463825192465846281055544300155752441883695 4453206740528713*7^(1/2)+140134978230425739234724846994068956308407534 9536308092739857726684926046447872/47753907612848395714539819870163641 2769111119929380250445492896919020710901, d[11,3] = 639978017158755641 84493318082204630969894988233105526044940120711359268346496000/1136144 3743773060297256038233087089626663750761151227392828226168956824513648 9-14155271912924794042200420814884825802303606479581550969609086005540 682816000/391773922199071044732966835623692745747025888315559565269938 8334123042935741*7^(1/2), d[6,6] = -1080869172953431585382817721722694 339997066780183516910999044101117987345314740625/223359235107257429276 0158595399860489758194207506460733164631671285940929007052+14192306768 2828920563119847486361207233302115067967059067848319570177574053125/12 0734721679598610419468032183776242689632119324673553144034144393834644 811192*7^(1/2), d[15,3] = -4937214129877186106425896282972812395746260 78285486310990557782016000000/4507910942887425836643435973558344513433 4982621471517380940476098620701-39621406312474090909906261896526147510 76655886507730008240955392000000/1218354308888493469363090803664417436 063107638418149118403796651314073*7^(1/2), d[9,3] = 752340952744132246 4254094025712164735019510064200828012237463400589042929543446346568431 304135381000456754120388736267/260831387654278646945738942072087155661 2458930471156488777864743134029773070126507697372464350399575300663531 9115764-96515231361682557852166269977346008384133406960606569165653563 334272479147954402956334707420502543277470497375823731/521662775308557 2938914778841441743113224917860942312977555729486268059546140253015394 7449287007991506013270638231528*7^(1/2), d[16,8] = 7110339759015994052 582830817369412546214417631795473228314208984375000/247724536003205696 17364138336984458034554868803326507028543000303983-9782902956551507203 1371539930075537441356295893257008870697021484375/15764288654749453392 868088032626473294716734693025959018163727466171*7^(1/2), d[10,3] = -2 5429002288261218106476512865255073247558390008138318236637288067686005 7295872/47753907612848395714539819870163641276911111992938025044549289 6919020710901+14828178091065932763473156534744523675672025144884450576 7851130038550829568/43412643284407632467763472609239673888101010902670 931858681172447183700991*7^(1/2), d[18,4] = -2431924248455269236581234 0828996788176321551825776874651063515808628625/90081649455711162244960 50304357984739838134110300548010379272837812+1209045837149275500889081 96496320589515215615932384629964716267046125/9008164945571116224496050 304357984739838134110300548010379272837812*7^(1/2), d[20,4] = -1619377 219492862530029779306523117110075604247520921668262179354900/750680412 130926352041337525363165394986511175858379000864939403151*7^(1/2)+1100 293247233717302786271379512216276794012408909541135348264281676300/525 4762884916484464289362677542157764905578231008653006054575822057, d[14 ,3] = -159557855597710601831830580576739962243346251237933462802775276 6016116/12183543088884934693630908036644174360631076384181491184037966 51314073*7^(1/2)-19455043786457900953144638995436763781703431658673931 91276923106181484/1218354308888493469363090803664417436063107638418149 118403796651314073, d[19,3] = 3796728006908811819698460215941648395071 4543369867486036043111227817000/15764288654749453392868088032626473294 716734693025959018163727466171-393723654144638609600228236789917252466 60945656160325539794836096500/2252041236392779056124012576089496184959 533527575137002594818209453*7^(1/2), d[7,3] = 201702823667046000116925 392146511140931190938274281144650150458879263782763266601562500000/144 5406335060715594420779923131546085144151194566335438747672911332449135 2289066486389197-12937880505855154359058286195159955148039830700663482 21924574580339655098098144531250000/1445406335060715594420779923131546 0851441511945663354387476729113324491352289066486389197*7^(1/2), d[14, 2] = 21018392110145211892926119609474828570095440061764792324748391338 204/135372700987610385484787867073824159562567515379794346489310739034 897*7^(1/2)+2562792895981259523378715740835007007385427529548451697663 8855616196/13537270098761038548478786707382415956256751537979434648931 0739034897, d[18,2] = -44798064238286382839034805015821876382073189465 669087580331678954625/100090721617456846938845003381755385998201490114 4505334486585870868+41822368492606150509444732516221625881413348074518 0549171230431125/10009072161745684693884500338175538599820149011445053 34486585870868*7^(1/2), d[1,4] = -241767597058162460996419210497909997 07227066292682918692275770608396231053144901/1374024274652665497061302 11622886028817366183196199789087842383237221084715528-7347151694487370 68413339595390484221549572154003273105199543454432698772885/4461117774 84631654890033154619759833822617477909739574960527218302665859466*7^(1 /2), d[10,8] = 2139467475782193593244539190487521009673247154567673326 69807711679092495360000/6821986801835485102077117124309091610987301713 2768607206498985274145815843-21144152320329907083056197208872881219061 9636392784313929735659109297920000/43412643284407632467763472609239673 888101010902670931858681172447183700991*7^(1/2), d[15,6] = 36646405823 156847459055721098172615322484997999971968229945753600000000/121835430 8888493469363090803664417436063107638418149118403796651314073*7^(1/2)+ 4566500018005182323167077408338028624776354300352880025303958732800000 000/450791094288742583664343597355834451343349826214715173809404760986 20701, d[1,7] = 288461937437743583657119550141490719974276680565672736 45747464049451336250/1173978361801662249710613564788841667954256520815 1041446329663639543838407*7^(1/2)+182211357938702175709933501083033820 5947328754403254962573306674886788953267000/92482403101621716148356873 20771175016553493099744216573220160410197573009699, d[18,8] = -3263283 116423624144306156731971093114558211187118608527443305062812500/225204 1236392779056124012576089496184959533527575137002594818209453+79226023 672976330933651625150442821017311142441193603112616245859375/157642886 54749453392868088032626473294716734693025959018163727466171*7^(1/2), d [16,3] = -498640865982140898343434624252323632799615008859769911184222 9003906250/24772453600320569617364138336984458034554868803326507028543 000303983+274425997485621255382009037937011522451814591457808962110006 103515625/630571546189978135714723521305058931788669387721038360726549 09864684*7^(1/2), d[2,4] = 0, d[7,5] = -102471028969814112087549570679 04747006614174490462607662485698488409991200876464843750000/1445406335 0607155944207799231315460851441511945663354387476729113324491352289066 486389197*7^(1/2)+3183781100278923523951847382539205125855515087346383 093549498094643381127118955078125000000/144540633506071559442077992313 15460851441511945663354387476729113324491352289066486389197, d[12,6] = 673233038986120793413778418972925317905425430152200225278061397693243 0273325/40900574586023475539418794171831476357790485866956574438417345 422156187996-327074739507908508173265788296707860397843090480285824333 14149231272281525/8180114917204695107883758834366295271558097173391314 8876834690844312375992*7^(1/2), d[7,2] = -1861127582613759470715628350 5269784636206345033007201281002211617534500541992187500000/17648428999 5203369282146510760872537868638729495279052350143212616904656316105817 90463+1872850202325251202103205380918907033152837271591753825844324510 06208078613281250000/1764842899952033692821465107608725378686387294952 7905235014321261690465631610581790463*7^(1/2), d[18,7] = 1270714072517 1571452121209934083420611459385902722387638036232273187500/22520412363 92779056124012576089496184959533527575137002594818209453-3156704437671 69393000149235660096036906790988780351377578943962015625/1576428865474 9453392868088032626473294716734693025959018163727466171*7^(1/2), d[11, 1] = 0, d[16,5] = 2173517859103192967015253651832967928598862243264021 108157840576171875/630571546189978135714723521305058931788669387721038 36072654909864684*7^(1/2)-39493518741702005475803674323125756824705515 241816471075655378417968750/247724536003205696173641383369844580345548 68803326507028543000303983, d[15,5] = -3138107723465909110747353775326 7836153463692608127416395216150528000000/12183543088884934693630908036 64417436063107638418149118403796651314073*7^(1/2)-39103886599034627004 72982340164129774235134242286677265798761734144000000/4507910942887425 8366434359735583445134334982621471517380940476098620701, d[18,1] = 0, \+ d[10,2] = 211171836997377541588953432974924883928883383259817629904789 183855071990784/524768215525806546313624394177622431614407824098220055 4346075790318908911-19318357974368153731293811726933807350398776890244 6821098998120901073408/47706201411436958755784035834329311964946165827 1109141304188708210809901*7^(1/2), d[13,2] = 9702358177161019601573145 1753362759625793708638711372455629402956213/30552692773728702528082437 2822808315759510048574360253352030337082457-93498270742024923842682588 50702233721270562092239650150014827133/3002721648523705408165350101452 661579946044703433516003459757612604*7^(1/2), d[12,5] = -4351065014954 093377034295353840201363867678725210673643915987290515429731971/409005 74586023475539418794171831476357790485866956574438417345422156187996+5 6016176383212401377445939312074384457922165032806983192686157404956240 289/163602298344093902157675176687325905431161943467826297753669381688 624751984*7^(1/2), d[8,6] = 789813245889387687194716304042276262776073 785597983613846153229518697127490234375/110461337959134347496499134689 719707314808424621821530630072154949242794698338-797739175371709413542 6135566958620200139372147017459897506207988825537109375/45929870253278 3149673593075632930175945149374726908651268491288770240310596*7^(1/2), d[7,7] = 270132843462431691166501782884125155481715090620566908835133 43071521131021391601562500000000/1011784434542500916094545946192082259 60090583619643480712337103793271439466023465404724379-7350750505261837 5368604493267420767478506678809108218387670768569229437707519531250000 00/1445406335060715594420779923131546085144151194566335438747672911332 4491352289066486389197*7^(1/2), d[17,2] = 2305077633563301546803694136 021241260570026324327439248655518701977600/975884535770204257653738782 9721150134824645286158927011244212240963-18275671922619105182906957485 17878116392646969950057249714267750400/7506804121309263520413375253631 65394986511175858379000864939403151*7^(1/2), d[9,7] = 2518953343378643 4106255249633181076867266871879228914700719195636847523386134189797772 5629468755249478081861864619682750/45645492839498763215504314862615252 2407180312832452385536126330048455210287272138847040181261319925677616 11808452587-1321982344730674348399193346456805977097907374662203236494 63569894267986345766319934723730260230342753691122506375/1257624819933 8411135281530475992630456183504968520523089575046977502554354243618648 492634832933459861623257145186*7^(1/2), d[11,2] = -5373665344366965136 46358288643959400800576485368292469427826726167820265088000/1262382638 1970066996951153592318988474070834179056919325364695743285360570721+54 0751226377072169252730029323450529595451950293726824600426708789875961 6000/12623826381970066996951153592318988474070834179056919325364695743 285360570721*7^(1/2), d[11,6] = -1563025728957657097579337657954854256 115104197944139324142177929754642161427200000/113614437437730602972560 382330870896266637507611512273928282261689568245136489+379679994547566 9217977406947495018473736080241774935100559569774038875811200000/11361 4437437730602972560382330870896266637507611512273928282261689568245136 489*7^(1/2), d[16,2] = 65685448103411251832171663781409404233805263145 739293835358886718750/275249484448006329081823759299827311495054097814 7389669838111144887-51642648893391414364144037984523596144978053548990 5148546142578125/10009072161745684693884500338175538599820149011445053 34486585870868*7^(1/2), d[11,5] = 101017421500625211398759090633558164 7427681464418487422500687750969913745075456000/11361443743773060297256 0382330870896266637507611512273928282261689568245136489-32512785266988 45887594133922085215242980075442942848177106696150597536598976000/1136 1443743773060297256038233087089626663750761151227392828226168956824513 6489*7^(1/2), d[6,4] = -1951098866784535430261436366351454780080971668 754853743185361821808342775901829625/178687388085805943420812687631988 83918065553660051685865317053370287527432056416+1669547908357196053661 3070219196566270422739905653146942921539695588415382375/34495634765599 602976990866338221783625609176949906729469724041255381327088912*7^(1/2 ), d[19,5] = 410057190208357565703511334931994575276920269533176347643 951103754444000/157642886547494533928680880326264732947167346930259590 18163727466171-3118383103917038083808290846282094353660476784628101904 75810269493500/2252041236392779056124012576089496184959533527575137002 594818209453*7^(1/2), d[13,6] = 18460045526316302061960088365035988144 2382300071728307974083408971036000/17498360406771893266083577716215385 35713557550925881451016173748744981-3282417236792801573663215006024135 068368078806229779285021568438425/135122474183566743367440754565369771 09757201165450822015568909256718*7^(1/2), d[17,8] = 433559334076191257 38966512963250110580873371765022438664279879680000000/6756123709178337 168372037728268488554878600582725411007784454628359-257180414505838005 4908221535691777827422208100854591092725399552000000/87829608219318383 188836490467490351213421807575430343101197910168667*7^(1/2), d[7,8] = \+ 1844869372609236492908436376223903442020075758569964675059797749453583 563232421875000000/144540633506071559442077992313154608514415119456633 54387476729113324491352289066486389197*7^(1/2)-25328737501882703227852 35155620618578470601940501580447042625376505649438476562500000000/3081 8898402147454038822599640331473030792136344697983768607098322653499685 051314469913, d[7,6] = 92049468211097455995004753711976480535816641438 2138289742171685991269307495117187500000/11118510269697812264775230177 93496988572423995820258029805902239486499334791466652799169*7^(1/2)-49 2621144074072092252695279682678245282271269570374360925363790845969160 2090576171875000000/14454063350607155944207799231315460851441511945663 354387476729113324491352289066486389197, d[6,2] = -9555470700191264538 26454916873895202203877744679223408705084338612462278116625/6381692431 63592655074331027257102997073769773573274495189894763224554551144872+5 1976526550563049738919099496889712177688887113612059568640919857176855 6375/34495634765599602976990866338221783625609176949906729469724041255 381327088912*7^(1/2), d[6,5] = 698559305892442368822785152069637500969 905948840373672577583243850664955527697375/223359235107257429276015859 5399860489758194207506460733164631671285940929007052-24306333176719413 8058900642261649229464483672897314758089052013901147904456625/24146944 3359197220838936064367552485379264238649347106288068288787669289622384 *7^(1/2), d[9,5] = 593766825561784837280187624406711407946758449571601 87308205895439835379901375509676447975176725074528268184637240152531/1 3041569382713932347286947103604357783062294652355782443889323715670148 865350632538486862321751997876503317659557882-764423126679479650194111 5985159559705889245749734039641950049463307498965691218402921548914246 42359299087864124493329/5216627753085572938914778841441743113224917860 9423129775557294862680595461402530153947449287007991506013270638231528 *7^(1/2), d[8,4] = 142570790951088852051393028173745814755103667092852 2240301201176074926980724609375/88369070367307477997199307751775765851 8467396974572245040577239593942357586704-85398099970325736383267798339 936564138231250599093870734618554049173865234375/119417662658523618915 13419966456184574573883742899624932980773508026248075496*7^(1/2), d[11 ,4] = 1563261581658508453792853186794051041737599175813433634158922205 582642856896000/113614437437730602972560382330870896266637507611512273 928282261689568245136489*7^(1/2)-3526811806686325505318373294404251438 21552122218234606527970301923831373382496000/1136144374377306029725603 82330870896266637507611512273928282261689568245136489, d[13,4] = 44978 0199550533747040960823918460125437344839074607085806442449771655170/19 248196447449082592691935487836923892849133060184695961177911236194791- 2702948185282388499404658712425365974862300040850732058062959386773/27 024494836713348673488150913073954219514402330901644031137818513436*7^( 1/2), d[9,8] = -158245394456362772829759685380886018192577992234715236 8428785011100110339243480369180077441211774951537441146103396250/93154 0670193852310520496221686025555933021046596841603134951693976439204667 902324177633022982285562607379832825563+344063299720693929096526899906 7845029867974719844523686561038700669237855156918168827414979667573920 6158031620738125/13041569382713932347286947103604357783062294652355782 443889323715670148865350632538486862321751997876503317659557882*7^(1/2 ), d[9,6] = -918725514438951297560136324132244285686664732557596355041 09212049187289585415128517333378983662445427803679752792495325/1304156 9382713932347286947103604357783062294652355782443889323715670148865350 632538486862321751997876503317659557882+446341594831599017447045122823 8674319454491500163442377920188482107006563412442247657126500761960232 41091303713655525/2608313876542786469457389420720871556612458930471156 4887778647431340297730701265076973724643503995753006635319115764*7^(1/ 2), d[8,2] = 417749121938701143672284927506237945286474316518941912950 00926656740642578125/1888227999301441837546993755379824056663391873877 291121881575298277654610228-227232639910524612263506183944332601493857 68836071295813104863512935546875/1020663783406184777052429056962067057 65588749939313033615220286393386735688*7^(1/2), d[20,6] = 460407111661 6295781538387126260504437531975209210058615811924427782500/52547628849 16484464289362677542157764905578231008653006054575822057-2753164273701 6428636637521707658265156563690923705750123418761835000/52547628849164 84464289362677542157764905578231008653006054575822057*7^(1/2), d[18,3] = 2473394951482722963301678377817314545524943207420794465184769160588 375/450408247278555811224802515217899236991906705515027400518963641890 6-22224160527643584434016576786529209242780257231683040645093905072462 5/63057154618997813571472352130505893178866938772103836072654909864684 *7^(1/2), d[12,8] = 57980335878923601213722076139961113361856525644820 058802284384762211748125/146073480664369555497924264899398129849251735 2391306229943476622219863857-25212620879045726922247603626624250888815 46134867159993214919889830388125/4090057458602347553941879417183147635 7790485866956574438417345422156187996*7^(1/2), d[15,7] = -280511396309 1267854720074768011254220401887730900261423685308416000000000/45079109 428874258366434359735583445134334982621471517380940476098620701-225111 88933828612575121820264359204228194691662123582303752192000000000/1218 354308888493469363090803664417436063107638418149118403796651314073*7^( 1/2), d[16,6] = -12691026654847075364606831167252496889293262103727519 65147308349609375/3152857730949890678573617606525294658943346938605191 8036327454932342*7^(1/2)+419272794809195896014687085548381598453116691 0526273081181274414062500/22520412363927790561240125760894961849595335 27575137002594818209453, d[3,5] = 0, d[4,1] = 0, d[17,7] = -2222702614 413765624451736212569736108618072977775930081805540792729600000/878296 08219318383188836490467490351213421807575430343101197910168667+1024717 0287175988315845639821584448641799750497702327259852082380800000/87829 608219318383188836490467490351213421807575430343101197910168667*7^(1/2 ), d[19,8] = 561427901946157830371520748690161768507249609904710632249 13178750000/2252041236392779056124012576089496184959533527575137002594 818209453*7^(1/2)-1155921932742012560023169374694174233826039400891171 8306976184701250000/22520412363927790561240125760894961849595335275751 37002594818209453, d[13,5] = -1298181170124138933022791158267093641773 841073704766009094724204936408480/192481964474490825926919354878369238 92849133060184695961177911236194791+5621603893229926124693543921751357 945938733181371553595719704653813/270244948367133486734881509130739542 19514402330901644031137818513436*7^(1/2), d[8,5] = -510451594549193878 460631357327392551448087052512651918152034064915844772416015625/110461 3379591343474964991346897197073148084246218215306300721549492427946983 38+1776113555863321864349342850797093014893595844993188137346770203191 29177734375/1194176626585236189151341996645618457457388374289962493298 0773508026248075496*7^(1/2), d[17,6] = 3487755850061989916389312843156 191104251336113200697117615019136942080000/878296082193183831888364904 67490351213421807575430343101197910168667-1283197647543486574211335691 610176629645867350301017715486628577280000/675612370917833716837203772 8268488554878600582725411007784454628359*7^(1/2), d[1,8] = -1146306239 99240232848198622240973661521724872784376962684165806230239497235000/2 4536147761654741018951823504086790860243961285035676622828997006646622 27063-1375548968784611399406418684136616789076950998454437494486221887 41115581250/2230558887423158274450165773098799169113087389548697874802 63609151332929733*7^(1/2), d[20,5] = -30713451639195212798252880513209 33574653190414930956966684399994785600/5254762884916484464289362677542 157764905578231008653006054575822057+235759165932013352976376574675608 77490273431047774224162861593548300/5254762884916484464289362677542157 764905578231008653006054575822057*7^(1/2), d[7,4] = -11115505234212488 9259448757507929676741264416636574812417181135883474004692021850585937 5000/14454063350607155944207799231315460851441511945663354387476729113 324491352289066486389197+703850683029488619401418304954959267368195560 479807234733238582994060156096191406250000/206486619294387942060111417 5902208693063073135094764912496675587617784478898438069484171*7^(1/2), d[6,7] = 148175868318440749872291406928623412699192658000696927708564 3335185370545584046875/39087866143770050123302775419497558570768398631 36306283038105424750396625762341-7145953823948592410558990166468900906 37173976644562571963045983326017484375/9896288662262181181923609195391 49530242886223972734042164214298310120039436*7^(1/2), d[14,5] = -15408 848107299474308359573744910931688118979350388479664992500704312356/121 8354308888493469363090803664417436063107638418149118403796651314073-12 637354036401135419339997318285509618390756427112136409046317266066044/ 1218354308888493469363090803664417436063107638418149118403796651314073 *7^(1/2), d[10,6] = 62105547022900030066197779463146939410849213898086 69166600182285462536551270400/4775390761284839571453981987016364127691 11119929380250445492896919020710901-1054984166227502118220921241121376 33633552240418992262438591868901996108800/3339434098800587112904882508 403051837546231607897763989129320957475669307*7^(1/2), d[17,3] = -2469 09756289710321676748541180301721235505787987210822308890342208307200/8 7829608219318383188836490467490351213421807575430343101197910168667+18 03579982802176927816989581951816199920176377022329150547699322060800/8 7829608219318383188836490467490351213421807575430343101197910168667*7^ (1/2), d[19,4] = -1755958896341365686809054845076827195048355312945212 71186768235374774500/1576428865474945339286808803262647329471673469302 5959018163727466171+14993635467448394900063201041907905630039872806485 7543972801591013500/22520412363927790561240125760894961849595335275751 37002594818209453*7^(1/2), d[18,5] = 150332858855211623470138625630560 53539654983335433437999127211421585875/2252041236392779056124012576089 496184959533527575137002594818209453-176020531046591259203940211473033 2052639447358842535783959206536302875/63057154618997813571472352130505 893178866938772103836072654909864684*7^(1/2), d[9,4] = 525065225992201 8125760018026495766099428999243119917050926734057767270497849280290247 7767968701500214983198708897287/74523253615508184841639697734882044474 6416837277473282507961355181151363734321859342106418385828450085903866 2604504*7^(1/2)-165841006012769942042436765163901958690860691729964373 012303220484083527378899564945169306255460599525765839379134598693/104 3325550617114587782955768288348622644983572188462595511145897253611909 22805060307894898574015983012026541276463056, d[16,7] = 38979279464698 5533717717583668812601457440464708764983979644775390625/15764288654749 453392868088032626473294716734693025959018163727466171*7^(1/2)-2833064 1915448358102738021353817904023875160889639273815465087890625000/24772 453600320569617364138336984458034554868803326507028543000303983, d[13, 8] = 72668762426747850778929437728391535863889404814086648826675920073 550000/274974234963558322752741935540527484183559043716924228016827303 3742113-25302578084328250098620102018695846149294541365376444765894291 0625/67561237091783371683720377282684885548786005827254110077844546283 59*7^(1/2), d[20,8] = 146353692252118455617219060985342711304418993480 549071169845478750000/750680412130926352041337525363165394986511175858 379000864939403151-424456423354542772099593591388812190214801765922029 1703491071750000/52547628849164844642893626775421577649055782310086530 06054575822057*7^(1/2), d[11,7] = 857097974066628632899721751892717573 4060764262917264353497409416424208243328000000/79530106206411422080792 2676316096273866462553280585917497975831826977715955423-12275273012412 4127559012530188787312112097472189176200059244887924149856000000/59797 0723356476857750317801741425769824407934797433020675169798366148658613 1*7^(1/2), d[1,5] = 15280639412466567486301013936828044347315164836340 12493728265068587807677685/4461117774846316548900331546197598338226174 77909739574960527218302665859466*7^(1/2)+55292839151575641194434990588 65773087995365460457061457904631040838537690531792/1717530343315831871 3266276452860753602170772899524973635980297904652635589441, 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[2,7] = 0, d[3,8] = 0, 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,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-$!)%RG2\"!\"'F+F+F+F+$!)0Yd9!\"($!)/[E5F2$\")U[`@F/$!)wp@@F/$\") \"fp\"RF/$!):VVTF/7-$\")+=@eF/F+F+F+F+$\")ivZ>F/$\")Fzr8F/$!)_#z(G!\"& $\")bWNGFF$!)0kM_FF$\")#*HPbFF7-$!)@8.=FFF+F+F+F+$!)75z5FF$!)I/+wF/$\" )fV%f\"!\"%$!)7!4d\"FV$\")(4,!HFV$!)&*ynIFV7-$\")s%*4LFFF+F+F+F+$\")E) 35$FF$\")C$R=#FF$!)xt\"e%FV$\"))3T^%FV$!)ZpL$)FV$\")e`:))FV7-$!)J$Ha$F FF+F+F+F+$!)303[FF$!)0G'Q$FF$\"))zT5(FV$!)%=$**pFV$\")\\<#H\"!\"$$!)l) oO\"Fcp7-$\")pBN?FFF+F+F+F+$\")gtrPFF$\")/TcEFF$!)L'Hd&FV$\")Nq!\\&FV$ !)Hm85Fcp$\")7Fs5Fcp7-$!)r1N[F/F+F+F+F+$!)#H@;\"FF$!)Q![=)F/$\")Y6<lVF7 $!)\"))***>F:$\")GkCGF47+$!)mBDmF4$!)F&z9%F4$!)Ivh]F4$!)Ikb>F7$!)3u(*= F:$!)b!pv#!\"%$\")K?)R&F:$\")/=iBFN$!)H\\NPF77+$\"):`qOF7$\")qF5BF7$\" )wfF>F7$\")cSZuF7$\")\\!pA(F:$\")3`08!\"$$!)&yTm#FN$!)\"pi4\"F\\o$\")D #o.#F:7+$!)iva5F:$!)OR*o'F7$!)k-4SF7$!)[\"*[:F:$!)]0.:FN$!)Qw2JF\\o$\" )(\\:g'FN$\")CakDF\\o$!)\\$F\\o$\"))*3B')F:7+$!)^%H G\"F:$!)Z&*e&FN$ \")@-9?F\\o$!)tn;lF:7+$\"))[H&RF7$\")%RGj#F7$\")kxF:Q(pprint16\"" }}}{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 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 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. " }}{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 380 294 294 {PLOTDATA 2 "6%-%'CURVESG6$7do7$$\"3`*****\\n5;\"o!#@$\"3u=2cZ 13Yi!#G7$$\"3#******\\8ABO\"!#?$\"39NAQ9BguC!#F7$$\"33+++-K[V?F1$\"3nF Yq/ao9bF47$$\"3#)******pUkCFF1$\"3#p'G[(e^,r*F47$$\"3s*****\\Smp3%F1$ \"3k'4**[&H8V@!#E7$$\"3k******R&)G\\aF1$\"3\"f&ztcy?PPFD7$$\"3Y******4 G$R<)F1$\"39M9h-aJ*3)FD7$$\"3%******zqd)*3\"!#>$\"3wbHM^`A$Q\"!#D7$$\" 3*)*****>c'yM;FR$\"3arU+')ouxGFU7$$\"3')*****fT:(z@FR$\"3]RE.rFEFZFU7$ $\"3#*******zZ*z7$FR$\"3SBgt;SPr%)FU7$$\"33+++XTFwSFR$\"3QzTR#f4(\\7!# C7$$\"3&******4z_\"4iFR$\"3)RqzT-qM5#Fdo7$$\"3o******R&phN)FR$\"3!34(3 s]JcFFdo7$$\"3++++*=)H\\5!#=$\"3c$f0>tu\"zJFdo7$$\"3%******z/3uC\"Fbp$ \"3z&QKA'z]CMFdo7$$\"35+++J$RDX\"Fbp$\"3c,[68=*Hf$Fdo7$$\"37+++)R'ok;F bp$\"3$H\"y$oJSCq$Fdo7$$\"3-+++1J:w=Fbp$\"3VBv1S%*p*ppFFbp$\"3 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***H,M^\\(Fbp$\"3KGj]vP\"R(QF_v7$$\"3S+++0#=bq(Fbp$\"3],#oA3W-Y$F_v7$$ \"3S+++0FO3yFbp$\"3-1f2bMuTKF_v7$$\"3Y*****p?27\"zFbp$\"3[]y@HX*f,$F_v 7$$\"3d******pe()=!)Fbp$\"3;T]Fb)\\_x#F_v7$$\"3a+++IXaE\")Fbp$\"3'\\&* R8[yR`#F_v7$$\"3a+++]ACI#)Fbp$\"3ieZ3#F_v7$$\"3i*****HvJga)Fbp$\"3q'z-3!)y!y;F_v7$$\"3s*****HJnjv)F bp$\"3Er'G\"G?1i8F_v7$$\"3k******[Qk\\*)Fbp$\"3/4^vu!4;;\"F_v7$$\"3w** ****o0;r\"*Fbp$\"3v@wOUTR<5F_v7$$\"3[*****\\w(Gp$*Fbp$\"3W')GUiaY*H*Fd o7$$\"37+++!oK0e*Fbp$\"3D8&>>rK7i)Fdo7$$\"33+++<5s#y*Fbp$\"3:IYgl(R'Q$ )Fdo7$$\"\"\"\"\"!$\"31+^3*H[4N)Fdo-%'COLOURG6&%$RGBG$\"#5!\"\"$FablFa blF[cl-%+AXESLABELSG6$Q\"u6\"Q!F`cl-%%VIEWG6$;F[clF_bl%(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 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "abreviated calculation for stages 15 to 20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "An alternative inte rpolation scheme ( a variation of Verner's dverk78 scheme ), which app ears to have a more favourable principal error curve, was obtained by \+ trial and error." }}{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 4966 "e5 := \{a[14,2] = 0, a[9,3] = 0, a[14,12] = 1100613127343/4843 9922837376-29746300739/1424703612864*7^(1/2), a[7,6] = 124391943/67562 50000, a[9,8] = 1366042683489166351293315549358278750/1446314182242677 18165055326464180836641, a[9,2] = 0, a[8,2] = 0, a[14,4] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[4,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[10,7] \+ = 187023075231349900768014890274453125/2522469884980817801075257565337 4848, a[12,5] = -697069297560926452045586710000/4110796775524543059403 6502319, a[7,2] = 0, a[11,4] = -9623541317323077848129/386444956497779 2573440, a[11,5] = -4823348333146829406881375/576413233634141239944816 , a[12,1] = -3218022174758599831659045535578571/1453396753634469525663 775847094384, a[12,3] = 0, a[12,4] = 26290092604284231996745/576087612 6062860430544, a[12,7] = 643504802814241550941949227194107500000/24212 4609118836550860494007545333945331, a[11,1] = -10116106591826909534781 157993685116703/9562819945036894030442231411871744000, a[11,2] = 0, a[ 11,3] = 0, a[12,6] = 1827357820434213461438077550902273440/13938101391 4245317709567680839641697, a[10,8] = 1908158550070998850625/1170870670 39189929394176, a[10,9] = -52956818288156668227044990077324877908565/2 912779959477433986349822224412353951940608, a[12,8] = 1622599381513802 66113750/59091082835244183497007, a[12,9] = -2302825163287352381854541 4856857015616678575554130463402/20013169183191444503443905240405603349 978424504151629055, a[12,2] = 0, a[12,10] = 7958341351371843889152/328 4467988443203581305, a[12,11] = -507974327957860843878400/121555654819 179042718967, a[11,8] = 39747262782380466933662225/1756032802431424164 410720256, a[11,9] = 4817577141926095533524468380517154803896686654512 2229/1989786420513815146528880165952064118903852843612160000, a[11,10] = -2378292068163246/47768728487211875, a[11,6] = 65661192465149968840 67001154977284529/970305487021846325473990863582315520, a[11,7] = 2226 455130519213549256016892506730559375/364880443159675255577435648380047 355776, a[3,2] = 100352/1134225, a[13,3] = 0, a[3,1] = 18928/1134225, \+ a[13,6] = 14327219974204125/40489566827933216, a[13,7] = 2720762324010 009765625000/10917367480696813922225349, a[13,8] = -498533005859375/95 352091037424, a[10,1] = -63077736705254280154824845013881/783693578537 86633855112190394368, a[10,2] = 0, a[13,11] = 863264105888000/85814662 253313, a[6,1] = 12089/252720, a[6,2] = 0, a[5,4] = 178077159/12544000 0, a[13,12] = -29746300739/247142463456, c[11] = 19/20, c[12] = 1, c[1 3] = 1, a[2,1] = 1/16, a[10,6] = 1001587844183325981198091450220795/18 4232684207722503701669953872896, a[10,3] = 0, a[7,5] = -21643947/59260 9375, a[7,4] = 3064329829899/27126050000000, a[9,4] = -147649608040486 57303638372252908780219281424435/2981692102565021975611711269209606363 661854518, a[8,7] = 1993321838240/380523459069, a[9,1] = -184091125228 2376584438157336464708426954728061551/29919236151711519215962538134831 18262195533733898, a[13,4] = 0, a[13,5] = 0, a[9,5] = -875325048502130 441118613421785266742862694404520560000/170212030428894418395571677575 961339495435011888324169, a[8,1] = -15365458811/13609565775, a[8,3] = \+ 0, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8,6] = 34193 6800488/47951126225, a[10,4] = -31948346510820970247215/69560092169600 26632192, a[10,5] = -3378604805394255292453489375/51704267056982469223 0499952, a[13,9] = 405932030463777247926705030596175437402459637909765 779/78803919436321841083201886041201537229769115088303952, a[13,10] = \+ -10290327637248/1082076946951, a[4,3] = 42/355, a[5,1] = 94495479/2508 80000, a[4,1] = 14/355, a[9,6] = 7632051964154290925661849798370645637 589377834346780/1734087257418811583049800347581865260479233950396659, \+ a[9,7] = 7519834791971137517048532179652347729899303513750000/10456773 03502317596597890707812349832637339039997351, a[6,3] = 0, a[6,4] = 250 5377/10685520, a[6,5] = 960400/5209191, a[7,1] = 21400899/350000000, a [7,3] = 0, c[2] = 1/16, a[5,2] = 0, a[5,3] = -352806597/250880000, a[1 3,1] = 4631674879841/103782082379976, a[13,2] = 0, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, a[14,3] = 0, a[14,5 ] = 0, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[1 4,7] = 129834778101757666015625000/534951006554143882189042101-8276641 1529674389648437500/26212599321153050227263062949*7^(1/2), a[14,1] = 9 23507123432989/20341288146475296+3728619917660047/7973784953418316032* 7^(1/2), a[14,11] = -5071676622092000/4204918450412337-380378998046332 000/206041004070204513*7^(1/2), a[14,6] = 1533012537239841375/79359550 98274910336-11049456415304617875/194430899907735303232*7^(1/2), a[14,8 ] = 13061564753515625/18689009843335104+31047286856168359375/256413215 05055762688*7^(1/2), a[14,9] = -32117994288265668498823115861026111114 68427562764815533/1544556820951908085230756966407550129703474655730757 4592-35609926058546725508440519811766382833790503793377686129/63599398 509784450568325286852075593576025427000678248320*7^(1/2), a[14,10] = 3 7302437685024/53021770400599+15742143124443936/12990333748146755*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 1975 "SO7_14 := \+ SimpleOrderConditions(7,14,'expanded'):\nSO7_15 := SimpleOrderConditio ns(7,15,'expanded'):\nSO7_16 := SimpleOrderConditions(7,16,'expanded') :\nSO7_17 := SimpleOrderConditions(7,17,'expanded'):\nSO7_18 := Simple OrderConditions(7,18,'expanded'):\nSO7_19 := SimpleOrderConditions(7,1 9,'expanded'):\nSO8_20 := SimpleOrderConditions(8,20,'expanded'):\nerr terms8_20 := PrincipalErrorTerms(8,20,'expanded'):\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],'interpolation_order_condition'):\n \+ interp_order_eqns15 := [op(interp_order_eqns15),temp_eqn];\nend do:\n whch := [1,2,4,8,16,17,25,27,32,63,64]:\ninterp_order_eqns16 := []:\nf or ct in whch do\n temp_eqn := convert(SO7_15[ct],'interpolation_ord er_condition'):\n interp_order_eqns16 := [op(interp_order_eqns16),te mp_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(i nterp_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 _order_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 c t in whch do\n temp_eqn := convert(SO7_18[ct],'interpolation_order_c ondition'):\n interp_order_eqns19 := [op(interp_order_eqns19),temp_e qn];\nend do:\nwhch := [1,2,4,8,16,17,25,27,32,61,63,64]:\ninterp_orde r_eqns20 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_19[ct], 'interpolation_order_condition'):\n interp_order_eqns20 := [op(inter p_order_eqns20),temp_eqn];\nend do:\nwhch := [1,2,4,8,16,17,25,27,32,5 8,61,63,64,102,117,121,123,125,127,128]:\nordeqns := []:\nfor ct in wh ch 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 "" }}{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 eqns,pols,e_u,eu,ct,eqs_15,eqs _16,eqs_17,eqs_18,eqs_19,eqs_20;\n global dd,sm,e6,e7,e8,e9,e10,e11, e12,e13,e14,e15,e16,e17,e18,e19,e20,e21,e22,e23;\n\n e6 := `union`(e 5,\{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 := `uni on`(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,i nterp_order_eqns17)):\n e13 := solve(\{op(eqs_17)\}):\n e14 := `un ion`(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 e1 8 := `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 := `unio n`(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_order_eqns20)):\n e22 := solve (\{op(eqs_20)\}):\n e23 := `union`(e21,e22):\n eqns := []:\n for ct to nops(ordeqns) do\n eqns := [op(eqns),expand(subs(e23,ordeq ns[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,er rterms8_20[ct]))^2;\n end 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);\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 interpolat ion scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 149 "c_15 := 9/20:\nc_16 := 1/10:\nc_17 := 13/40:\nc_18 := 2/5:\nc_19 := 3/10:\nc_20 := 7/10:\ncalc_coeffs();\nssmA := sqrt(s m)*u^9:plot(ssmA,u=0..1,color=blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6(/&%\"cG6#\"#:#\"\"*\"#?/&F%6#\"#;#\"\"\"\"#5/&F%6#\"#<#\"#8\"#S/&F%6 #\"#=#\"\"#\"\"&/&F%6#\"#>#\"\"$F1/&F%6#F*#\"\"(F1" }}{PARA 13 "" 1 " " {GLPLOT2D 403 333 333 {PLOTDATA 2 "6&-%'CURVESG6#7_o7$$\"3`*****\\n5 ;\"o!#@$\"3I=J0nhpmg!#G7$$\"3#******\\8ABO\"!#?$\"3W[r'Q!Q^.C!#F7$$\"3 3+++-K[V?F1$\"3ee\\'4&pAc`F47$$\"3#)******pUkCFF1$\"3o(Q\"fPn3J%*F47$$ \"3s*****\\Smp3%F1$\"31'3LFVB:3#!#E7$$\"3k******R&)G\\aF1$\"3o+%Rrwf(H OFD7$$\"3Y******4G$R<)F1$\"3!3$e35UzcyFD7$$\"3%******zqd)*3\"!#>$\"3S: >+-3]V8!#D7$$\"3*)*****>c'yM;FR$\"3o=K2]!=az#FU7$$\"3')*****fT:(z@FR$ \"3m3%\\[SLHf%FU7$$\"3#*******zZ*z7$FR$\"31R$Q`ik_B)FU7$$\"33+++XTFwSF 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Iq&F[u7$$\"3q*****\\f]tH'Fbp$\"3d!R\"fY=(Gq&F[u7$$\"3K+++5kh`jFbp$\"3W `E)fe&e'p&F[u7$$\"3#)*****\\A#))4kFbp$\"3oa5e:g8%o&F[u7$$\"35+++O![hY' Fbp$\"3qf(eS>![lcF[u7$$\"3G+++5FEnlFbp$\"3t>n$oihih&F[u7$$\"3I+++#Qx$o mFbp$\"3u#)fGjS\"oa&F[u7$$\"3s*****RP+V)oFbp$\"3$oh)*=%[\"F[u7$$\"3w******o0;r\"*Fbp$\"3MgX-4'Qt.\"F[u7$ $\"3[*****\\w(Gp$*Fbp$\"3PS()*=)3'GS*Fdo7$$\"37+++!oK0e*Fbp$\"3Y='zo3n " 0 "" {MPLTEXT 1 0 165 "c_15 := 3/40:\nc_16 := 1/10:\nc_17 := 13/4 0:\nc_18 := 2/5:\nc_19 := 3/10:\nc_20 := 7/10:\ncalc_coeffs();\nssmB : = sqrt(sm)*u^9:\nplot([ssmA,ssmB],u=0..1,color=[blue,brown]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(/&%\"cG6#\"#:#\"\"$\"#S/&F%6#\"#;#\"\" \"\"#5/&F%6#\"#<#\"#8F*/&F%6#\"#=#\"\"#\"\"&/&F%6#\"#>#F)F1/&F%6#\"#?# \"\"(F1" }}{PARA 13 "" 1 "" {GLPLOT2D 482 369 369 {PLOTDATA 2 "6&-%'CU RVESG6$7_o7$$\"3`*****\\n5;\"o!#@$\"3I=J0nhpmg!#G7$$\"3#******\\8ABO\" !#?$\"3W[r'Q!Q^.C!#F7$$\"33+++-K[V?F1$\"3ee\\'4&pAc`F47$$\"3#)******pU 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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 "nodes and linking coefficients: ee" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15534 "ee := \{c[2] = 1/16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/ 100, c[6] = 7/15, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/1617886 1, c[10] = 11/12, c[11] = 19/20, c[12] = 1, c[13] = 1, c[14] = 1/2-1/1 4*7^(1/2), c[15] = 3/40, c[16] = 3/25, c[17] = 13/40, c[18] = 2/5, c[1 9] = 3/10, c[20] = 7/10,\na[2,1] = 1/16, a[3,1] = 18928/1134225, a[3,2 ] = 100352/1134225, a[4,1] = 14/355, a[4,2] = 0, a[4,3] = 42/355, a[5, 1] = 94495479/250880000, a[5,2] = 0, a[5,3] = -352806597/250880000, a[ 5,4] = 178077159/125440000, a[6,1] = 12089/252720, a[6,2] = 0, a[6,3] \+ = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a[7,1] = 2140 0899/350000000, a[7,2] = 0, a[7,3] = 0, a[7,4] = 3064329829899/2712605 0000000, a[7,5] = -21643947/592609375, a[7,6] = 124391943/6756250000, \+ a[8,1] = -15365458811/13609565775, a[8,2] = 0, a[8,3] = 0, a[8,4] = -7 /5, a[8,5] = -8339128164608/939060038475, a[8,6] = 341936800488/479511 26225, a[8,7] = 1993321838240/380523459069, a[9,1] = -1840911252282376 584438157336464708426954728061551/299192361517115192159625381348311826 2195533733898, a[9,2] = 0, a[9,3] = 0, a[9,4] = -147649608040486573036 38372252908780219281424435/2981692102565021975611711269209606363661854 518, a[9,5] = -875325048502130441118613421785266742862694404520560000/ 170212030428894418395571677575961339495435011888324169, a[9,6] = 76320 51964154290925661849798370645637589377834346780/1734087257418811583049 800347581865260479233950396659, a[9,7] = 75198347919711375170485321796 52347729899303513750000/1045677303502317596597890707812349832637339039 997351, a[9,8] = 1366042683489166351293315549358278750/144631418224267 718165055326464180836641, a[10,1] = -63077736705254280154824845013881/ 78369357853786633855112190394368, a[10,2] = 0, a[10,3] = 0, a[10,4] = \+ -31948346510820970247215/6956009216960026632192, a[10,5] = -3378604805 394255292453489375/517042670569824692230499952, a[10,6] = 100158784418 3325981198091450220795/184232684207722503701669953872896, a[10,7] = 18 7023075231349900768014890274453125/25224698849808178010752575653374848 , a[10,8] = 1908158550070998850625/117087067039189929394176, a[10,9] = -52956818288156668227044990077324877908565/29127799594774339863498222 24412353951940608, a[11,1] = -10116106591826909534781157993685116703/9 562819945036894030442231411871744000, a[11,2] = 0, a[11,3] = 0, a[11,4 ] = -9623541317323077848129/3864449564977792573440, a[11,5] = -4823348 333146829406881375/576413233634141239944816, a[11,6] = 656611924651499 6884067001154977284529/970305487021846325473990863582315520, a[11,7] = 2226455130519213549256016892506730559375/3648804431596752555774356483 80047355776, a[11,8] = 39747262782380466933662225/17560328024314241644 10720256, a[11,9] = 48175771419260955335244683805171548038966866545122 229/1989786420513815146528880165952064118903852843612160000, a[11,10] \+ = -2378292068163246/47768728487211875, a[12,1] = -32180221747585998316 59045535578571/1453396753634469525663775847094384, a[12,2] = 0, a[12,3 ] = 0, a[12,4] = 26290092604284231996745/5760876126062860430544, a[12, 5] = -697069297560926452045586710000/41107967755245430594036502319, a[ 12,6] = 1827357820434213461438077550902273440/139381013914245317709567 680839641697, a[12,7] = 643504802814241550941949227194107500000/242124 609118836550860494007545333945331, a[12,8] = 162259938151380266113750/ 59091082835244183497007, a[12,9] = -2302825163287352381854541485685701 5616678575554130463402/20013169183191444503443905240405603349978424504 151629055, a[12,10] = 7958341351371843889152/3284467988443203581305, a [12,11] = -507974327957860843878400/121555654819179042718967, a[13,1] \+ = 4631674879841/103782082379976, a[13,2] = 0, a[13,3] = 0, a[13,4] = 0 , a[13,5] = 0, a[13,6] = 14327219974204125/40489566827933216, a[13,7] \+ = 2720762324010009765625000/10917367480696813922225349, a[13,8] = -498 533005859375/95352091037424, a[13,9] = 4059320304637772479267050305961 75437402459637909765779/7880391943632184108320188604120153722976911508 8303952, a[13,10] = -10290327637248/1082076946951, a[13,11] = 86326410 5888000/85814662253313, a[13,12] = -29746300739/247142463456, a[14,1] \+ = 923507123432989/20341288146475296+3728619917660047/79737849534183160 32*7^(1/2), a[14,2] = 0, a[14,3] = 0, a[14,4] = 0, a[14,5] = 0, a[14,6 ] = 1533012537239841375/7935955098274910336-11049456415304617875/19443 0899907735303232*7^(1/2), a[14,7] = 129834778101757666015625000/534951 006554143882189042101-82766411529674389648437500/262125993211530502272 63062949*7^(1/2), a[14,8] = 13061564753515625/18689009843335104+310472 86856168359375/25641321505055762688*7^(1/2), a[14,9] = -32117994288265 66849882311586102611111468427562764815533/1544556820951908085230756966 4075501297034746557307574592-35609926058546725508440519811766382833790 503793377686129/635993985097844505683252868520755935760254270006782483 20*7^(1/2), a[14,10] = 37302437685024/53021770400599+15742143124443936 /12990333748146755*7^(1/2), a[14,11] = -5071676622092000/4204918450412 337-380378998046332000/206041004070204513*7^(1/2), a[14,12] = 11006131 27343/48439922837376-29746300739/1424703612864*7^(1/2), a[14,13] = 3/3 92-3/392*7^(1/2), a[15,1] = 53968119671846752033/121108663654809600000 0-18013979306423851/41103546452541440000*7^(1/2), a[15,2] = 0, a[15,3] = 0, a[15,4] = 0, a[15,5] = 0, a[15,6] = -3197585392137639804267/3316 90531454428905472000+37078101149895800937/3922016419224666112000*7^(1/ 2), a[15,7] = 12599626362153730936328125/254800781771704557409886208+1 013426797972694921875/352212771011439949026304*7^(1/2), a[15,8] = -853 2637324759244825/3240219442044469248+132594730877413325/27245088401575 1168*7^(1/2), a[15,9] = 1437829216002805549145888241731712075764900749 803936348249893/149435580560728824572590243159611803932006618241524531 2000000-4367810325925642786942443213218202333579800829827668077/266731 24624013005776924627213516128165085289185280000000*7^(1/2), a[15,10] = -43024834368326969649/17313231151216000000+24278487789419081919/55087 553662960000000*7^(1/2), a[15,11] = 15187119353760445983/3661425589474 688000-19473860153393911521/25629979126322816000*7^(1/2), a[15,12] = - 36585897414219009/16871592171929600000, a[15,13] = 609279/512000000-22 21887/4096000000*7^(1/2), a[15,14] = -15553209/1024000000*7^(1/2), a[1 6,1] = 33317884221304756710581664536827945414542207027177538799/139174 8247282161490991675299723566089376318996169218750000-30228589849835343 14773428506933040222615091059789769/6659082522881155459290312438868737 269743153091718750000*7^(1/2), a[16,2] = 0, a[16,3] = 0, a[16,4] = 0, \+ a[16,5] = 0, a[16,6] = -6167678449409636588173213325011964076581612784 2489932383/58641515918369718256965079710079269237067699352911722500000 -27634079534573129698460778434691027193623514337349372/300417602040828 47467707520343278314158333862373417890625*7^(1/2), a[16,7] = -53623120 676210693478915318396423432061873602463681326025000/749594156753291060 7209275449283868805039087051861676693501311-24408621804868375085536711 81096852807790831050835529920000/1070848795361844372458467921326266972 148441007408810956214473*7^(1/2), a[16,8] = 12871682863191528707812574 253462271793937198554473/818367733822252968284538517562950447076898208 684200+7923793831170279449496891966550656386059347318121/1636735467644 505936569077035125900894153796417368400*7^(1/2), a[16,9] = 40611546028 7238735025830165985301412067123513597095836215277820544199798339360779 698993414799/528391868034181107309001742753778980704457203417916281347 492795784532666801613048775622343750000-431379155895084396489932754084 472156781047031795210079948966119713148914776761016489118643/173243235 4210429860029513910668127805588384273501364856877025559949287432136436 22549384375000*7^(1/2), a[16,10] = 15682219803354958595748193603184832 17200155100501135104/1958980963515503114549668523058637200949413352197 55859375+591872623465406532819676818747091838359436881921536/101765244 857948213742839923275773361088281213101171875*7^(1/2), a[16,11] = -203 308023958329587199829150033803325261178399821537536/828575898567110520 1066755800819103710987390140968171875-18395594530987044403702119081901 330600389569409698304/218046289096608031607019889495239571341773424762 3203125*7^(1/2), a[16,12] = 831464376531791898078674019732785579512976 6290647999/14914139040691643843786103420465233322847554416562500000+12 743937478203669740869195012551757724982855520824/932133690043227740236 63146377907708267797215103515625*7^(1/2), a[16,13] = 81244633141010175 6322947639951373288835154/66984419041833493354505016109118090494238281 25+386918100083180096375404793490759651703962/669844190418334933545050 1610911809049423828125*7^(1/2), a[16,14] = 348098152674365735187679001 267334640046338304/97942082977383567310235712743332126911845703125+757 8809409397518388644113904993904548128984/39176833190953426924094285097 33285076473828125*7^(1/2), a[16,15] = 29010441228214901202494331363634 75030176169984/28990856561305535923829770972026309565906328125+2079777 281538426380995265453726087303397376/115963426245222143695319083888105 2382636253125*7^(1/2), a[17,1] = 1204472757949318149125704321217/20800 267292079442987248844800000-2198100637293808474641843/4168724405178660 21068800000*7^(1/2), a[17,2] = 0, a[17,3] = 0, a[17,4] = 0, a[17,5] = \+ 0, a[17,6] = 48686024023005238414132359876273/168792443739398321152387 1237734400-90947086755739272427273888113/65170827698609390406327074816 00*7^(1/2), a[17,7] = 10629566348189721232485330989365234375/350093822 003577650477950866758750994432-9677609936618471677294132306640625/6174 49421523064639290918636258820096*7^(1/2), a[17,8] = -13981356309740608 6392326875/1528856569422457169284694016+7802957598378772386375/6801719 80612816002351104*7^(1/2), a[17,9] = 138982280655945534149961463082732 31365548079803650544776770188499895421/1026615247764494734714599938528 07430051568429917145049778991367782400000-1817510267133383600678887792 23183710172814810723200929646376184583/4828290406887688346688300710302 524634994399996103235733285896192000*7^(1/2), a[17,10] = -148975873252 8250543113521169/6777268656058713348166400000+182993414470798047123969 /3520659042108422518528000*7^(1/2), a[17,11] = 10565149828794636491259 146771/55897372479169766976172492800-78967956837666628354617891/266177 9641865226998865356800*7^(1/2), a[17,12] = -19066122638589247450672386 19/1287855848637222919589068800000-1218769836385461326457543/122652937 96544980186562560000*7^(1/2), a[17,13] = -30165509761493343939/1696696 9514364108800000+159134879536428159/458566743631462400000*7^(1/2), a[1 7,14] = 27217572159567342015099/893668418551873945600000+9899315565749 11765269/35746736742074957824000*7^(1/2), a[17,15] = -1204548165456597 /10186357540916840+350890378839/13765348028266*7^(1/2), a[17,16] = 132 95441661397228125/46517010473975545856-682342228168453125/465170104739 75545856*7^(1/2), a[18,1] = 29131180830546450644731399/515753697961869 733494253125-143343963156330219505552/15628899938238476772553125*7^(1/ 2), a[18,2] = 0, a[18,3] = 0, a[18,4] = 0, a[18,5] = 0, a[18,6] = 5657 5761137587272324976173/804865091988555532228408450-3907466402887273678 6931649/1609730183977111064456816900*7^(1/2), a[18,7] = 11129944106509 439530972892187500000/138892202332962324713542519160881437-12622051408 01795010221127437500000/46297400777654108237847506386960479*7^(1/2), a [18,8] = -2770031482617170508125/151635249347625299077614+387929392394 51158625/1944041658302888449713*7^(1/2), a[18,9] = 6110065569363880271 6486435678887510402373992546715729966430155473/39162263784961499584754 9415026883812147401542347507666698422881250-17094902496639758288013710 5076434230151670370842813631056736667/26108175856640999723169961001792 25414316010282316717777989485875*7^(1/2), a[18,10] = -8442401866475056 490423808/43019771742560192151446875+1087953059968188774912/1203350258 5331522280125*7^(1/2), a[18,11] = 2045460261009505062889984/2729363890 5844612781334225-469489191889390453370368/9097879635281537593778075*7^ (1/2), a[18,12] = 911312938075044590941/614097523039447269243750-14152 280688871027947/81879669738592969232500*7^(1/2), a[18,13] = -567003395 81774/48542889159290625+29301850944203/48542889159290625*7^(1/2), a[18 ,14] = 1083595819903600472/20454446835751509375+39411512915496232/8181 77873430060375*7^(1/2), a[18,15] = -3206129319411712/34378956700594335 +304869207965696/6875791340118867*7^(1/2), a[18,16] = 259448245437500/ 1197776110455837-30561362328125/1197776110455837*7^(1/2), a[18,17] = 0 , a[19,1] = 54669805406555815509959179/1000249596047262513443400000-20 405065149351317359441/5103314265547257721650000*7^(1/2), a[19,2] = 0, \+ a[19,3] = 0, a[19,4] = 0, a[19,5] = 0, a[19,6] = 816147521608308392480 1944517/412090927098140432500945126400-88996681564173938312881497/8410 018920370212908182553600*7^(1/2), a[19,7] = 74755448178182559525995895 60546875/370379206221232865902780051095683832-898376798115247150214280 60546875/7558759310637405426587347981544568*7^(1/2), a[19,8] = -549387 050628922094174375/6469770638832012760644864+4329396153890301811625/49 7674664525539443126528*7^(1/2), a[19,9] = 1867425894703702010525783959 157970802614636176859438775066185829501/167092325482502398228287750411 47042651622465806826993779132709600000-3893544913757614479421305579697 61865140472157707449139336703651/1364018983530631822271736738052819808 2957114944348566350312416000*7^(1/2), a[19,10] = -73383659778667485618 6786/3910888340232744741040625+9679407044454704106/2455816854149290261 25*7^(1/2), a[19,11] = 1560146031091654580841646/909787963528153759377 8075-29238981082342572586738/1299697090754505370539725*7^(1/2), a[19,1 2] = -164244455362347392642157/104805977265399000617600000-32233316630 530693491/427779499042444900480000*7^(1/2), a[19,13] = -19472213594618 603/12426979624778400000+297287747500681/1129725420434400000*7^(1/2), \+ a[19,14] = 15116525284746085123/654542298744048300000+5498038295775232 13/26181691949761932000*7^(1/2), a[19,15] = 132907117156352/6875791340 118867*7^(1/2)-3531632929957888/34378956700594335, a[19,16] = 85069109 306103125/306630684276694272-310066370359375/27875516752426752*7^(1/2) , a[19,17] = 0, a[19,18] = 0, a[20,1] = 25772202909830044727181487/673 637483052238019257800000+2313448267872880980787/3925626358113275170500 00*7^(1/2), a[20,2] = 0, a[20,3] = 0, a[20,4] = 0, a[20,5] = 0, a[20,6 ] = 3128676148761100227755815437/8410018920370212908182553600+13117134 5200484783698014927/8410018920370212908182553600*7^(1/2), a[20,7] = 10 683684211320620397871859541015625/22676277931912216279762043944633704+ 10185452642688448704857556640625/581443023895185032814411383195736*7^( 1/2), a[20,8] = 28537844110255116480186875/19409311916496038281934592- 6381054972281168645375/497674664525539443126528*7^(1/2), a[20,9] = -51 7827546394048821394457686498439852369824124297313387912381872377/10230 14237647973866703802553539614856221783620826142476273431200000+5738658 05960213827517536309484932171046626421224269648745064341/1364018983530 6318222717367380528198082957114944348566350312416000*7^(1/2), a[20,10] = 1370691905165773958915514/877954525358371268396875-1426638409937388 6246/245581685414929026125*7^(1/2), a[20,11] = -9970960321488888197319 914/3899091272263516111619175+43095050438446097412958/1299697090754505 370539725*7^(1/2), a[20,12] = 300637995762234907715489/641669248563667 3507200000+47508372541408403781/427779499042444900480000*7^(1/2), a[20 ,13] = 206707689553129373/12426979624778400000-4819864782877781/124269 79624778400000*7^(1/2), a[20,14] = -22280099903809149493/6545422987440 48300000-810350528295354683/26181691949761932000*7^(1/2), a[20,15] = 3 185294836283392/34378956700594335-195890510047232/6875791340118867*7^( 1/2), a[20,16] = -83291528425371875/306630684276694272+502704195317187 5/306630684276694272*7^(1/2), a[20,17] = 0, a[20,18] = 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 31 "interpolation coefficients: dd " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37233 "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] = -80859340236513354645057216119881786 498583192039803960911417945806609253663737/763346819251480831700723397 9049223823187010177566654949324576846512282484196-19549763706002525611 7862699941600460943029557474980519944371434271498545/38129211750823218 36666950039485126784808696392390936538124164258997144098*7^(1/2), d[2, 2] = 0, d[3,2] = 0, d[4,2] = 0, d[5,2] = 0, d[6,2] = -9555470700191264 53826454916873895202203877744679223408705084338612462278116625/6381692 4316359265507433102725710299707376977357327449518989476322455455114487 2+51976526550563049738919099496889712177688887113612059568640919857176 8556375/34495634765599602976990866338221783625609176949906729469724041 255381327088912*7^(1/2), d[7,2] = -18611275826137594707156283505269784 636206345033007201281002211617534500541992187500000/176484289995203369 28214651076087253786863872949527905235014321261690465631610581790463+1 8728502023252512021032053809189070331528372715917538258443245100620807 8613281250000/17648428999520336928214651076087253786863872949527905235 014321261690465631610581790463*7^(1/2), d[8,2] = 417749121938701143672 28492750623794528647431651894191295000926656740642578125/1888227999301 441837546993755379824056663391873877291121881575298277654610228-227232 63991052461226350618394433260149385768836071295813104863512935546875/1 0206637834061847770524290569620670576558874993931303361522028639338673 5688*7^(1/2), d[9,2] = -9024480744460783094427320915168210922688742289 831869506964781851318609947081887822066370347859939623672563906675893/ 4140180756417121380091094318604558026368982429319296013933118639895285 35407956588523392454658793583381057703478028+1816264585623659115295397 0061004454329559881109569554360766300150105113239841494818821201340595 0150907492266416127/82803615128342427601821886372091160527379648586385 9202786623727979057070815913177046784909317587166762115406956056*7^(1/ 2), d[10,2] = 21117183699737754158895343297492488392888338325981762990 4789183855071990784/52476821552580654631362439417762243161440782409822 00554346075790318908911-1931835797436815373129381172693380735039877689 02446821098998120901073408/4770620141143695875578403583432931196494616 58271109141304188708210809901*7^(1/2), d[11,2] = -53736653443669651364 6358288643959400800576485368292469427826726167820265088000/12623826381 970066996951153592318988474070834179056919325364695743285360570721+540 7512263770721692527300293234505295954519502937268246004267087898759616 000/126238263819700669969511535923189884740708341790569193253646957432 85360570721*7^(1/2), d[12,2] = 661305090734886841270782062890403445628 082324744813509222072805967406213/129843093923884049331488235466131670 9771126535458938871060868108639878984-13309408603167030446478290092404 437718941128474129897768024725744010607/259686187847768098662976470932 2633419542253070917877742121736217279757968*7^(1/2), d[13,2] = 9702358 1771610196015731451753362759625793708638711372455629402956213/30552692 7737287025280824372822808315759510048574360253352030337082457-93498270 74202492384268258850702233721270562092239650150014827133/3002721648523 705408165350101452661579946044703433516003459757612604*7^(1/2), d[14,2 ] = 210183921101452118929261196094748285700954400617647923247483913382 04/1353727009876103854847878670738241595625675153797943464893107390348 97*7^(1/2)+25627928959812595233787157408350070073854275295484516976638 855616196/135372700987610385484787867073824159562567515379794346489310 739034897, d[15,2] = 6503741362327325693514817822275293878921942613966 692501486690304000000/500878993654158426293715108173149390381499806905 2390820104497344291189+52192870774763167012978480095237129754473425595 550397746536448000000/135372700987610385484787867073824159562567515379 794346489310739034897*7^(1/2), d[16,2] = 65685448103411251832171663781 409404233805263145739293835358886718750/275249484448006329081823759299 8273114950540978147389669838111144887-51642648893391414364144037984523 5961449780535489905148546142578125/10009072161745684693884500338175538 59982014901144505334486585870868*7^(1/2), d[17,2] = 230507763356330154 6803694136021241260570026324327439248655518701977600/97588453577020425 76537387829721150134824645286158927011244212240963-1827567192261910518 290695748517878116392646969950057249714267750400/750680412130926352041 337525363165394986511175858379000864939403151*7^(1/2), d[18,2] = -4479 8064238286382839034805015821876382073189465669087580331678954625/10009 07216174568469388450033817553859982014901144505334486585870868+4182236 84926061505094447325162216258814133480745180549171230431125/1000907216 174568469388450033817553859982014901144505334486585870868*7^(1/2), d[1 9,2] = -51417275879498704605153750740940201155476439440081271114580632 173000/250226804043642117347112508454388464995503725286126333621646467 717+518648117627995591648772716717168248917051095645439658289278833500 /250226804043642117347112508454388464995503725286126333621646467717*7^ (1/2), d[20,2] = 17529466021841406425637718620952486628603562242741180 00141723011400/5838625427684982738099291863935730849895086923342947784 50508424673-5601623091631438080327498033171665021924765064524089718297 822900/834089346812140391157041694847961549985012417620421112072154892 39*7^(1/2), d[1,3] = 9801530410895616763555307686291964232082887276793 38084631427508618558897811172/1717530343315831871326627645286075360217 0772899524973635980297904652635589441+19293168884816342206474145251795 0348120145548012868666916234183477113213215/44611177748463165489003315 4619759833822617477909739574960527218302665859466*7^(1/2), d[2,3] = 0, d[3,3] = 0, d[4,3] = 0, d[5,3] = 0, d[6,3] = 885119799756668662383781 40063318146886422272356119272545520606383895827826000375/4467184702145 148585520317190799720979516388415012921466329263342571881858014104-105 8238328390613718438966144998989607660365801549350854247491823877298115 375/832653252962749037375641601267422363376773236721886573407132030302 3078952496*7^(1/2), d[7,3] = 20170282366704600011692539214651114093119 0938274281144650150458879263782763266601562500000/14454063350607155944 207799231315460851441511945663354387476729113324491352289066486389197- 1293788050585515435905828619515995514803983070066348221924574580339655 098098144531250000/144540633506071559442077992313154608514415119456633 54387476729113324491352289066486389197*7^(1/2), d[8,3] = -646775169039 72117674496548388029830868024150344778011253622168763840726931640625/2 2092267591826869499299826937943941462961684924364306126014430989848558 9396676+22425016301298527926213525844180542214095509749486981554285278 278176408203125/119417662658523618915134199664561845745738837428996249 32980773508026248075496*7^(1/2), d[9,3] = 7523409527441322464254094025 7121647350195100642008280122374634005890429295434463465684313041353810 00456754120388736267/2608313876542786469457389420720871556612458930471 1564887778647431340297730701265076973724643503995753006635319115764-96 5152313616825578521662699773460083841334069606065691656535633342724791 47954402956334707420502543277470497375823731/5216627753085572938914778 8414417431132249178609423129775557294862680595461402530153947449287007 991506013270638231528*7^(1/2), d[10,3] = -2542900228826121810647651286 52550732475583900081383182366372880676860057295872/4775390761284839571 45398198701636412769111119929380250445492896919020710901+1482817809106 59327634731565347445236756720251448844505767851130038550829568/4341264 3284407632467763472609239673888101010902670931858681172447183700991*7^ (1/2), d[11,3] = 63997801715875564184493318082204630969894988233105526 044940120711359268346496000/113614437437730602972560382330870896266637 507611512273928282261689568245136489-141552719129247940422004208148848 25802303606479581550969609086005540682816000/3917739221990710447329668 356236927457470258883155595652699388334123042935741*7^(1/2), d[12,3] = -55130806536809454006713383318850790786772881240227475692468011377890 8844347/81801149172046951078837588343662952715580971733913148876834690 844312375992+707254141709066429829485744545928848726419287934548599631 9652139700891171/16360229834409390215767517668732590543116194346782629 7753669381688624751984*7^(1/2), d[13,3] = -811781385040969851945127885 16639307689576497046518215860691542694024109/1924819644744908259269193 5487836923892849133060184695961177911236194791+70977758448472463737514 6468338854710232746252951093504726875327007/27024494836713348673488150 913073954219514402330901644031137818513436*7^(1/2), d[14,3] = -1595578 555977106018318305805767399622433462512379334628027752766016116/121835 4308888493469363090803664417436063107638418149118403796651314073*7^(1/ 2)-1945504378645790095314463899543676378170343165867393191276923106181 484/121835430888849346936309080366441743606310763841814911840379665131 4073, d[15,3] = -49372141298771861064258962829728123957462607828548631 0990557782016000000/45079109428874258366434359735583445134334982621471 517380940476098620701-396214063124740909099062618965261475107665588650 7730008240955392000000/12183543088884934693630908036644174360631076384 18149118403796651314073*7^(1/2), d[16,3] = -49864086598214089834343462 42523236327996150088597699111842229003906250/2477245360032056961736413 8336984458034554868803326507028543000303983+27442599748562125538200903 7937011522451814591457808962110006103515625/63057154618997813571472352 130505893178866938772103836072654909864684*7^(1/2), d[17,3] = -2469097 56289710321676748541180301721235505787987210822308890342208307200/8782 9608219318383188836490467490351213421807575430343101197910168667+18035 79982802176927816989581951816199920176377022329150547699322060800/8782 9608219318383188836490467490351213421807575430343101197910168667*7^(1/ 2), d[18,3] = 24733949514827229633016783778173145455249432074207944651 84769160588375/4504082472785558112248025152178992369919067055150274005 189636418906-222241605276435844340165767865292092427802572316830406450 939050724625/630571546189978135714723521305058931788669387721038360726 54909864684*7^(1/2), d[19,3] = 379672800690881181969846021594164839507 14543369867486036043111227817000/1576428865474945339286808803262647329 4716734693025959018163727466171-39372365414463860960022823678991725246 660945656160325539794836096500/225204123639277905612401257608949618495 9533527575137002594818209453*7^(1/2), d[20,3] = -204166820452965567460 365367263043332326353670451314975076344116875800/525476288491648446428 9362677542157764905578231008653006054575822057+29766695500705866115736 99544020730516840700340550363347243520913700/5254762884916484464289362 677542157764905578231008653006054575822057*7^(1/2), d[1,4] = -24176759 7058162460996419210497909997072270662926829186922757706083962310531449 01/1374024274652665497061302116228860288173661831961997890878423832372 21084715528-7347151694487370684133395953904842215495721540032731051995 43454432698772885/4461117774846316548900331546197598338226174779097395 74960527218302665859466*7^(1/2), d[2,4] = 0, d[3,4] = 0, d[4,4] = 0, d [5,4] = 0, d[6,4] = -1951098866784535430261436366351454780080971668754 853743185361821808342775901829625/178687388085805943420812687631988839 18065553660051685865317053370287527432056416+1669547908357196053661307 0219196566270422739905653146942921539695588415382375/34495634765599602 976990866338221783625609176949906729469724041255381327088912*7^(1/2), \+ d[7,4] = -111155052342124889259448757507929676741264416636574812417181 1358834740046920218505859375000/14454063350607155944207799231315460851 441511945663354387476729113324491352289066486389197+703850683029488619 401418304954959267368195560479807234733238582994060156096191406250000/ 2064866192943879420601114175902208693063073135094764912496675587617784 478898438069484171*7^(1/2), d[8,4] = 142570790951088852051393028173745 8147551036670928522240301201176074926980724609375/88369070367307477997 1993077517757658518467396974572245040577239593942357586704-85398099970 325736383267798339936564138231250599093870734618554049173865234375/119 4176626585236189151341996645618457457388374289962493298077350802624807 5496*7^(1/2), d[9,4] = 52506522599220181257600180264957660994289992431 199170509267340577672704978492802902477767968701500214983198708897287/ 7452325361550818484163969773488204447464168372774732825079613551811513 637343218593421064183858284500859038662604504*7^(1/2)-1658410060127699 4204243676516390195869086069172996437301230322048408352737889956494516 9306255460599525765839379134598693/10433255506171145877829557682883486 2264498357218846259551114589725361190922805060307894898574015983012026 541276463056, d[10,4] = -806687252425648628560896258674273628820416885 21020271661420196742074036736/6201806183486804638251924658462810555443 001557524418836954453206740528713*7^(1/2)+1401349782304257392347248469 940689563084075349536308092739857726684926046447872/477539076128483957 145398198701636412769111119929380250445492896919020710901, d[11,4] = 1 5632615816585084537928531867940510417375991758134336341589222055826428 56896000/1136144374377306029725603823308708962666375076115122739282822 61689568245136489*7^(1/2)-35268118066863255053183732944042514382155212 2218234606527970301923831373382496000/11361443743773060297256038233087 0896266637507611512273928282261689568245136489, d[12,4] = 121526661349 63609339643471567902272688421348215270844813219503690759850401013/3272 0459668818780431535035337465181086232388693565259550733876337724950396 8-38476264368964968805560252770565072980662958360503079847597388502918 90167/2337175690629912887966788238390370077588027763826089967909562595 5517821712*7^(1/2), d[13,4] = 4497801995505337470409608239184601254373 44839074607085806442449771655170/1924819644744908259269193548783692389 2849133060184695961177911236194791-27029481852823884994046587124253659 74862300040850732058062959386773/2702449483671334867348815091307395421 9514402330901644031137818513436*7^(1/2), d[14,4] = 7408796283045694306 410397470048681014672681412521125135646448795379876/121835430888849346 9363090803664417436063107638418149118403796651314073+60762219836580780 92011304752061426360605804136249743761664766103126524/1218354308888493 469363090803664417436063107638418149118403796651314073*7^(1/2), d[15,4 ] = 188017123454105219389200373772778165549148183683194083258654594662 4000000/45079109428874258366434359735583445134334982621471517380940476 098620701+150884742814730063539208210877344376980643719842352058981002 36288000000/1218354308888493469363090803664417436063107638418149118403 796651314073*7^(1/2), d[16,4] = 18989053096013528681288086717742824162 415325204157313237136237792968750/247724536003205696173641383369844580 34554868803326507028543000303983-1492941024489206293102671267634540662 40717845071784859435802001953125/9008164945571116224496050304357984739 838134110300548010379272837812*7^(1/2), d[17,4] = 11648144406048160728 95331764271503128104872521702292846814934538466099200/8782960821931838 3188836490467490351213421807575430343101197910168667-68683251600089122 61494693911012589565049932528664537266243896062771200/8782960821931838 3188836490467490351213421807575430343101197910168667*7^(1/2), d[18,4] \+ = -2431924248455269236581234082899678817632155182577687465106351580862 8625/90081649455711162244960503043579847398381341103005480103792728378 12+1209045837149275500889081964963205895152156159323846299647162670461 25/9008164945571116224496050304357984739838134110300548010379272837812 *7^(1/2), d[19,4] = -1755958896341365686809054845076827195048355312945 21271186768235374774500/1576428865474945339286808803262647329471673469 3025959018163727466171+14993635467448394900063201041907905630039872806 4857543972801591013500/22520412363927790561240125760894961849595335275 75137002594818209453*7^(1/2), d[20,4] = -16193772194928625300297793065 23117110075604247520921668262179354900/7506804121309263520413375253631 65394986511175858379000864939403151*7^(1/2)+11002932472337173027862713 79512216276794012408909541135348264281676300/5254762884916484464289362 677542157764905578231008653006054575822057, d[1,5] = 15280639412466567 48630101393682804434731516483634012493728265068587807677685/4461117774 84631654890033154619759833822617477909739574960527218302665859466*7^(1 /2)+552928391515756411944349905886577308799536546045706145790463104083 8537690531792/17175303433158318713266276452860753602170772899524973635 980297904652635589441, d[2,5] = 0, d[3,5] = 0, d[4,5] = 0, d[5,5] = 0, d[6,5] = 698559305892442368822785152069637500969905948840373672577583 243850664955527697375/223359235107257429276015859539986048975819420750 6460733164631671285940929007052-24306333176719413805890064226164922946 4483672897314758089052013901147904456625/24146944335919722083893606436 7552485379264238649347106288068288787669289622384*7^(1/2), d[7,5] = -1 0247102896981411208754957067904747006614174490462607662485698488409991 200876464843750000/144540633506071559442077992313154608514415119456633 54387476729113324491352289066486389197*7^(1/2)+31837811002789235239518 47382539205125855515087346383093549498094643381127118955078125000000/1 4454063350607155944207799231315460851441511945663354387476729113324491 352289066486389197, d[8,5] = -5104515945491938784606313573273925514480 87052512651918152034064915844772416015625/1104613379591343474964991346 89719707314808424621821530630072154949242794698338+1776113555863321864 34934285079709301489359584499318813734677020319129177734375/1194176626 5852361891513419966456184574573883742899624932980773508026248075496*7^ (1/2), d[9,5] = 593766825561784837280187624406711407946758449571601873 08205895439835379901375509676447975176725074528268184637240152531/1304 1569382713932347286947103604357783062294652355782443889323715670148865 350632538486862321751997876503317659557882-764423126679479650194111598 5159559705889245749734039641950049463307498965691218402921548914246423 59299087864124493329/5216627753085572938914778841441743113224917860942 3129775557294862680595461402530153947449287007991506013270638231528*7^ (1/2), d[10,5] = -4013844497187512011429456845977150770131602282196775 219367078287009030286702592/477539076128483957145398198701636412769111 119929380250445492896919020710901+117442626406352679588648918327640146 0992934188525388041595211196527995674112/43412643284407632467763472609 239673888101010902670931858681172447183700991*7^(1/2), d[11,5] = 10101 7421500625211398759090633558164742768146441848742250068775096991374507 5456000/11361443743773060297256038233087089626663750761151227392828226 1689568245136489-32512785266988458875941339220852152429800754429428481 77106696150597536598976000/1136144374377306029725603823308708962666375 07611512273928282261689568245136489*7^(1/2), d[12,5] = -43510650149540 93377034295353840201363867678725210673643915987290515429731971/4090057 4586023475539418794171831476357790485866956574438417345422156187996+56 0161763832124013774459393120743844579221650328069831926861574049562402 89/1636022983440939021576751766873259054311619434678262977536693816886 24751984*7^(1/2), d[13,5] = -12981811701241389330227911582670936417738 41073704766009094724204936408480/1924819644744908259269193548783692389 2849133060184695961177911236194791+56216038932299261246935439217513579 45938733181371553595719704653813/2702449483671334867348815091307395421 9514402330901644031137818513436*7^(1/2), d[14,5] = -154088481072994743 08359573744910931688118979350388479664992500704312356/1218354308888493 469363090803664417436063107638418149118403796651314073-126373540364011 35419339997318285509618390756427112136409046317266066044/1218354308888 493469363090803664417436063107638418149118403796651314073*7^(1/2), d[1 5,5] = -31381077234659091107473537753267836153463692608127416395216150 528000000/121835430888849346936309080366441743606310763841814911840379 6651314073*7^(1/2)-391038865990346270047298234016412977423513424228667 7265798761734144000000/45079109428874258366434359735583445134334982621 471517380940476098620701, d[16,5] = 2173517859103192967015253651832967 928598862243264021108157840576171875/630571546189978135714723521305058 93178866938772103836072654909864684*7^(1/2)-39493518741702005475803674 323125756824705515241816471075655378417968750/247724536003205696173641 38336984458034554868803326507028543000303983, d[17,5] = 14284773814649 096068127661436676990475766429004191858948112254743347200/878296082193 18383188836490467490351213421807575430343101197910168667*7^(1/2)-27673 30752964448391683957769424099011399113939004087716189060644628070400/8 7829608219318383188836490467490351213421807575430343101197910168667, d [18,5] = 1503328588552116234701386256305605353965498333543343799912721 1421585875/22520412363927790561240125760894961849595335275751370025948 18209453-1760205310465912592039402114730332052639447358842535783959206 536302875/630571546189978135714723521305058931788669387721038360726549 09864684*7^(1/2), d[19,5] = 410057190208357565703511334931994575276920 269533176347643951103754444000/157642886547494533928680880326264732947 16734693025959018163727466171-3118383103917038083808290846282094353660 47678462810190475810269493500/2252041236392779056124012576089496184959 533527575137002594818209453*7^(1/2), d[20,5] = -3071345163919521279825 288051320933574653190414930956966684399994785600/525476288491648446428 9362677542157764905578231008653006054575822057+23575916593201335297637 657467560877490273431047774224162861593548300/525476288491648446428936 2677542157764905578231008653006054575822057*7^(1/2), d[1,6] = -6863280 0216021672512749379562594026363792946379911701317408599854591400125/17 1581452878704482650012751776830705316391337657592144215587391654871484 41*7^(1/2)-59033281177651767670109780290890365337408979465032456583256 42265376667526822275/1717530343315831871326627645286075360217077289952 4973635980297904652635589441, d[2,6] = 0, d[3,6] = 0, d[4,6] = 0, d[5, 6] = 0, d[6,6] = -1080869172953431585382817721722694339997066780183516 910999044101117987345314740625/223359235107257429276015859539986048975 8194207506460733164631671285940929007052+14192306768282892056311984748 6361207233302115067967059067848319570177574053125/12073472167959861041 9468032183776242689632119324673553144034144393834644811192*7^(1/2), d[ 7,6] = 920494682110974559950047537119764805358166414382138289742171685 991269307495117187500000/111185102696978122647752301779349698857242399 5820258029805902239486499334791466652799169*7^(1/2)-492621144074072092 2526952796826782452822712695703743609253637908459691602090576171875000 000/144540633506071559442077992313154608514415119456633543874767291133 24491352289066486389197, d[8,6] = 789813245889387687194716304042276262 776073785597983613846153229518697127490234375/110461337959134347496499 134689719707314808424621821530630072154949242794698338-797739175371709 4135426135566958620200139372147017459897506207988825537109375/45929870 2532783149673593075632930175945149374726908651268491288770240310596*7^ (1/2), d[9,6] = -91872551443895129756013632413224428568666473255759635 504109212049187289585415128517333378983662445427803679752792495325/130 4156938271393234728694710360435778306229465235578244388932371567014886 5350632538486862321751997876503317659557882+44634159483159901744704512 2823867431945449150016344237792018848210700656341244224765712650076196 023241091303713655525/260831387654278646945738942072087155661245893047 11564887778647431340297730701265076973724643503995753006635319115764*7 ^(1/2), d[10,6] = 6210554702290003006619777946314693941084921389808669 166600182285462536551270400/477539076128483957145398198701636412769111 119929380250445492896919020710901-105498416622750211822092124112137633 633552240418992262438591868901996108800/333943409880058711290488250840 3051837546231607897763989129320957475669307*7^(1/2), d[11,6] = -156302 5728957657097579337657954854256115104197944139324142177929754642161427 200000/113614437437730602972560382330870896266637507611512273928282261 689568245136489+379679994547566921797740694749501847373608024177493510 0559569774038875811200000/11361443743773060297256038233087089626663750 7611512273928282261689568245136489*7^(1/2), d[12,6] = 6732330389861207 934137784189729253179054254301522002252780613976932430273325/409005745 86023475539418794171831476357790485866956574438417345422156187996-3270 7473950790850817326578829670786039784309048028582433314149231272281525 /818011491720469510788375883436629527155809717339131488768346908443123 75992*7^(1/2), d[13,6] = 184600455263163020619600883650359881442382300 071728307974083408971036000/174983604067718932660835777162153853571355 7550925881451016173748744981-32824172367928015736632150060241350683680 78806229779285021568438425/1351224741835667433674407545653697710975720 1165450822015568909256718*7^(1/2), d[14,6] = 1799424846970563922035374 3162669928256465752258551209088349757672692200/12183543088884934693630 90803664417436063107638418149118403796651314073+1475773444887913525993 6429969395914778314308271027965880151310742387800/12183543088884934693 63090803664417436063107638418149118403796651314073*7^(1/2), d[15,6] = \+ 3664640582315684745905572109817261532248499799997196822994575360000000 0/12183543088884934693630908036644174360631076384181491184037966513140 73*7^(1/2)+45665000180051823231670774083380286247763543003528800253039 58732800000000/4507910942887425836643435973558344513433498262147151738 0940476098620701, d[16,6] = -12691026654847075364606831167252496889293 26210372751965147308349609375/3152857730949890678573617606525294658943 3469386051918036327454932342*7^(1/2)+419272794809195896014687085548381 5984531166910526273081181274414062500/22520412363927790561240125760894 96184959533527575137002594818209453, d[17,6] = 34877558500619899163893 12843156191104251336113200697117615019136942080000/8782960821931838318 8836490467490351213421807575430343101197910168667-12831976475434865742 11335691610176629645867350301017715486628577280000/6756123709178337168 372037728268488554878600582725411007784454628359*7^(1/2), d[18,6] = -1 9533234704336153683638841435542242043378577022005640232490897982450000 /2252041236392779056124012576089496184959533527575137002594818209453+1 027772208982142979601125383837386628856684797851179251962327532584375/ 31528577309498906785736176065252946589433469386051918036327454932342*7 ^(1/2), d[19,6] = -509913925180556685403243049723802960846039798301015 253619510340070087500/157642886547494533928680880326264732947167346930 25959018163727466171+3641606433191671950128107353318922084670006170421 15329794290741575000/2252041236392779056124012576089496184959533527575 137002594818209453*7^(1/2), d[20,6] = 46040711166162957815383871262605 04437531975209210058615811924427782500/5254762884916484464289362677542 157764905578231008653006054575822057-275316427370164286366375217076582 65156563690923705750123418761835000/5254762884916484464289362677542157 764905578231008653006054575822057*7^(1/2), d[1,7] = 288461937437743583 65711955014149071997427668056567273645747464049451336250/1173978361801 6622497106135647888416679542565208151041446329663639543838407*7^(1/2)+ 1822113579387021757099335010830338205947328754403254962573306674886788 953267000/924824031016217161483568732077117501655349309974421657322016 0410197573009699, d[2,7] = 0, d[3,7] = 0, d[4,7] = 0, d[5,7] = 0, d[6, 7] = 14817586831844074987229140692862341269919265800069692770856433351 85370545584046875/3908786614377005012330277541949755857076839863136306 283038105424750396625762341-714595382394859241055899016646890090637173 976644562571963045983326017484375/989628866226218118192360919539149530 242886223972734042164214298310120039436*7^(1/2), d[7,7] = 270132843462 4316911665017828841251554817150906205669088351334307152113102139160156 2500000000/10117844345425009160945459461920822596009058361964348071233 7103793271439466023465404724379-73507505052618375368604493267420767478 50667880910821838767076856922943770751953125000000/1445406335060715594 4207799231315460851441511945663354387476729113324491352289066486389197 *7^(1/2), d[8,7] = -30935754945460883012225796776543488204727454923249 8955047310280884427935058593750/55230668979567173748249567344859853657 404212310910765315036077474621397349169+318523385327053673550644731236 14010011621215379703524573534798363358642578125/2985441566463090472878 354991614046143643470935724906233245193377006562018874*7^(1/2), d[9,7] = 2518953343378643410625524963318107686726687187922891470071919563684 75233861341897977725629468755249478081861864619682750/4564549283949876 3215504314862615252240718031283245238553612633004845521028727213884704 018126131992567761611808452587-132198234473067434839919334645680597709 7907374662203236494635698942679863457663199347237302602303427536911225 06375/1257624819933841113528153047599263045618350496852052308957504697 7502554354243618648492634832933459861623257145186*7^(1/2), d[10,7] = - 3405608592708545784350469093333906983380630746710147916477791802792999 4730496000/33427735328993877000177873909114548893837778395056617531184 50278433144976307+8424736767794947608846435866661726565402990284827157 40618317834465216768000/4341264328440763246776347260923967388810101090 2670931858681172447183700991*7^(1/2), d[11,7] = 8570979740666286328997 217518927175734060764262917264353497409416424208243328000000/795301062 064114220807922676316096273866462553280585917497975831826977715955423- 1227527301241241275590125301887873121120974721891762000592448879241498 56000000/5979707233564768577503178017414257698244079347974330206751697 983661486586131*7^(1/2), d[12,7] = -9229321422855407030562516932164067 886881661000752317997779792964310093271375/715760055255410821939828898 00705083626133350267174005267230354488773328993+5909287684289989383798 78111289332796362732451023465810340969090457718875/2405916152119027972 906987892460675079870028580409210261083373260126834588*7^(1/2), d[13,7 ] = -16158197214559374534407637428478884767437834064551796068173802937 23710000/1924819644744908259269193548783692389284913306018469596117791 1236194791+10081631870485733300164041861968943512225487331082145424113 55342375/6756123709178337168372037728268488554878600582725411007784454 628359*7^(1/2), d[14,7] = -1105352402029656240732254197313430740040170 7152640677684661691086334000/12183543088884934693630908036644174360631 07638418149118403796651314073-9065395117249487980524568422255648774497 405007157123225220383401266000/121835430888849346936309080366441743606 3107638418149118403796651314073*7^(1/2), d[15,7] = -280511396309126785 4720074768011254220401887730900261423685308416000000000/45079109428874 258366434359735583445134334982621471517380940476098620701-225111889338 28612575121820264359204228194691662123582303752192000000000/1218354308 888493469363090803664417436063107638418149118403796651314073*7^(1/2), \+ d[16,7] = 389792794646985533717717583668812601457440464708764983979644 775390625/157642886547494533928680880326264732947167346930259590181637 27466171*7^(1/2)-28330641915448358102738021353817904023875160889639273 815465087890625000/247724536003205696173641383369844580345548688033265 07028543000303983, d[17,7] = -2222702614413765624451736212569736108618 072977775930081805540792729600000/878296082193183831888364904674903512 13421807575430343101197910168667+1024717028717598831584563982158444864 1799750497702327259852082380800000/87829608219318383188836490467490351 213421807575430343101197910168667*7^(1/2), d[18,7] = 12707140725171571 452121209934083420611459385902722387638036232273187500/225204123639277 9056124012576089496184959533527575137002594818209453-31567044376716939 3000149235660096036906790988780351377578943962015625/15764288654749453 392868088032626473294716734693025959018163727466171*7^(1/2), d[19,7] = 321639168209596867775399139665346050163858290439599839355336233198250 000/157642886547494533928680880326264732947167346930259590181637274661 71-2236969454407512180345818667632407952456691418392823779010039152500 00/2252041236392779056124012576089496184959533527575137002594818209453 *7^(1/2), d[20,7] = -3469104744662012692142612460845000024443119693119 638369589638456150000/525476288491648446428936267754215776490557823100 8653006054575822057+16912163618512752757691188131716413718508067126312 923646892737450000/525476288491648446428936267754215776490557823100865 3006054575822057*7^(1/2), d[1,8] = -1146306239992402328481986222409736 61521724872784376962684165806230239497235000/2453614776165474101895182 350408679086024396128503567662282899700664662227063-137554896878461139 940641868413661678907695099845443749448622188741115581250/223055888742 315827445016577309879916911308738954869787480263609151332929733*7^(1/2 ), d[2,8] = 0, d[3,8] = 0, d[4,8] = 0, d[5,8] = 0, d[6,8] = -930868719 4465852793565801560973618467448909879788578029468256078519053597890625 /797711553954490818842913784071378746342212216966593118987368454030693 18893109+1094016769641665308675362426414511299752577496751505237881505 3831415345078125/60367360839799305209734016091888121344816059662336776 572017072196917322405596*7^(1/2), d[7,8] = 184486937260923649290843637 6223903442020075758569964675059797749453583563232421875000000/14454063 3506071559442077992313154608514415119456633543874767291133244913522890 66486389197*7^(1/2)-25328737501882703227852351556206185784706019405015 80447042625376505649438476562500000000/3081889840214745403882259964033 1473030792136344697983768607098322653499685051314469913, d[8,8] = 1421 3235500447742572017578910082893822877820697574417487816347355769506835 93750/8243383429786145335559636917143261739911076464315039599259116040 98826826107-7994204641132215478393839496334125742651723598062930975797 065315272216796875/298544156646309047287835499161404614364347093572490 6233245193377006562018874*7^(1/2), d[9,8] = -1582453944563627728297596 8538088601819257799223471523684287850111001103392434803691800774412117 74951537441146103396250/9315406701938523105204962216860255559330210465 96841603134951693976439204667902324177633022982285562607379832825563+3 4406329972069392909652689990678450298679747198445236865610387006692378 551569181688274149796675739206158031620738125/130415693827139323472869 4710360435778306229465235578244388932371567014886535063253848686232175 1997876503317659557882*7^(1/2), d[10,8] = 2139467475782193593244539190 48752100967324715456767332669807711679092495360000/6821986801835485102 0771171243090916109873017132768607206498985274145815843-21144152320329 9070830561972088728812190619636392784313929735659109297920000/43412643 284407632467763472609239673888101010902670931858681172447183700991*7^( 1/2), d[11,8] = -37691156585060503332995528627066006036470156205432680 1090127358264206943360000000/11361443743773060297256038233087089626663 7507611512273928282261689568245136489+58535414702390917143617836237565 4058239461909330256807418482131904456160000000/11361443743773060297256 0382330870896266637507611512273928282261689568245136489*7^(1/2), d[12, 8] = 57980335878923601213722076139961113361856525644820058802284384762 211748125/146073480664369555497924264899398129849251735239130622994347 6622219863857-25212620879045726922247603626624250888815461348671599932 14919889830388125/4090057458602347553941879417183147635779048586695657 4438417345422156187996*7^(1/2), d[13,8] = 7266876242674785077892943772 8391535863889404814086648826675920073550000/27497423496355832275274193 55405274841835590437169242280168273033742113-2530257808432825009862010 20186958461492945413653764447658942910625/6756123709178337168372037728 268488554878600582725411007784454628359*7^(1/2), d[14,8] = 27741803928 52179927128354568195155564887907520164855664145158728210000/1218354308 888493469363090803664417436063107638418149118403796651314073+227520574 8099209159198801748365943419270652578815001489555641065790000/12183543 08888493469363090803664417436063107638418149118403796651314073*7^(1/2) , d[15,8] = 5649790858132390457492635700515467944395409342191600999444 480000000000/121835430888849346936309080366441743606310763841814911840 3796651314073*7^(1/2)+704019111175268717534932230006377309033514430761 903910070743040000000000/450791094288742583664343597355834451343349826 21471517380940476098620701, d[16,8] = 71103397590159940525828308173694 12546214417631795473228314208984375000/2477245360032056961736413833698 4458034554868803326507028543000303983-97829029565515072031371539930075 537441356295893257008870697021484375/157642886547494533928680880326264 73294716734693025959018163727466171*7^(1/2), d[17,8] = 433559334076191 25738966512963250110580873371765022438664279879680000000/6756123709178 337168372037728268488554878600582725411007784454628359-257180414505838 0054908221535691777827422208100854591092725399552000000/87829608219318 383188836490467490351213421807575430343101197910168667*7^(1/2), d[18,8 ] = -32632831164236241443061567319710931145582111871186085274433050628 12500/2252041236392779056124012576089496184959533527575137002594818209 453+792260236729763309336516251504428210173111424411936031126162458593 75/1576428865474945339286808803262647329471673469302595901816372746617 1*7^(1/2), d[19,8] = 5614279019461578303715207486901617685072496099047 1063224913178750000/22520412363927790561240125760894961849595335275751 37002594818209453*7^(1/2)-11559219327420125600231693746941742338260394 008911718306976184701250000/225204123639277905612401257608949618495953 3527575137002594818209453, d[20,8] = 146353692252118455617219060985342 711304418993480549071169845478750000/750680412130926352041337525363165 394986511175858379000864939403151-424456423354542772099593591388812190 2148017659220291703491071750000/52547628849164844642893626775421577649 05578231008653006054575822057*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 272 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*,$* (\"$7\"F*\"%l5F.F-F.F*,$*(\"&G*=F*\"(DU8\"F.F-F.F*,$*(\"'_.5F*F8F.F-F. F*F/F/F/F/F/7*,$*(\"#cF*\"$b$F.F-F.F*,$*(\"#9F*F@F.F-F.F*\"\"!,$*(\"#U F*F@F.F-F.F*F/F/F/F/7*,$*(\"#RF*\"$+\"F.F-F.F*,$*(\")za\\%*F*\"*++)3DF .F-F.F*FD,$*(\"*(f1GNF*FPF.F-F.F.,$*(\"*fr2y\"F*\"*++WD\"F.F-F.F*F/F/F /7*,$*(\"\"(F*\"#:F.F-F.F*,$*(\"&*37F*\"'?FDF.F-F.F*FDFD,$*(\"(x`]#F* \")?bo5F.F-F.F*,$*(\"'+/'*F*\"(\">4_F.F-F.F*F/F/7*,$*(FKF*\"$]#F.F-F.F *,$*(\")**3S@F*\"*+++]$F.F-F.F*FDFD,$*(\".**)H)HV1$F*\"/+++]g7FF.F-F.F *,$*(\")ZRk@F*\"*v$4EfF.F-F.F.,$*(\"*V>RC\"F*\"+++DcnF.F-F.F*F/7*,$*( \"#CF*\"#DF.F-F.F*,$*(\",6)eaO:F*\",vdc4O\"F.F-F.F.FDFD,$*(FenF*\"\"&F .F-F.F.,$*(\".3Y;G\"R$)F*\"-v%Q+1R*F.F-F.F.,$*(\"-)[+o$>MF*\",Di7^z%F. F-F.F*,$*(\".S#Q=K$*>F*\"-p!fM_!QF.F-F.F*Q(pprint16\"" }}}{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'*&$\"'%G2\"!\"%F'%\"uGF'!\"\"*&$\"'=@eF.F')F/\"\"#F'F'*& $\"'8.=!\"$F')F/\"\"$F'F0*&$\"'%*4LF9F')F/\"\"%F'F'*&$\"'$Ha$F9F')F/\" \"&F'F0*&$\"'CN?F9F')F/\"\"'F'F'*&$\"'2N[F.F')F/\"\"(F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',0*&$\"'Yd9!\"&\"\"\"%\"uGF-!\"\" *&$\"'vZ>!\"%F-)F.\"\"#F-F-*&$\"'5z5!\"$F-)F.\"\"$F-F/*&$\"')35$F9F-)F .\"\"%F-F-*&$\"'03[F9F-)F.\"\"&F-F/*&$\"'urPF9F-)F.F'F-F-*&$\"'8i6F9F- )F.\"\"(F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"(,0*&$\"' [E5!\"&\"\"\"%\"uGF-!\"\"*&$\"'zr8!\"%F-)F.\"\"#F-F-*&$\"'/+wF3F-)F.\" \"$F-F/*&$\"'$R=#!\"$F-)F.\"\"%F-F-*&$\"'G'Q$F>F-)F.\"\"&F-F/*&$\"'TcE F>F-)F.\"\"'F-F-*&$\"'![=)F3F-)F.F'F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"),0*&$\"'\\`@!\"%\"\"\"%\"uGF-F-*&$\"'$z(G!\"$F-)F. \"\"#F-!\"\"*&$\"'W%f\"!\"#F-)F.\"\"$F-F-*&$\"'u\"e%F9F-)F.\"\"%F-F5*& $\"'=/rF9F-)F.\"\"&F-F-*&$\"''Hd&F9F-)F.\"\"'F-F5*&$\"'7< " 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) f or 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 95 "S tage by stage construction of an interpolation scheme C .. [8 stage sc heme] .. (longer method)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 149 "In this section an alternative interpolation sche me is constructed - one which requires 8 stages - in an effort to impr ove the principal error curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 125 "Start with linking coefficients using th e weights of the 12 stage scheme as the linking coefficients for the f irst 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 3978 "e1 := \{c[2]=1/16,c[3 ]=112/1065,c[4]=56/355,c[5]=39/100,c[6]=7/15,c[7]=39/250,c[8]=24/25,\n c[9]=14435868/16178861,c[10]=11/12,c[11]=19/20,c[12]=1,c[13]=1,\n a [2,1]=1/16,a[3,1]=18928/1134225,a[3,2]=100352/1134225,a[4,1]=14/355,a[ 4,2]=0,\n a[4,3]=42/355,\n a[5,1]=94495479/250880000,a[5,2]=0,a[5,3] =-352806597/250880000,a[5,4]=178077159/125440000,\n a[6,1]=12089/2527 20,a[6,2]=0,a[6,3]=0,a[6,4]=2505377/10685520,a[6,5]=960400/5209191,\n \+ a[7,1]=21400899/350000000,a[7,2]=0,a[7,3]=0,a[7,4]=3064329829899/2712 6050000000,\n a[7,5]=-21643947/592609375,a[7,6]=124391943/6756250000, \n a[8,1]=-15365458811/13609565775,a[8,2]=0,a[8,3]=0,a[8,4]=-7/5,\n \+ a[8,5]=-8339128164608/939060038475,a[8,6]=341936800488/47951126225,\n \+ a[8,7]=1993321838240/380523459069,\n a[9,1]=-18409112522823765844381 57336464708426954728061551/\n 2991923615171151921 596253813483118262195533733898,a[9,2]=0,a[9,3]=0,\n a[9,4]=-147649608 04048657303638372252908780219281424435/\n 298169210256 5021975611711269209606363661854518,\n a[9,5]=-87532504850213044111861 3421785266742862694404520560000/\n 170212030428894418395 571677575961339495435011888324169,\n a[9,6]=7632051964154290925661849 798370645637589377834346780/\n 1734087257418811583049800 347581865260479233950396659,\n a[9,7]=7519834791971137517048532179652 347729899303513750000/\n 1045677303502317596597890707812 349832637339039997351,\n a[9,8]=1366042683489166351293315549358278750 /\n 144631418224267718165055326464180836641,\n a[10,1] =-63077736705254280154824845013881/78369357853786633855112190394368,\n a[10,2]=0,a[10,3]=0,a[10,4]=-31948346510820970247215/695600921696002 6632192,\n a[10,5]=-3378604805394255292453489375/51704267056982469223 0499952,\n a[10,6]=1001587844183325981198091450220795/184232684207722 503701669953872896,\n a[10,7]=187023075231349900768014890274453125/25 224698849808178010752575653374848,\n a[10,8]=1908158550070998850625/1 17087067039189929394176,\n a[10,9]=-529568182881566682270449900773248 77908565/\n 2912779959477433986349822224412353951940 608,\n a[11,1]=-10116106591826909534781157993685116703/95628199450368 94030442231411871744000,\n a[11,2]=0,a[11,3]=0,a[11,4]=-9623541317323 077848129/3864449564977792573440,\n a[11,5]=-482334833314682940688137 5/576413233634141239944816,\n a[11,6]=6566119246514996884067001154977 284529/970305487021846325473990863582315520,\n a[11,7]=22264551305192 13549256016892506730559375/364880443159675255577435648380047355776,\n \+ a[11,8]=39747262782380466933662225/1756032802431424164410720256,\n a [11,9]=48175771419260955335244683805171548038966866545122229/\n \+ 1989786420513815146528880165952064118903852843612160000,\n a[11, 10]=-2378292068163246/47768728487211875,\n a[12,1]=-32180221747585998 31659045535578571/1453396753634469525663775847094384,\n a[12,2]=0,a[1 2,3]=0,a[12,4]=26290092604284231996745/5760876126062860430544,\n a[12 ,5]=-697069297560926452045586710000/41107967755245430594036502319,\n \+ a[12,6]=1827357820434213461438077550902273440/139381013914245317709567 680839641697,\n a[12,7]=643504802814241550941949227194107500000/24212 4609118836550860494007545333945331,\n a[12,8]=16225993815138026611375 0/59091082835244183497007,\n a[12,9]=-2302825163287352381854541485685 7015616678575554130463402/\n 2001316918319144450344390524 0405603349978424504151629055,\n a[12,10]=7958341351371843889152/32844 67988443203581305,\n a[12,11]=-507974327957860843878400/1215556548191 79042718967,\n a[13,1]=4631674879841/103782082379976,a[13,2]=0,a[13,3 ]=0,a[13,4]=0,a[13,5]=0,\n a[13,6]=14327219974204125/4048956682793321 6,\n a[13,7]=2720762324010009765625000/10917367480696813922225349,\n \+ a[13,8]=-498533005859375/95352091037424,\n a[13,9]=40593203046377724 7926705030596175437402459637909765779/\n 788039194363218 41083201886041201537229769115088303952,\n a[13,10]=-10290327637248/10 82076946951,\n a[13,11]=863264105888000/85814662253313,\n a[13,12]=- 29746300739/247142463456\}:" }}}{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/$\"++++]i!#6F(%!GF+F+F +F+F+F+F+F+F+F+7/$\"+#>V;0\"!#5$\"+!p/)o;F*$\"+MsiZ))F*F+F+F+F+F+F+F+F +F+F+7/$\"+*ykud\"F/$\"+s>mVRF*$\"\"!F:$\"+#f)4$=\"F/F+F+F+F+F+F+F+F+F +7/$\"+++++RF/$\"+m3cmPF/F9$!+&HwiS\"!\"*$\"+4-i>9FDF+F+F+F+F+F+F+F+7/ $\"+nmmmYF/$\"+A\\b$y%F*F9F9$\"+unkWBF/$\"++WmV=F/F+F+F+F+F+F+F+7/$\"+ +++g:F/$\"+rDa9hF*F9F9$\"+@JmH6F/$!+AEJ_OF*$\"+U)Q6%=F*F+F+F+F+F+F+7/$ \"+++++'*F/$!+M!>!H6FDF9F9$!+++++9FD$!+qGH!)))FD$\"+(4W48(FD$\"+3yOQ_F DF+F+F+F+F+7/$\"+;EnA*)F/$!+K`$H:'F/F9F9$!+jI(=&\\FD$!+#)obU^FD$\"+wD> ,WFD$\"+rJN\">(FD$\"+gE*\\W*!#7F+F+F+F+7/$\"+nmmm\"*F/$!+&\\v([!)F/F9F 9$!+%>8Hf%FD$!+V'zW`'FD$\"+;s`OaFD$\"+oPG9uFD$\"+))>pH;F*$!+&4&3==F*F+ F+F+7/$\"+++++&*F/$!+1\"ey0\"FDF9F9$!+(yu-\\#FD$!+k`'yO)FD$\"+9R1nnFD$ \"+ed(=5'FD$\"+m#pME#F*$\"+!*G:@CF*$!+/Owy\\F*F+F+7/$\"\"\"F:$!+@)QT@# FDF9F9$\"+Z!eNc%FD$!+@Oq&p\"!\")$\"+z@068F_s$\"+ZAudEFD$\"+T$Hfu#FD$!+ A\\l]6FD$\"+hH-BCFD$!+TW%*yTFDF+7/Fgr$\"+(*[)GY%F*F9F9F9F9$\"+Gn\\QNF/ $\"+y89#\\#F/$!+r%Q$G_FD$\"+#\\l6:&FD$!+sGz4&*FD$\"+9M'f+\"F_s$!+i%4O? \"F/Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "cal culation for stage 14" }}{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\" coefficient s of the weight polynomials for a given stage (corresponding to an \"a pproximate\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO6_13 := SimpleOrderCondit ions(6,13,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "whch := [1,2,3,6,7,8,12,15,16,24,2 7,31]:\nordeqns1 := []:\nfor ct in whch do\n eqn_group := convert(SO 6_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 coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns1 := []:\nfor ct to nops(ordeq ns1) do\n eqns1 := [op(eqns1),expand(subs(e1,ordeqns1[ct]))];\nend d o:\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 the system of equ ations 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(eqns1)\},in dets(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 " and \+ " }{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 := `union`(dd,simpl ify(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#/&%\"cG 6#\"#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 minimze the 2-norm of the linking coefficients for this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "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..-292.93,0..0.2);" }}{PARA 13 "" 1 "" {GLPLOT2D 420 359 359 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!$$H\"\"!$\"37;d))zE_t;!#=7$$!3<$3#*>u %)*HH!#:$\"3kd**Q(>$4&e\"F-7$$!3@z43m9(*HHF1$\"3?(*4*\\3?;^\"F-7$$!3fe /$f`c*HHF1$\"3)p%)f2@[GV\"F-7$$!3K3K\"o]T*HHF1$\"3G*)*z#ftnd8F-7$$!3!* Gn7\\l#*HHF1$\"3!4<808rpG\"F-7$$!3qXmV\"o7*HHF1$\"3N'pWL&z2D7F-7$$!3l7 oCA$)*)HHF1$\"3;yKZTFqk6F-7$$!3,Y@&>Z$))HHF1$\"3Pzb'QwMi5\"F-7$$!3MiDG p'o)HHF1$\"3E7b[AQ(>0\"F-7$$!3$oT7$$!3$)\\sZL\\#)HHF1$\"3/\")pjZy*4:*F`o7$$!3-] PVt(4)HHF1$\"3S$[_Hd;$e()F`o7$$!3-]F#R;&zHHF1$\"3v4W[k@s>%)F`o7$$!3\"e 9(3(*=yHHF1$\"3M5@5$\\7h9)F`o7$$!3o;k`@hwHHF1$\"338;%y8>F'yF`o7$$!3#pm vvv_(HHF1$\"3_i')*o/3$ewF`o7$$!3O7B67stHHF1$\"3)QF>%oJohuF`o7$$!3smJu^ MsHHF1$\"3I6ZGPWcCtF`o7$$!3K7tVa$3(HHF1$\"36$Q\\=J>S@(F`o7$$!3JP>ByRpH HF1$\"3?[>EH_bZrF`o7$$!3*=zp\"y*y'HHF1$\"3hiBh\\\"e&=rF`o7$$!3@H-V._mH HF1$\"3s$z>'3hAGrF`o7$$!3L3F^X.lHHF1$\"39>.>/9hxrF`o7$$!3iX@J7\\jHHF1$ \"31B/Gi\"=fHHF1$\"3El6hA\"**Hw(F`o7$$!3=D;.8tdHHF1 $\"3;\"HKMa_\"3!)F`o7$$!3a7`1CJcHHF1$\"3o=p(G_TGG)F`o7$$!3.vtjptaHHF1$ \"3\\^wI)\\N5j)F`o7$$!3dmJe8K`HHF1$\"3+@Zbdln#)*)F`o7$$!3G]P(*)4=&HHF1 $\"3Yqd%zjT')R*F`o7$$!3nX6*GS/&HHF1$\"3Zw5I3>q6)*F`o7$$!30]_XH%*[HHF1$ \"3(G7\\7kf-.\"F-7$$!3)*y*=1Mv%HHF1$\"3O#zUQZ)>!3\"F-7$$!3Wibs81YHHF1$ \"3J<9U()\\GO6F-7$$!3%pT&\\:iWHHF1$\"3;mI#y\"*f\\>\"F-7$$!3V7G)=9J%HHF 1$\"3=GpDK_Xg7F-7$$!3ULB?CmTHHF1$\"34LBJ[*ouK\"F-7$$!3i3sxx!fZ\"F-7$$!39+0$\\_t$HHF1$\"3Q $Qn9Kb\"\\:F-7$$!3xT+w=!e$HHF1$\"36UhQb[BP;F-7$$!3RLj&)\\TMHHF1$\"3@;v <@\\u>F-7$$!32+++++IHHF1$\"3[b$>mfhe+#F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F* Fa[l-%+AXESLABELSG6$Q'd[1,6]6\"Q!Ff[l-%%VIEWG6$;F($!&$HH!\"#;Fa[l$\"\" #F`[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 44.000000 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$$\"0['*>V.x6(!#;<#7$<#/&%\"d G6$\"\"\"\"\"'$!0DbI#fnHH!#7F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "-292.967592305525;\nconvert( %,rational,8);\nevalf[20](eval(sm,d[1,6]=%));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0DbI#fnHH!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!&*> a\"$&=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5%)>UPEP!4x6(!#@" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "d[1,6]=-54199/185" "6#/&%\"dG6$\"\"\"\"\"',$*&\"&*>aF' \"$&=!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "dd := \{d[1,1]=1,d[1,2]=0,d [1,3]=0,d[1,4]=0,d[1,5]=0,d[1,6]=-54199/185\}:\nd_1 := `union`(dd,subs (dd,d1)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 67 "subs(d_1,matrix([seq([seq(d[j,i],j=1..13)],i=1..6)] )):\nevalf[6](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7(7/$ \"\"\"\"\"!$F*F*F+F+F+F+F+F+F+F+F+F+F+7/F+F+F+F+F+$\"'!QR%!\"%$!'Y55F/ $\"'baG!\"\"$!'P#>\"F4$\"'!=)GF4$!'K[XF4$\"']Fr!\"&$\"'KhBF=7/F+F+F+F+ F+$!'(F4$\"'c\"*HF4$!'$RD(F4$\"'8Z6F*$!'$QZ&F/$ !'b2>F/7/F+F+F+F+F+$\"'!R,\"!\"$$!'0c:F/$\"'(e'*)F4$!'17PF4$\"'BP!*F4$ !'%=V\"F*$\"'0w8FT$\"')>/&F/7/F+F+F+F+F+$!'$=W&F/$\"'/?vF=$!'CsaF4$\"' neAF4$!'!=_&F4$\"'\\f()F4$!'E-9FT$!'&eS&F/7/$!'oHHFTF+F+F+F+$!'I\"R#! \"#$\"'$o;*FT$!'V59F)$\"'x4fF*$!'SD9F)$\"'aYAF)$\"'A6]F/$\"'GN?F/Q)ppr int456\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can check which of the groups of order conditions are satisfied ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 318 "nm := NULL:\nfor ct to nops(SO6_13) do\n eqn_group := convert(SO6_13[ct],'polynom_order_co nditions',6):\n tt := expand(subs(\{op(e1),op(d1)\},eqn_group));\n \+ tt := map(_Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op(i,tt),i=1. .nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%]);\nop(\{ seq(i,i=1..32)\} minus \{nm\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6@\" \"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#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 " Evaluate 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 t he next 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 [14]=1/2-7^(1/2)/14\},d[j,i]*c[14]^i)),i=1..6),j=1..13)\}:\ne2 := `uni on`(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 4917 "e2 := \{a[7,2] = 0, a[14,13] = 3/ 392-3/392*7^(1/2), a[14,2] = 0, a[14,6] = -142489509197785007255/11337 07871182130048+996623278509306786175/20973595616869405888*7^(1/2), c[1 4] = 1/2-1/14*7^(1/2), a[10,7] = 187023075231349900768014890274453125/ 25224698849808178010752575653374848, a[10,8] = 1908158550070998850625/ 117087067039189929394176, a[10,9] = -529568182881566682270449900773248 77908565/2912779959477433986349822224412353951940608, a[11,1] = -10116 106591826909534781157993685116703/956281994503689403044223141187174400 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = -9623541317323077848129/3864449 564977792573440, a[11,5] = -4823348333146829406881375/5764132336341412 39944816, a[11,6] = 6566119246514996884067001154977284529/970305487021 846325473990863582315520, a[11,7] = 2226455130519213549256016892506730 559375/364880443159675255577435648380047355776, a[11,8] = 397472627823 80466933662225/1756032802431424164410720256, a[11,9] = 481757714192609 55335244683805171548038966866545122229/1989786420513815146528880165952 064118903852843612160000, a[11,10] = -2378292068163246/477687284872118 75, a[12,1] = -3218022174758599831659045535578571/14533967536344695256 63775847094384, a[12,2] = 0, a[12,3] = 0, a[12,4] = 262900926042842319 96745/5760876126062860430544, a[12,5] = -69706929756092645204558671000 0/41107967755245430594036502319, a[14,1] = -26242/1715+588937/101528*7 ^(1/2), a[14,12] = 1100613127343/48439922837376-29746300739/1424703612 864*7^(1/2), a[12,6] = 1827357820434213461438077550902273440/139381013 914245317709567680839641697, a[12,7] = 6435048028142415509419492271941 07500000/242124609118836550860494007545333945331, a[12,8] = 1622599381 51380266113750/59091082835244183497007, a[12,9] = -2302825163287352381 8545414856857015616678575554130463402/20013169183191444503443905240405 603349978424504151629055, a[12,10] = 7958341351371843889152/3284467988 443203581305, a[12,11] = -507974327957860843878400/1215556548191790427 18967, a[13,1] = 4631674879841/103782082379976, a[13,2] = 0, a[14,7] = 663073659206904988750000000/13716692475747279030488259-92274454250510 62363134765625/507517621602649324128065583*7^(1/2), a[8,2] = 0, a[14,4 ] = 0, a[14,9] = 17318895166382455115817006657422200600438218265372153 /554375052657198964316171246174366066645917060800-27496808765423852349 0991092528596412316293248643447707/23287013241323869436363405111359694 599520551359840*7^(1/2), a[14,10] = -9081365921021033056/1205040236377 25+25400013053688208472/891729774919165*7^(1/2), a[14,3] = 0, a[14,5] \+ = 0, c[2] = 1/16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] \+ = 7/15, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[3,1] = 1 8928/1134225, a[3,2] = 100352/1134225, a[13,3] = 0, a[13,4] = 0, a[13, 5] = 0, a[13,6] = 14327219974204125/40489566827933216, a[13,7] = 27207 62324010009765625000/10917367480696813922225349, a[13,8] = -4985330058 59375/95352091037424, a[13,9] = 40593203046377724792670503059617543740 2459637909765779/78803919436321841083201886041201537229769115088303952 , a[13,10] = -10290327637248/1082076946951, a[13,11] = 863264105888000 /85814662253313, a[13,12] = -29746300739/247142463456, a[14,11] = 2628 8142988644492320/221311497390123-367631244295366501400/818852540343455 1*7^(1/2), a[4,2] = 0, a[4,1] = 14/355, a[4,3] = 42/355, a[5,1] = 9449 5479/250880000, a[5,2] = 0, a[5,3] = -352806597/250880000, a[5,4] = 17 8077159/125440000, a[6,1] = 12089/252720, a[6,2] = 0, a[6,3] = 0, a[6, 4] = 2505377/10685520, a[6,5] = 960400/5209191, a[7,1] = 21400899/3500 00000, a[7,3] = 0, a[7,4] = 3064329829899/27126050000000, a[7,5] = -21 643947/592609375, a[7,6] = 124391943/6756250000, a[8,1] = -15365458811 /13609565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -8339128164608/93906 0038475, a[8,6] = 341936800488/47951126225, a[8,7] = 1993321838240/380 523459069, a[9,1] = -1840911252282376584438157336464708426954728061551 /2991923615171151921596253813483118262195533733898, a[9,3] = 0, a[9,4] = -14764960804048657303638372252908780219281424435/298169210256502197 5611711269209606363661854518, a[9,5] = -875325048502130441118613421785 266742862694404520560000/170212030428894418395571677575961339495435011 888324169, a[9,6] = 76320519641542909256618497983706456375893778343467 80/1734087257418811583049800347581865260479233950396659, a[9,7] = 7519 834791971137517048532179652347729899303513750000/104567730350231759659 7890707812349832637339039997351, a[9,8] = 1366042683489166351293315549 358278750/144631418224267718165055326464180836641, a[10,1] = -63077736 705254280154824845013881/78369357853786633855112190394368, a[10,2] = 0 , a[9,2] = 0, a[14,8] = -16848658354789365625/225946578416832+22737651 361282524171875/806742258237298656*7^(1/2), a[10,4] = -319483465108209 70247215/6956009216960026632192, a[10,5] = -33786048053942552924534893 75/517042670569824692230499952, a[10,6] = 1001587844183325981198091450 220795/184232684207722503701669953872896, a[10,3] = 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 52 "seq(a[14,i]=subs(e2,a[14,i]),i=1..13):\neva lf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6//&%\"aG6$\"#9\"\"\"$\"F( *GMlkot[!)GYu\"*>QoBF+/&F%6$F'\"\")$\"C?/!\\a1*y\"[9eDRT?E\\ %!#N/&F%6$F'\"\"*$!C6vGf>u7c9K1*pyDTK(FQ/&F%6$F'\"#5$\"C)fm;PN&)>\"*[l `?X;R7\"FQ/&F%6$F'\"#6$\"Bo#[9FB'[\\`>9Z%f=Aa!#M/&F%6$F'\"#7$!I54O6%y. XmXq1>dR2HG>D$!#T/&F%6$F'\"#8$!ILD;U)>XYK5Coi;'QaN]f7Feo" }}}{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 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$!'UIs!\"&F+F +F+F+$\"'>PVF/$\"'53:!\"%$!'/*p'!\"$$\"'=!p#F7$!'O4mF7$\"'wh5!\"#F+$\" '3dA!\"'$!'oM7F470$\"'J^CF4F+F+F+F+$!'XpCF4$!'VwwF4$\"'1BxF>$!'4XJF>$ \"'$!'vA7!\"\"$!'m@sF/$!'1Q[F/$\"'AJ#)F470$!'m\"H%F4F+F+F+F+$\"'k %\\%F4$\"'=o:F7$!'n6IFS$\"'YN7FS$!'I(*HFS$\"'CrZFS$\"'?pIF4$\"'cQ=F4$! '-_=F770$\"'V6RF4F+F+F+F+$!&yX*F4$!'$\"'#=H\"FS$!':O?FSF+F+F +Q)pprint486\"" }}}{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:\n for 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_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([%] );\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 47 "Eval uate the \"weight\" polynomials at the node " }{XPPEDIT 18 0 "c[15]=1 187/2500" "6#/&%\"cG6#\"#:*&\"%(=\"\"\"\"\"%+D!\"\"" }{TEXT -1 83 " t o obtain the linking coefficients in the next stage in the interpolati on scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "eqs15 := \{se q(a[15,j]=add(expand(subs(\{op(d2),c[15]=1187/2500\},d[j,i]*c[15]^i)), i=1..7),j=1..14)\}:\ne3 := `union`(eqs15,\{c[15]=1187/2500\},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 6412 "e3 := \{a[ 7,2] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[15,12] = 12163051029345238 407733418657/3016875774609375000000000000000, a[15,13] = -579945478542 537523/549316406250000000000+3727856565327672037/219726562500000000000 0*7^(1/2), a[14,2] = 0, a[15,14] = 26094995957293704259/54931640625000 0000000*7^(1/2), a[14,6] = -142489509197785007255/1133707871182130048+ 996623278509306786175/20973595616869405888*7^(1/2), c[14] = 1/2-1/14*7 ^(1/2), a[15,2] = 0, a[10,7] = 187023075231349900768014890274453125/25 224698849808178010752575653374848, a[10,8] = 1908158550070998850625/11 7087067039189929394176, a[10,9] = -52956818288156668227044990077324877 908565/2912779959477433986349822224412353951940608, a[11,1] = -1011610 6591826909534781157993685116703/9562819945036894030442231411871744000, a[11,2] = 0, a[11,3] = 0, a[11,4] = -9623541317323077848129/386444956 4977792573440, a[11,5] = -4823348333146829406881375/576413233634141239 944816, a[11,6] = 6566119246514996884067001154977284529/97030548702184 6325473990863582315520, a[11,7] = 222645513051921354925601689250673055 9375/364880443159675255577435648380047355776, a[11,8] = 39747262782380 466933662225/1756032802431424164410720256, a[11,9] = 48175771419260955 335244683805171548038966866545122229/198978642051381514652888016595206 4118903852843612160000, a[11,10] = -2378292068163246/47768728487211875 , a[12,1] = -3218022174758599831659045535578571/1453396753634469525663 775847094384, a[12,2] = 0, a[12,3] = 0, a[12,4] = 26290092604284231996 745/5760876126062860430544, a[12,5] = -697069297560926452045586710000/ 41107967755245430594036502319, a[14,1] = -26242/1715+588937/101528*7^( 1/2), a[15,3] = 0, a[14,12] = 1100613127343/48439922837376-29746300739 /1424703612864*7^(1/2), a[12,6] = 182735782043421346143807755090227344 0/139381013914245317709567680839641697, a[12,7] = 64350480281424155094 1949227194107500000/242124609118836550860494007545333945331, a[12,8] = 162259938151380266113750/59091082835244183497007, a[12,9] = -23028251 632873523818545414856857015616678575554130463402/200131691831914445034 43905240405603349978424504151629055, a[12,10] = 7958341351371843889152 /3284467988443203581305, a[12,11] = -507974327957860843878400/12155565 4819179042718967, a[13,1] = 4631674879841/103782082379976, a[13,2] = 0 , a[15,8] = 16161913072172934315785836819/9932509483065000000000000000 -75980668436324671626317237501/49917227145660000000000000000*7^(1/2), \+ a[15,4] = 0, a[14,7] = 663073659206904988750000000/1371669247574727903 0488259-9227445425051062363134765625/507517621602649324128065583*7^(1/ 2), a[8,2] = 0, a[14,4] = 0, a[15,5] = 0, a[14,9] = 173188951663824551 15817006657422200600438218265372153/5543750526571989643161712461743660 66645917060800-274968087654238523490991092528596412316293248643447707/ 23287013241323869436363405111359694599520551359840*7^(1/2), a[14,10] = -9081365921021033056/120504023637725+25400013053688208472/89172977491 9165*7^(1/2), a[15,6] = 3353943190402140976803568136084793/19770296302 701765625000000000000000-255748988917794999289135717669191/86495046324 32022460937500000000000*7^(1/2), a[15,11] = -2079418008367563117920929 15907/81839239362061500549316406250+10891006522099224595926000559657/4 582997404275444030761718750000*7^(1/2), a[15,10] = 1446205909971003323 5500874587/10319489926824569702148437500-34815528884845218754039633745 7/252574928278923034667968750000*7^(1/2), a[14,3] = 0, a[15,9] = -3727 850143078386229058076561522609367827766330696417108435354320431359/801 6349226513859159668160608032382937598583484731440429687500000000000+81 4251349796512127448124174135016152994969999173763717229208919303/15898 42118264019833619870377392538080518560957031250000000000000000*7^(1/2) , a[14,5] = 0, c[2] = 1/16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/ 100, c[6] = 7/15, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/1617886 1, c[10] = 11/12, c[11] = 19/20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, \+ a[3,1] = 18928/1134225, a[3,2] = 100352/1134225, a[13,3] = 0, a[13,4] \+ = 0, a[13,5] = 0, a[13,6] = 14327219974204125/40489566827933216, a[13, 7] = 2720762324010009765625000/10917367480696813922225349, a[13,8] = - 498533005859375/95352091037424, a[13,9] = 4059320304637772479267050305 96175437402459637909765779/7880391943632184108320188604120153722976911 5088303952, a[13,10] = -10290327637248/1082076946951, a[13,11] = 86326 4105888000/85814662253313, a[13,12] = -29746300739/247142463456, a[14, 11] = 26288142988644492320/221311497390123-367631244295366501400/81885 25403434551*7^(1/2), a[4,2] = 0, a[4,1] = 14/355, a[4,3] = 42/355, a[5 ,1] = 94495479/250880000, a[5,2] = 0, a[5,3] = -352806597/250880000, a [5,4] = 178077159/125440000, a[6,1] = 12089/252720, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a[7,1] = 214 00899/350000000, a[7,3] = 0, a[7,4] = 3064329829899/27126050000000, a[ 7,5] = -21643947/592609375, a[7,6] = 124391943/6756250000, a[15,7] = 1 0596614540600968744792362758669/45489031169570058009272287500000-13058 425932946007192467094947/1451071902055187778276093750000*7^(1/2), a[8, 1] = -15365458811/13609565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -83 39128164608/939060038475, a[8,6] = 341936800488/47951126225, a[8,7] = \+ 1993321838240/380523459069, a[9,1] = -18409112522823765844381573364647 08426954728061551/2991923615171151921596253813483118262195533733898, a [9,3] = 0, a[9,4] = -14764960804048657303638372252908780219281424435/2 981692102565021975611711269209606363661854518, a[9,5] = -8753250485021 30441118613421785266742862694404520560000/1702120304288944183955716775 75961339495435011888324169, a[9,6] = 763205196415429092566184979837064 5637589377834346780/17340872574188115830498003475818652604792339503966 59, a[9,7] = 7519834791971137517048532179652347729899303513750000/1045 677303502317596597890707812349832637339039997351, a[9,8] = 13660426834 89166351293315549358278750/144631418224267718165055326464180836641, a[ 10,1] = -63077736705254280154824845013881/7836935785378663385511219039 4368, a[10,2] = 0, a[9,2] = 0, a[14,8] = -16848658354789365625/2259465 78416832+22737651361282524171875/806742258237298656*7^(1/2), c[15] = 1 187/2500, a[10,4] = -31948346510820970247215/6956009216960026632192, a [10,5] = -3378604805394255292453489375/517042670569824692230499952, a[ 10,6] = 1001587844183325981198091450220795/184232684207722503701669953 872896, a[10,3] = 0, a[15,1] = 130946139152859396534950567713097/28152 69161783203125000000000000000+90670944595916412828478989403/6614898170 3437500000000000000000*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[15,i]=subs(e3,a[15,i]),i=1..14):\nevalf[40](%);" }}{PARA 12 "" 1 " " {XPPMATH 20 "60/&%\"aG6$\"#:\"\"\"$\"I$oS.U0Po*eL%zw*)\\\")QQR,&!#T/ &F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6 $F'\"\"'$\"IL,q'f7Z)Q(pTYj&=79ieT\"*F+/&F%6$F'\"\"($\"I3D9h)*\\!3[??sk s@x\\#R\"4#!#S/&F%6$F'\"\")$!Io?-\"pOe%R`Yw=ki!Hr7+S#!#R/&F%6$F'\"\"*$ \"It%R%[N&)))=BivrA+v=W8+*)FJ/&F%6$F'\"#5$!Iyj:b'Hs^+4F@*)\\:g&\\`XAFQ /&F%6$F'\"#6$\"I3$enY\\2Z-z9;mG(R)***[YPFQ/&F%6$F'\"#7$\"I/K-:&\\iY(=q )fkd&*=:r;.%!#U/&F%6$F'\"#8$\"IJ$=eF(G0n!y-uY_Lr*Q*HV$Fdo/&F%6$F'\"#9$ \"I6p+:#e_l.q+C$\\b0y1&oD\"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 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$!'\"[R(!\"&F+F+F+F+$!&Up#!\"%$\"']3;F 2$!'vB8!\"$$\"'ib[F2$!'i!R\"F7$\"'etAF7$!'PsSF/$!'92:F/$!'3f9F2$\"',)p *F/71$\"'KYEF2F+F+F+F+$\"&A(eF7$!'Xn))F2$\"'1Y8!\"#$!'E'H&F7$\"'*[Y\"F O$!'^GBFO$\"'44TF2$\"'(>d\"F2$\"'L*3\"F7$!'`]6F771$!'Y`]F2F+F+F+F+$!'5 4GF7$\"'WL?F7$!'w0_FO$\"'oP@FO$!'x(y&FO$\"'nV!*FO$!'L!e\"F7$!'-#>'F2$! '%>*GF7$\"'P%\\%F771$\"'`F_F2F+F+F+F+$\"')\\`&F7$!''zI#F7$\"'7e%*FO$!' CBSFO$\"'&*p5!\"\"$!'d[;F\\q$\"'0cGF7$\"'yX6F7$\"':DNF7$!'dkxF771$!'J^ FF2F+F+F+F+$!'4K[F7$\"'yv7F7$!')z0)FO$\"'CXNFO$!'hm#*FO$\"'!)39F\\q$!' `>CF7$!''Q(**F2$!'E-?F7$\"'CjhF771$\"'j[dF/F+F+F+F+$\"'Y\\:F7$!'sGFF2$ \"'o'e#FO$!'Xt6FO$\"'Z>IFO$!'$[`%FO$\"'KCxF2$\"'$oG$F2$\"'GcUF2$!']R=F 7Q)pprint516\"" }}}{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:\n for ct to nops(SO7_15) do\n eqn_group := convert(SO7_15[ct],'polynom _order_conditions',7):\n tt := expand(subs(\{op(e3),op(d3)\},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([%] );\nop(\{seq(i,i=1..64)\} minus \{nm\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6R\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\" #9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I\"#J \"#K\"#O\"#R\"#V\"#Y\"#Z\"#\\\"#^\"#`\"#a\"#c\"#d\"#f\"#g\"#i\"#j\"#k " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"#L\"#M\"#N\"#P\"#Q\"#S\"#T\"#U\"#W\"#X\"#[\"#]\"#_\"#b\"#e\"#h " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Eval uate the \"weight\" polynomials at the node " }{XPPEDIT 18 0 "c[16] = 277/2500" "6#/&%\"cG6#\"#;*&\"$x#\"\"\"\"%+D!\"\"" }{TEXT -1 83 " to obtain the linking coefficients in the next stage in the interpolatio n scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "eqs16 := \{seq (a[16,j]=add(expand(subs(\{op(d3),c[16]=277/2500\},d[j,i]*c[16]^i)),i= 1..7),j=1..15)\}:\ne4 := `union`(eqs16,\{c[16]=277/2500\},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 10025 "e4 := \{a [16,1] = 1013856404118125952763527205161498788180173939254661359435974 309585825318269273/198564667739239660374389171210942338307459063268324 46413024679687500000000000000-1931754243710985131731267585181486489297 59153027347863824763410934840962491/2778932721896691403250310788177330 67056972748113814414737187500000000000000000*7^(1/2), a[16,6] = -19039 8901498664954325849292034437666076647895943482485736813399594207394699 3/26105498804470984484513299908841941049181178531837828644206625000000 000000000+289290358369928611582932131505420731599890246264645126636225 54495682276029/8922777911684418524980131804779960319544348130999257837 37531127929687500000*7^(1/2), a[7,2] = 0, c[16] = 277/2500, a[16,7] = \+ 1471649975107913251117693334592045704450918568535357968485718078648304 2792373/16088988337674671465702156595052532701656799236967006367882563 9487979375000000+35097161131542028791945288573990002288024169106005874 1934303404427219789/74868036898960959337075562503093457016256710599371 634570393697387568750000*7^(1/2), a[14,13] = 3/392-3/392*7^(1/2), a[15 ,12] = 12163051029345238407733418657/3016875774609375000000000000000, \+ a[16,9] = 207904342034969170900628672444919837194509196335848531194132 84106748179318736252691355190285977916572557692108955233/1587766758359 4659668953564588772545305777807317441893201879580358667168054523455849 3166228541026757812500000000000000+16750291917128487119349998833576560 4001116436018919123161320355297232333887892701752121887704693497136971 8057209/73181019315333410819868281232202677952556148952818602411955267 4967375646957459693466445168584156250000000000000000*7^(1/2), a[15,13] = -579945478542537523/549316406250000000000+3727856565327672037/21972 65625000000000000*7^(1/2), a[14,2] = 0, a[16,12] = -124817112626650602 913170312644481143348777609285630839757851632016681433/398360478373506 9658159427207807518356647862336983160384375000000000000000+19488568273 447271249669733067688159410742449986073234566291555857980323/290471182 1473488292407915672359648801722399620716887780273437500000000000*7^(1/ 2), a[16,11] = 3307587237786214580372394397330733034806289881443017455 732775116590947887/216127682925333604166493245606691739590970049710504 0226143341064453125000-15924948134835543062332142541253213652166685490 3188329515308104047100201/53083992297450358918086060324450602706554047 2973167774842224121093750000*7^(1/2), a[16,13] = -46330543544553540874 179448395431083082899647522368296822645347/508745135208312439739125075 9096940921758056970520019531250000000+18909196831867663116267125639920 17565381198275789277564189007889/1139589102866619865015640170037714766 473804761396484375000000000000*7^(1/2), a[15,14] = 2609499595729370425 9/549316406250000000000*7^(1/2), a[14,6] = -142489509197785007255/1133 707871182130048+996623278509306786175/20973595616869405888*7^(1/2), a[ 16,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[15,2] = 0, a[10,7] = 1870230752 31349900768014890274453125/25224698849808178010752575653374848, a[10,8 ] = 1908158550070998850625/117087067039189929394176, a[10,9] = -529568 18288156668227044990077324877908565/2912779959477433986349822224412353 951940608, a[11,1] = -10116106591826909534781157993685116703/956281994 5036894030442231411871744000, a[11,2] = 0, a[11,3] = 0, a[11,4] = -962 3541317323077848129/3864449564977792573440, a[11,5] = -482334833314682 9406881375/576413233634141239944816, a[11,6] = 65661192465149968840670 01154977284529/970305487021846325473990863582315520, a[11,7] = 2226455 130519213549256016892506730559375/364880443159675255577435648380047355 776, a[11,8] = 39747262782380466933662225/1756032802431424164410720256 , a[11,9] = 48175771419260955335244683805171548038966866545122229/1989 786420513815146528880165952064118903852843612160000, a[11,10] = -23782 92068163246/47768728487211875, a[12,1] = -3218022174758599831659045535 578571/1453396753634469525663775847094384, a[12,2] = 0, a[12,3] = 0, a [12,4] = 26290092604284231996745/5760876126062860430544, a[12,5] = -69 7069297560926452045586710000/41107967755245430594036502319, a[14,1] = \+ -26242/1715+588937/101528*7^(1/2), a[15,3] = 0, a[16,14] = 37807079247 0829109357504373975352653612596622387047615352035143989/38652465726311 665474213406887371187527810692671413726806640625000000-736694485119203 754900544593199058955707390426548138628337098463663/247375780648394659 03496580407917560017798843309704785156250000000000*7^(1/2), a[14,12] = 1100613127343/48439922837376-29746300739/1424703612864*7^(1/2), a[12, 6] = 1827357820434213461438077550902273440/139381013914245317709567680 839641697, a[12,7] = 643504802814241550941949227194107500000/242124609 118836550860494007545333945331, a[12,8] = 162259938151380266113750/590 91082835244183497007, a[12,9] = -2302825163287352381854541485685701561 6678575554130463402/20013169183191444503443905240405603349978424504151 629055, a[12,10] = 7958341351371843889152/3284467988443203581305, a[12 ,11] = -507974327957860843878400/121555654819179042718967, a[13,1] = 4 631674879841/103782082379976, a[13,2] = 0, a[15,8] = 16161913072172934 315785836819/9932509483065000000000000000-7598066843632467162631723750 1/49917227145660000000000000000*7^(1/2), a[15,4] = 0, a[16,15] = 46088 82876307429368775251443876403372924872509408621400283/7592227894936087 1648575860105318720457058558153486785997500-84914702986428741912331429 025972417475312941024930181294/531455952645526101540031020737231043199 4099070744075019825*7^(1/2), a[16,3] = 0, a[14,7] = 663073659206904988 750000000/13716692475747279030488259-9227445425051062363134765625/5075 17621602649324128065583*7^(1/2), a[8,2] = 0, a[14,4] = 0, a[15,5] = 0, a[14,9] = 17318895166382455115817006657422200600438218265372153/55437 5052657198964316171246174366066645917060800-27496808765423852349099109 2528596412316293248643447707/23287013241323869436363405111359694599520 551359840*7^(1/2), a[14,10] = -9081365921021033056/120504023637725+254 00013053688208472/891729774919165*7^(1/2), a[15,6] = 33539431904021409 76803568136084793/19770296302701765625000000000000000-2557489889177949 99289135717669191/8649504632432022460937500000000000*7^(1/2), a[15,11] = -207941800836756311792092915907/81839239362061500549316406250+10891 006522099224595926000559657/4582997404275444030761718750000*7^(1/2), a [15,10] = 14462059099710033235500874587/10319489926824569702148437500- 348155288848452187540396337457/252574928278923034667968750000*7^(1/2), a[16,8] = -2859135378249218370182625112630676639646122987466023691908 9505718702483/30738954422006868689408605302697869750592763266032965939 343750000000000+326217186457072779972278244440069764006022537683484345 700377402630409/174372289725344440302058311815240843915993136349259496 5600000000000000*7^(1/2), a[14,3] = 0, a[15,9] = -37278501430783862290 58076561522609367827766330696417108435354320431359/8016349226513859159 668160608032382937598583484731440429687500000000000+814251349796512127 448124174135016152994969999173763717229208919303/158984211826401983361 9870377392538080518560957031250000000000000000*7^(1/2), a[16,4] = 0, a [16,10] = -17520323069888524301525598935749031032957349045468285405735 16708228831279/2433262830102477113639615580994907630153630681641719830 799102783203125000+818839996665861726917359740393249069399644466975014 50842550606413648603/7337223303078238681128686982692336854001717132335 03210517883300781250000*7^(1/2), a[14,5] = 0, c[2] = 1/16, c[3] = 112/ 1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/20, c[12] \+ = 1, c[13] = 1, a[2,1] = 1/16, a[3,1] = 18928/1134225, a[3,2] = 100352 /1134225, a[13,3] = 0, a[13,4] = 0, a[13,5] = 0, a[13,6] = 14327219974 204125/40489566827933216, a[13,7] = 2720762324010009765625000/10917367 480696813922225349, a[13,8] = -498533005859375/95352091037424, a[13,9] = 405932030463777247926705030596175437402459637909765779/788039194363 21841083201886041201537229769115088303952, a[13,10] = -10290327637248/ 1082076946951, a[13,11] = 863264105888000/85814662253313, a[13,12] = - 29746300739/247142463456, a[14,11] = 26288142988644492320/221311497390 123-367631244295366501400/8188525403434551*7^(1/2), a[4,2] = 0, a[4,1] = 14/355, a[4,3] = 42/355, a[5,1] = 94495479/250880000, a[5,2] = 0, a [5,3] = -352806597/250880000, a[5,4] = 178077159/125440000, a[6,1] = 1 2089/252720, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a[7,1] = 21400899/350000000, a[7,3] = 0, a[7,4] = 3 064329829899/27126050000000, a[7,5] = -21643947/592609375, a[7,6] = 12 4391943/6756250000, a[15,7] = 10596614540600968744792362758669/4548903 1169570058009272287500000-13058425932946007192467094947/14510719020551 87778276093750000*7^(1/2), a[8,1] = -15365458811/13609565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8,6] = 3419 36800488/47951126225, a[8,7] = 1993321838240/380523459069, a[9,1] = -1 840911252282376584438157336464708426954728061551/299192361517115192159 6253813483118262195533733898, a[9,3] = 0, a[9,4] = -147649608040486573 03638372252908780219281424435/2981692102565021975611711269209606363661 854518, a[9,5] = -8753250485021304411186134217852667428626944045205600 00/170212030428894418395571677575961339495435011888324169, a[9,6] = 76 32051964154290925661849798370645637589377834346780/1734087257418811583 049800347581865260479233950396659, a[9,7] = 75198347919711375170485321 79652347729899303513750000/1045677303502317596597890707812349832637339 039997351, a[9,8] = 1366042683489166351293315549358278750/144631418224 267718165055326464180836641, a[10,1] = -630777367052542801548248450138 81/78369357853786633855112190394368, a[10,2] = 0, a[9,2] = 0, a[14,8] \+ = -16848658354789365625/225946578416832+22737651361282524171875/806742 258237298656*7^(1/2), a[16,5] = 0, c[15] = 1187/2500, a[10,4] = -31948 346510820970247215/6956009216960026632192, a[10,5] = -3378604805394255 292453489375/517042670569824692230499952, a[10,6] = 100158784418332598 1198091450220795/184232684207722503701669953872896, a[10,3] = 0, a[15, 1] = 130946139152859396534950567713097/2815269161783203125000000000000 000+90670944595916412828478989403/66148981703437500000000000000000*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$\"#;\"\"\"$\"IFunUTX#\\/M19(Gq@+\"3?#\\!#T/&F%6$F'\"\"#$\"\"!F1/&F %6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"Ieu\\`LXyf\" 3f!*)3S5#R(\\%G\"F+/&F%6$F'\"\"($\"IY3XY`dl;kJ*y+D/PHB(Q5!#S/&F%6$F'\" \")$!If8!yZQbg)QV)pQ0r+\"zk^VFJ/&F%6$F'\"\"*$\"Ia!f/J>&3/=s>v*p'H>=(*p 8FJ/&F%6$F'\"#5$!I/M>:P&o`d1TEhH$*)Qwu h^q!QysmtFJ/&F%6$F'\"#7$!I$>FqH\\+,SG%H!fWTQ*y:e8F+/&F%6$F'\"#8$!I:m&y ><&y0/x<4MSCtQt;Z!#U/&F%6$F'\"#9$!I$f_'4j)\\+%>])4Z2em_>5!pF+/&F%6$F' \"#:$\"I_/tllatnr*>BjsJWH7K%=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 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_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 know n coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns4 := []:\nfor ct to nops(ordeqns4) do\n eqns4 := [op(eqns4),expand(subs( e4,ordeqns4[ct]))];\nend do:\nnops(eqns4);\nnops(indets(eqns4));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$7\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "d4 := solve(\{op(eqns4)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs(d4,matrix([seq([seq(d[j,i],j=1..16)],i=1..7)])):\nevalf[6 ](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)72$\"\"\"\"\"!$ F*F*F+F+F+F+F+F+F+F+F+F+F+F+F+F+72$!'&H,\"!\"%F+F+F+F+$!'Q69F/$!'*>(=F /$!'5&R)F/$\"'m9VF/$!'nf5!\"$$\"'>,:F:$!'eyC!\"&$!'\"G3\"F?$\"'trw!\"' $\"'\">W\"F/$\"'2*z#F/72$\"'En\\F/F+F+F+F+$\"'Tc:F:$\"'9n?F:$\"'%3N*F: $!';P[F:$\"'+%=\"!\"#$!'+t;FV$\"'VcFF/$\"'$=@\"F/$!')49#F/$!'@^:F:$!'d vBF:72$!'+68F:F+F+F+F+$!'PthF:$!'B?#)F:$!';zPFV$\"'Jy>FV$!'z7[FV$\"'?o nFV$!'z56F:$!'(=%\\F/$\"'fK;F:$\"'A&)eF:$\"'@Y#)F:72$\"'?\")=F:F+F+F+F +$\"'w?6FV$\"'7)\\\"FV$\"'r_qFV$!'`aPFV$\"'fb!*FV$!'*[E\"!\"\"$\"'Mk?F :$\"'()\\$*F/$!'(Q5%F:$!'(4,\"FV$!'V!R\"FV72$!'qm8F:F+F+F+F+$!'M!R*F:$ !'ph7FV$!'4DhFV$\"'KHLFV$!'dXzFV$\"']+6Feq$!'O$y\"F:$!'1!G)F/$\"'+GTF: $\"'qZ!)F:$\"'G<6FV72$\"'@:RF/F+F+F+F+$\"'MWHF:$\"'VyRF:$\"'>&*>FV$!'Q 26FV$\"'A:EFV$!'R\"f$FV$\"'cxdF/$\"']oFF/$!'J]9F:$!';;CF:$!')*=MF:Q)pp rint526\"" }}}{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 0 285 "nm := NULL:\nfor 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_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[17] = 3971 /10000" "6#/&%\"cG6#\"#<*&\"%rR\"\"\"\"&++\"!\"\"" }{TEXT -1 83 " to \+ obtain the linking coefficients in the next stage in the interpolation scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "eqs17 := \{seq( a[17,j]=add(expand(subs(\{op(d4),c[17]=3971/10000\},d[j,i]*c[17]^i)),i =1..7),j=1..16)\}:\ne5 := `union`(eqs17,\{c[17]=3971/10000\},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 13121 "e5 := \{a [16,1] = 1013856404118125952763527205161498788180173939254661359435974 309585825318269273/198564667739239660374389171210942338307459063268324 46413024679687500000000000000-1931754243710985131731267585181486489297 59153027347863824763410934840962491/2778932721896691403250310788177330 67056972748113814414737187500000000000000000*7^(1/2), a[16,6] = -19039 8901498664954325849292034437666076647895943482485736813399594207394699 3/26105498804470984484513299908841941049181178531837828644206625000000 000000000+289290358369928611582932131505420731599890246264645126636225 54495682276029/8922777911684418524980131804779960319544348130999257837 37531127929687500000*7^(1/2), a[17,2] = 0, a[7,2] = 0, c[16] = 277/250 0, a[16,7] = 147164997510791325111769333459204570445091856853535796848 57180786483042792373/1608898833767467146570215659505253270165679923696 70063678825639487979375000000+3509716113154202879194528857399000228802 41691060058741934303404427219789/7486803689896095933707556250309345701 6256710599371634570393697387568750000*7^(1/2), a[14,13] = 3/392-3/392* 7^(1/2), a[15,12] = 12163051029345238407733418657/30168757746093750000 00000000000, a[16,9] = 20790434203496917090062867244491983719450919633 584853119413284106748179318736252691355190285977916572557692108955233/ 1587766758359465966895356458877254530577780731744189320187958035866716 80545234558493166228541026757812500000000000000+1675029191712848711934 9998833576560400111643601891912316132035529723233388789270175212188770 46934971369718057209/7318101931533341081986828123220267795255614895281 86024119552674967375646957459693466445168584156250000000000000000*7^(1 /2), a[15,13] = -579945478542537523/549316406250000000000+372785656532 7672037/2197265625000000000000*7^(1/2), a[14,2] = 0, a[17,12] = -11349 0358552214825884402065259188700089375815491942289/31252940536457380106 686585123539663360000000000000000000*7^(1/2)+1289068908207103177247139 1877143529539151923688844177161/10938529187760083037340304793238882176 00000000000000000000, a[17,5] = 0, a[16,12] = -12481711262665060291317 0312644481143348777609285630839757851632016681433/39836047837350696581 59427207807518356647862336983160384375000000000000000+1948856827344727 1249669733067688159410742449986073234566291555857980323/29047118214734 88292407915672359648801722399620716887780273437500000000000*7^(1/2), a [16,11] = 330758723778621458037239439733073303480628988144301745573277 5116590947887/21612768292533360416649324560669173959097004971050402261 43341064453125000-1592494813483554306233214254125321365216668549031883 29515308104047100201/5308399229745035891808606032445060270655404729731 67774842224121093750000*7^(1/2), a[16,13] = -4633054354455354087417944 8395431083082899647522368296822645347/50874513520831243973912507590969 40921758056970520019531250000000+1890919683186766311626712563992017565 381198275789277564189007889/113958910286661986501564017003771476647380 4761396484375000000000000*7^(1/2), a[15,14] = 26094995957293704259/549 316406250000000000*7^(1/2), a[14,6] = -142489509197785007255/113370787 1182130048+996623278509306786175/20973595616869405888*7^(1/2), a[16,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[15,2] = 0, a[17,13] = -9661716767957 01349259082624752343417581690261/1131534950955995511243121650862500000 000000000000*7^(1/2)+1942338020448191858221320768023291254935293468947 9/6119341014770023724802801887864400000000000000000000, a[10,7] = 1870 23075231349900768014890274453125/25224698849808178010752575653374848, \+ a[10,8] = 1908158550070998850625/117087067039189929394176, a[10,9] = - 52956818288156668227044990077324877908565/2912779959477433986349822224 412353951940608, a[11,1] = -10116106591826909534781157993685116703/956 2819945036894030442231411871744000, a[11,2] = 0, a[11,3] = 0, a[11,4] \+ = -9623541317323077848129/3864449564977792573440, a[11,5] = -482334833 3146829406881375/576413233634141239944816, a[11,6] = 65661192465149968 84067001154977284529/970305487021846325473990863582315520, a[11,7] = 2 226455130519213549256016892506730559375/364880443159675255577435648380 047355776, a[11,8] = 39747262782380466933662225/1756032802431424164410 720256, a[11,9] = 4817577141926095533524468380517154803896686654512222 9/1989786420513815146528880165952064118903852843612160000, a[11,10] = \+ -2378292068163246/47768728487211875, a[12,1] = -3218022174758599831659 045535578571/1453396753634469525663775847094384, a[12,2] = 0, a[12,3] \+ = 0, a[12,4] = 26290092604284231996745/5760876126062860430544, a[12,5] = -697069297560926452045586710000/41107967755245430594036502319, a[14 ,1] = -26242/1715+588937/101528*7^(1/2), a[15,3] = 0, a[16,14] = 37807 0792470829109357504373975352653612596622387047615352035143989/38652465 726311665474213406887371187527810692671413726806640625000000-736694485 119203754900544593199058955707390426548138628337098463663/247375780648 39465903496580407917560017798843309704785156250000000000*7^(1/2), a[14 ,12] = 1100613127343/48439922837376-29746300739/1424703612864*7^(1/2), a[17,9] = -2647017627964898493331644600569876862731826433080138765378 22807951183236955127319033004381711/2657419178388574677233558327633544 8051371609165162367507927820737503487973304320000000000000000*7^(1/2)- 1159377718029256387942578465291225079322378160773707689649770373007738 863712156809629044640218707/116262089054500142128968176833967585224750 790097585357847184215726577759883206400000000000000000000, a[12,6] = 1 827357820434213461438077550902273440/139381013914245317709567680839641 697, a[12,7] = 643504802814241550941949227194107500000/242124609118836 550860494007545333945331, a[12,8] = 162259938151380266113750/590910828 35244183497007, a[12,9] = -2302825163287352381854541485685701561667857 5554130463402/20013169183191444503443905240405603349978424504151629055 , a[12,10] = 7958341351371843889152/3284467988443203581305, a[12,11] = -507974327957860843878400/121555654819179042718967, a[13,1] = 4631674 879841/103782082379976, a[13,2] = 0, a[15,8] = 16161913072172934315785 836819/9932509483065000000000000000-75980668436324671626317237501/4991 7227145660000000000000000*7^(1/2), a[15,4] = 0, a[16,15] = 46088828763 07429368775251443876403372924872509408621400283/7592227894936087164857 5860105318720457058558153486785997500-84914702986428741912331429025972 417475312941024930181294/531455952645526101540031020737231043199409907 0744075019825*7^(1/2), a[16,3] = 0, a[17,14] = 15739031823875445778785 537999179335674270343962861810116243/309619125620788378533311524058377 597436400000000000000000000+606652050483146218029195891134085469196882 580940598209/12384765024831535141332460962335103897456000000000000000* 7^(1/2), a[14,7] = 663073659206904988750000000/13716692475747279030488 259-9227445425051062363134765625/507517621602649324128065583*7^(1/2), \+ a[8,2] = 0, a[14,4] = 0, a[15,5] = 0, a[14,9] = 1731889516638245511581 7006657422200600438218265372153/55437505265719896431617124617436606664 5917060800-274968087654238523490991092528596412316293248643447707/2328 7013241323869436363405111359694599520551359840*7^(1/2), a[17,8] = -264 3363415762169747333266167756151785333485317345751/27702383577536990068 404949009388454690816000000000000*7^(1/2)+2345191818743325882006422291 3608502147503898444041667/72026197301596174177852867424409982196121600 000000000, a[14,10] = -9081365921021033056/120504023637725+25400013053 688208472/891729774919165*7^(1/2), a[15,6] = 3353943190402140976803568 136084793/19770296302701765625000000000000000-255748988917794999289135 717669191/8649504632432022460937500000000000*7^(1/2), a[15,11] = -2079 41800836756311792092915907/81839239362061500549316406250+1089100652209 9224595926000559657/4582997404275444030761718750000*7^(1/2), a[15,10] \+ = 14462059099710033235500874587/10319489926824569702148437500-34815528 8848452187540396337457/252574928278923034667968750000*7^(1/2), a[16,8] = -285913537824921837018262511263067663964612298746602369190895057187 02483/3073895442200686868940860530269786975059276326603296593934375000 0000000+32621718645707277997227824444006976400602253768348434570037740 2630409/17437228972534444030205831181524084391599313634925949656000000 00000000*7^(1/2), a[17,15] = 14093108260308803339864661009803994368997 /667116832617261294418298462366640626401280*7^(1/2)-127742518491569943 31101322175199160614539284227/1972998032465550278242117702449339652581 78560000, a[14,3] = 0, c[17] = 3971/10000, a[15,9] = -3727850143078386 229058076561522609367827766330696417108435354320431359/801634922651385 9159668160608032382937598583484731440429687500000000000+81425134979651 2127448124174135016152994969999173763717229208919303/15898421182640198 33619870377392538080518560957031250000000000000000*7^(1/2), a[17,10] = -75888621039765715002449611997758928569635156851427/14951530380129748 52497236867354599875000000000000000*7^(1/2)+38286958923199060861303166 0622192244633710639124838997/17007365807397588947156069366158573578125 00000000000000, a[16,4] = 0, a[16,10] = -17520323069888524301525598935 74903103295734904546828540573516708228831279/2433262830102477113639615 580994907630153630681641719830799102783203125000+818839996665861726917 35974039324906939964446697501450842550606413648603/7337223303078238681 12868698269233685400171713233503210517883300781250000*7^(1/2), a[14,5] = 0, c[2] = 1/16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] \+ = 11/12, c[11] = 19/20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[3,1] = \+ 18928/1134225, a[3,2] = 100352/1134225, a[17,16] = 5632103863128032194 320344991957663639629/137337912974056400486943321291590340771840*7^(1/ 2)+2186548150923933141497467590323092999998671947/40617687762077180444 013487271987843283271680000, a[17,4] = 0, a[13,3] = 0, a[13,4] = 0, a[ 13,5] = 0, a[13,6] = 14327219974204125/40489566827933216, a[13,7] = 27 20762324010009765625000/10917367480696813922225349, a[13,8] = -4985330 05859375/95352091037424, a[13,9] = 40593203046377724792670503059617543 7402459637909765779/78803919436321841083201886041201537229769115088303 952, a[13,10] = -10290327637248/1082076946951, a[13,11] = 863264105888 000/85814662253313, a[13,12] = -29746300739/247142463456, a[14,11] = 2 6288142988644492320/221311497390123-367631244295366501400/818852540343 4551*7^(1/2), a[4,2] = 0, a[4,1] = 14/355, a[4,3] = 42/355, a[5,1] = 9 4495479/250880000, a[5,2] = 0, a[5,3] = -352806597/250880000, a[5,4] = 178077159/125440000, a[17,7] = -9997698942504931672941874400127034714 1667794343723953623/18124531831249098313283387727955682002635155113451 52000000*7^(1/2)+21504714622972422164050837057219547262581159136987494 25643/13194659173149343572070306265951736497918392922592706560000, a[6 ,1] = 12089/252720, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a[7,1] = 21400899/350000000, a[7,3] = 0, a[7 ,4] = 3064329829899/27126050000000, a[7,5] = -21643947/592609375, a[7, 6] = 124391943/6756250000, a[15,7] = 10596614540600968744792362758669/ 45489031169570058009272287500000-13058425932946007192467094947/1451071 902055187778276093750000*7^(1/2), a[8,1] = -15365458811/13609565775, a [8,3] = 0, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8,6] = 341936800488/47951126225, a[8,7] = 1993321838240/380523459069, a[9, 1] = -1840911252282376584438157336464708426954728061551/29919236151711 51921596253813483118262195533733898, a[9,3] = 0, a[9,4] = -14764960804 048657303638372252908780219281424435/298169210256502197561171126920960 6363661854518, a[9,5] = -875325048502130441118613421785266742862694404 520560000/170212030428894418395571677575961339495435011888324169, a[9, 6] = 7632051964154290925661849798370645637589377834346780/173408725741 8811583049800347581865260479233950396659, a[9,7] = 7519834791971137517 048532179652347729899303513750000/104567730350231759659789070781234983 2637339039997351, a[9,8] = 1366042683489166351293315549358278750/14463 1418224267718165055326464180836641, a[10,1] = -63077736705254280154824 845013881/78369357853786633855112190394368, a[10,2] = 0, a[9,2] = 0, a [17,11] = 27418190460279475664846671472967149325313851242007749/178484 880415390961272564938626684644725000000000000000*7^(1/2)-3353756327413 70301724806991185800538844107770117192189/6246970814538683644539772851 93396256537500000000000000, a[14,8] = -16848658354789365625/2259465784 16832+22737651361282524171875/806742258237298656*7^(1/2), a[16,5] = 0, a[17,1] = 33621176084706347878029792643771803161870755474678361955380 83/80292620412472779397993102954202712163478400000000000000000000-2989 723142262948660161298577240408579957089549899835987317/882336488049151 421955968164331897935862400000000000000000000*7^(1/2), a[17,3] = 0, c[ 15] = 1187/2500, a[10,4] = -31948346510820970247215/695600921696002663 2192, a[10,5] = -3378604805394255292453489375/517042670569824692230499 952, a[10,6] = 1001587844183325981198091450220795/18423268420772250370 1669953872896, a[10,3] = 0, a[15,1] = 13094613915285939653495056771309 7/2815269161783203125000000000000000+90670944595916412828478989403/661 48981703437500000000000000000*7^(1/2), a[17,6] = -93315437771922194880 90069025678637265860768694394667197161/2048078677739008165828999089148 94308064358400000000000000000*7^(1/2)+19136107903291168666032476370571 5648161934674584562401842333/14336550744173057160802993624042601564505 08800000000000000000\}: " }{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[17,i ]=subs(e5,a[17,i]),i=1..16):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "62/&%\"aG6$\"#<\"\"\"$\"I^=d74t;\\A_8!HSt<:S3H$!#T/&F%6$F '\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\" \"'$\"Hm6IIQfZ@0;ljx@)\\84$H\"!#S/&F%6$F'\"\"($\"H\"*[4suyKqlw(G!Ga47y Pq\"FD/&F%6$F'\"\")$\"HV\\\"[7g/+*[`z)f@(ei$[9tFD/&F%6$F'\"\"*$!I`AGB< ]kv*Q*p0b0v%*egKOF+/&F%6$F'\"#5$\"Hm)3#G2:z$o9uQ4VTT!*4$3*FD/&F%6$F'\" #6$!IPb&)*=R;Dv)[/>(>yNL1VI\"FD/&F%6$F'\"#7$\"I@[\"yTQ9df(*e%)4%\\>Yb, x@!#U/&F%6$F'\"#8$\"Hrrd(ReE.a^&eIpw#yE(*\\\"*Fco/&F%6$F'\"#9$\"Ig%[G? d&yq=**3U,7ouIK/=FD/&F%6$F'\"#:$!H5o#o)H**30\"3,)z4SWaPG&))F+/&F%6$F' \"#;$\"I6q=V_s#z?G-qKj#pmGKB;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 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_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]=4531/1 0000" "6#/&%\"cG6#\"#=*&\"%JX\"\"\"\"&++\"!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "c_18 := 4531/10000;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c_18G#\" %JX\"&++\"" }}}{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[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(rationalize(solve(subs(dd, eq))))\}:\nsol;\ndd_5 := `union`(subs(sol,dd),sol):\nd_5 := `union`(su bs(dd_5,d5),dd_5):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/&%\"dG6$\"\" \"\"\"(,&#\"[o+]i!*z,?EPyjz1C()[%[LIn;cOl<(Rt+^P%*\"in(>7i%GN87*fV8D++ [RTmfB$3t]5Rr$>L,*!\"\"*(\"Y+]7`_#*z<,\\a$G4FJ6H#p`$G)yY;JbGF(\"Y6uspg PN_M20FKQ\"f(zB)*e0RzS\")=)z#F.F)#F(\"\"#F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "subs(d_5,matrix([s eq([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+$\"'**3B!\"$$\"'b2IF/$\"'D!>\"!\"#$!'\"=]&F/$\"'qH 9F4$!'l5@F4$\"'-*f$!\"%$\"'(RW\"F=$!'!)f[F/$!'&)fZF/$!'H'o\"F/72F+F+F+ F+F+$!'&e/\"F4$!'\"*f8F4$!'E8`F4$\"'!eU#F4$!'aYjF4$\"'M8%*F4$!'x5;F/$! '-+kF=$\"'[lBF4$\"'s\\AF4$\"'>msF/72F+F+F+F+F+$\"'s`DF4$\"'w7LF4$\"'@r 7!\"\"$!'x+dF4$\"'D1:F^o$!'C\\AF^o$\"'+oQF/$\"'![^\"F/$!'SOhF4$!'`edF4 $!'6?F^o$\"'N MHF^o$!'#*z]F/$!'xZ>F/$\"'AH')F4$\"'m'4)F4$\"'6hAF472F+F+F+F+F+$\"'xmB F4$\"'y[IF4$\"'n16F^o$!'6wYF4$\"'Ux7F^o$!'r\\>F^o$\"'(pS$F/$\"'Nm7F/$! 'bahF4$!',7eF4$!'fN:F472$!'1u5F/F+F+F+F+$!'`]KF4$!'YCUF4$!'/W;F^o$\"'9 xuF4$!'Ng>F^o$\"'x6HF^o$!'4))\\F/$!',p>F/$\"'Y(*oF4$\"'$Q&oF4$\"'!H]#F 4Q)pprint546\"" }}}{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(SO7_17) do\n eqn_group := convert(SO7_17[ct],'polynom _order_conditions',7):\n tt := expand(subs(\{op(e5),op(d_5)\},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\\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 "Eval uate the \"weight\" polynomials at the node " }{XPPEDIT 18 0 "c[18]=4 531/10000" "6#/&%\"cG6#\"#=*&\"%JX\"\"\"\"&++\"!\"\"" }{TEXT -1 83 " \+ to obtain the linking coefficients in the next stage in the interpolat ion scheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "eqs18 := \{s eq(a[18,j]=add(expand(subs(\{op(d_5),c[18]=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 16224 "e6 := \{a[16,1] = \+ 1013856404118125952763527205161498788180173939254661359435974309585825 318269273/198564667739239660374389171210942338307459063268324464130246 79687500000000000000-1931754243710985131731267585181486489297591530273 47863824763410934840962491/2778932721896691403250310788177330670569727 48113814414737187500000000000000000*7^(1/2), a[18,6] = 194471406905941 485117086317959160425736682183804756519030733/143365507441730571608029 9362404260156450508800000000000000000-37992144340278788096731287327181 0009475439610374506979309/70623402680655453994103416867204933815296000 00000000000000*7^(1/2), a[16,6] = -19039890149866495432584929203443766 60766478959434824857368133995942073946993/2610549880447098448451329990 8841941049181178531837828644206625000000000000000+28929035836992861158 293213150542073159989024626464512663622554495682276029/892277791168441 852498013180477996031954434813099925783737531127929687500000*7^(1/2), \+ c[18] = 4531/10000, a[17,2] = 0, a[7,2] = 0, c[16] = 277/2500, a[16,7] = 1471649975107913251117693334592045704450918568535357968485718078648 3042792373/16088988337674671465702156595052532701656799236967006367882 5639487979375000000+35097161131542028791945288573990002288024169106005 8741934303404427219789/74868036898960959337075562503093457016256710599 371634570393697387568750000*7^(1/2), a[14,13] = 3/392-3/392*7^(1/2), a [18,1] = 6431334448960379167546546351751113222150988727702037620106614 523/151030418995861298047625026656855301579502870400000000000000000000 -4314390544903719789469749993269597109886551871788377645343/1078411263 171185071279516645294541921609600000000000000000000*7^(1/2), a[15,12] \+ = 12163051029345238407733418657/3016875774609375000000000000000, a[16, 9] = 20790434203496917090062867244491983719450919633584853119413284106 748179318736252691355190285977916572557692108955233/158776675835946596 6895356458877254530577780731744189320187958035866716805452345584931662 28541026757812500000000000000+1675029191712848711934999883357656040011 1643601891912316132035529723233388789270175212188770469349713697180572 09/7318101931533341081986828123220267795255614895281860241195526749673 75646957459693466445168584156250000000000000000*7^(1/2), a[15,13] = -5 79945478542537523/549316406250000000000+3727856565327672037/2197265625 000000000000*7^(1/2), a[14,2] = 0, a[18,2] = 0, a[17,12] = -1134903585 52214825884402065259188700089375815491942289/3125294053645738010668658 5123539663360000000000000000000*7^(1/2)+128906890820710317724713918771 43529539151923688844177161/1093852918776008303734030479323888217600000 000000000000000, a[17,5] = 0, a[16,12] = -1248171126266506029131703126 44481143348777609285630839757851632016681433/3983604783735069658159427 207807518356647862336983160384375000000000000000+194885682734472712496 69733067688159410742449986073234566291555857980323/2904711821473488292 407915672359648801722399620716887780273437500000000000*7^(1/2), a[16,1 1] = 33075872377862145803723943973307330348062898814430174557327751165 90947887/2161276829253336041664932456066917395909700497105040226143341 064453125000-159249481348355430623321425412532136521666854903188329515 308104047100201/530839922974503589180860603244506027065540472973167774 842224121093750000*7^(1/2), a[18,3] = 0, a[16,13] = -46330543544553540 874179448395431083082899647522368296822645347/508745135208312439739125 0759096940921758056970520019531250000000+18909196831867663116267125639 92017565381198275789277564189007889/1139589102866619865015640170037714 766473804761396484375000000000000*7^(1/2), a[18,4] = 0, a[15,14] = 260 94995957293704259/549316406250000000000*7^(1/2), a[18,5] = 0, a[18,7] \+ = 6485285588294739183483110191854194357484624835466068526129/395839775 19448030716210918797855209493755178767778119680000-4371944069111595199 522286141913363272705812493584890149/671278956712929567158643989924284 51861611685605376000000*7^(1/2), a[14,6] = -142489509197785007255/1133 707871182130048+996623278509306786175/20973595616869405888*7^(1/2), a[ 16,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[15,2] = 0, a[17,13] = -96617167 6795701349259082624752343417581690261/11315349509559955112431216508625 00000000000000000*7^(1/2)+19423380204481918582213207680232912549352934 689479/6119341014770023724802801887864400000000000000000000, a[10,7] = 187023075231349900768014890274453125/25224698849808178010752575653374 848, a[10,8] = 1908158550070998850625/117087067039189929394176, a[10,9 ] = -52956818288156668227044990077324877908565/29127799594774339863498 22224412353951940608, a[11,1] = -1011610659182690953478115799368511670 3/9562819945036894030442231411871744000, a[11,2] = 0, a[11,3] = 0, a[1 1,4] = -9623541317323077848129/3864449564977792573440, a[11,5] = -4823 348333146829406881375/576413233634141239944816, a[11,6] = 656611924651 4996884067001154977284529/970305487021846325473990863582315520, a[11,7 ] = 2226455130519213549256016892506730559375/3648804431596752555774356 48380047355776, a[11,8] = 39747262782380466933662225/17560328024314241 64410720256, a[11,9] = 48175771419260955335244683805171548038966866545 122229/1989786420513815146528880165952064118903852843612160000, a[11,1 0] = -2378292068163246/47768728487211875, a[12,1] = -32180221747585998 31659045535578571/1453396753634469525663775847094384, a[12,2] = 0, a[1 2,3] = 0, a[12,4] = 26290092604284231996745/5760876126062860430544, a[ 12,5] = -697069297560926452045586710000/41107967755245430594036502319, a[14,1] = -26242/1715+588937/101528*7^(1/2), a[18,8] = -1502708592933 305431413728452036942690686722169823969/133381846854807729958986791526 68515221504000000000000*7^(1/2)+58589616623242646214825661347124714930 789685430959801/216078591904788522533558602273229946588364800000000000 , a[15,3] = 0, a[16,14] = 37807079247082910935750437397535265361259662 2387047615352035143989/38652465726311665474213406887371187527810692671 413726806640625000000-736694485119203754900544593199058955707390426548 138628337098463663/247375780648394659034965804079175600177988433097047 85156250000000000*7^(1/2), a[14,12] = 1100613127343/48439922837376-297 46300739/1424703612864*7^(1/2), a[17,9] = -264701762796489849333164460 056987686273182643308013876537822807951183236955127319033004381711/265 7419178388574677233558327633544805137160916516236750792782073750348797 3304320000000000000000*7^(1/2)-115937771802925638794257846529122507932 2378160773707689649770373007738863712156809629044640218707/11626208905 4500142128968176833967585224750790097585357847184215726577759883206400 000000000000000000, a[12,6] = 1827357820434213461438077550902273440/13 9381013914245317709567680839641697, a[12,7] = 643504802814241550941949 227194107500000/242124609118836550860494007545333945331, a[12,8] = 162 259938151380266113750/59091082835244183497007, a[12,9] = -230282516328 73523818545414856857015616678575554130463402/2001316918319144450344390 5240405603349978424504151629055, a[12,10] = 7958341351371843889152/328 4467988443203581305, a[12,11] = -507974327957860843878400/121555654819 179042718967, a[13,1] = 4631674879841/103782082379976, a[13,2] = 0, a[ 15,8] = 16161913072172934315785836819/9932509483065000000000000000-759 80668436324671626317237501/49917227145660000000000000000*7^(1/2), a[15 ,4] = 0, a[16,15] = 46088828763074293687752514438764033729248725094086 21400283/75922278949360871648575860105318720457058558153486785997500-8 4914702986428741912331429025972417475312941024930181294/53145595264552 61015400310207372310431994099070744075019825*7^(1/2), a[16,3] = 0, a[1 7,14] = 15739031823875445778785537999179335674270343962861810116243/30 9619125620788378533311524058377597436400000000000000000000+60665205048 3146218029195891134085469196882580940598209/12384765024831535141332460 962335103897456000000000000000*7^(1/2), a[14,7] = 66307365920690498875 0000000/13716692475747279030488259-9227445425051062363134765625/507517 621602649324128065583*7^(1/2), a[8,2] = 0, a[14,4] = 0, a[18,16] = 288 266095365211419064692734565913406048103319/571656346281086243286115746 7909400165793792000+60833408527191271313421144941358828000769/12563875 74244145589639814828111956080394240*7^(1/2), a[15,5] = 0, a[14,9] = 17 318895166382455115817006657422200600438218265372153/554375052657198964 316171246174366066645917060800-274968087654238523490991092528596412316 293248643447707/23287013241323869436363405111359694599520551359840*7^( 1/2), a[18,13] = -10435801212172743074622612541578159307187516521/1035 1449366152995973224112880112500000000000000000*7^(1/2)+101126309543475 847280329109976443851552294375363/426356726368281814342696134468000000 00000000000000, a[17,8] = -2643363415762169747333266167756151785333485 317345751/27702383577536990068404949009388454690816000000000000*7^(1/2 )+23451918187433258820064222913608502147503898444041667/72026197301596 174177852867424409982196121600000000000, a[14,10] = -90813659210210330 56/120504023637725+25400013053688208472/891729774919165*7^(1/2), a[18, 14] = 688260040952087965212862705907587124167147667584978421209/114673 75022992162167900426816976948053200000000000000000000+2652859272295366 8251624457086005829986967133546824867/45869500091968648671601707267907 7922128000000000000000*7^(1/2), a[18,9] = 1417729703147854028657911907 6615020298107454902607424914474325122930069419330134278983169782333079 /348786267163500426386904530501902755674252370292756073541552647179733 279649619200000000000000000000-115752765571029020017132140741067326567 20056039985689215131617039362310394019419472694886093/9842293253291017 3230872530653094252042117070982082842621954891620383288790016000000000 0000000*7^(1/2), a[15,6] = 3353943190402140976803568136084793/19770296 302701765625000000000000000-255748988917794999289135717669191/86495046 32432022460937500000000000*7^(1/2), a[15,11] = -2079418008367563117920 92915907/81839239362061500549316406250+1089100652209922459592600055965 7/4582997404275444030761718750000*7^(1/2), a[15,10] = 1446205909971003 3235500874587/10319489926824569702148437500-34815528884845218754039633 7457/252574928278923034667968750000*7^(1/2), a[16,8] = -28591353782492 183701826251126306766396461229874660236919089505718702483/307389544220 06868689408605302697869750592763266032965939343750000000000+3262171864 57072779972278244440069764006022537683484345700377402630409/1743722897 253444403020583118152408439159931363492594965600000000000000*7^(1/2), \+ a[17,15] = 14093108260308803339864661009803994368997/66711683261726129 4418298462366640626401280*7^(1/2)-127742518491569943311013221751991606 14539284227/197299803246555027824211770244933965258178560000, a[14,3] \+ = 0, a[18,15] = -40554328659658584237572971829594948977715703/19534633 98480742849744670992524098665922560000+2163159021025400630153842731233 25235039761/8672518824024396827437880010766328143216640*7^(1/2), c[17] = 3971/10000, a[15,9] = -37278501430783862290580765615226093678277663 30696417108435354320431359/8016349226513859159668160608032382937598583 484731440429687500000000000+814251349796512127448124174135016152994969 999173763717229208919303/158984211826401983361987037739253808051856095 7031250000000000000000*7^(1/2), a[17,10] = -75888621039765715002449611 997758928569635156851427/149515303801297485249723686735459987500000000 0000000*7^(1/2)+382869589231990608613031660622192244633710639124838997 /1700736580739758894715606936615857357812500000000000000, a[18,17] = 0 , a[16,4] = 0, a[18,12] = -4466589287639107513221454335284860361425205 7936533163/10417646845485793368895528374513221120000000000000000000*7^ (1/2)+11473752696537480050786046301681238985163187194373379961/1093852 918776008303734030479323888217600000000000000000000, a[16,10] = -17520 32306988852430152559893574903103295734904546828540573516708228831279/2 4332628301024771136396155809949076301536306816417198307991027832031250 00+8188399966658617269173597403932490693996444669750145084255060641364 8603/73372233030782386811286869826923368540017171323350321051788330078 1250000*7^(1/2), a[14,5] = 0, c[2] = 1/16, c[3] = 112/1065, c[4] = 56/ 355, c[5] = 39/100, c[6] = 7/15, c[7] = 39/250, c[8] = 24/25, c[9] = 1 4435868/16178861, c[10] = 11/12, c[11] = 19/20, c[12] = 1, c[13] = 1, \+ a[2,1] = 1/16, a[3,1] = 18928/1134225, a[3,2] = 100352/1134225, a[17,1 6] = 5632103863128032194320344991957663639629/137337912974056400486943 321291590340771840*7^(1/2)+2186548150923933141497467590323092999998671 947/40617687762077180444013487271987843283271680000, a[17,4] = 0, a[13 ,3] = 0, a[13,4] = 0, a[13,5] = 0, a[13,6] = 14327219974204125/4048956 6827933216, a[13,7] = 2720762324010009765625000/1091736748069681392222 5349, a[13,8] = -498533005859375/95352091037424, a[13,9] = 40593203046 3777247926705030596175437402459637909765779/78803919436321841083201886 041201537229769115088303952, a[13,10] = -10290327637248/1082076946951, a[13,11] = 863264105888000/85814662253313, a[13,12] = -29746300739/24 7142463456, a[14,11] = 26288142988644492320/221311497390123-3676312442 95366501400/8188525403434551*7^(1/2), a[4,2] = 0, a[4,1] = 14/355, a[4 ,3] = 42/355, a[5,1] = 94495479/250880000, a[5,2] = 0, a[5,3] = -35280 6597/250880000, a[5,4] = 178077159/125440000, a[17,7] = -9997698942504 9316729418744001270347141667794343723953623/18124531831249098313283387 72795568200263515511345152000000*7^(1/2)+21504714622972422164050837057 21954726258115913698749425643/1319465917314934357207030626595173649791 8392922592706560000, a[6,1] = 12089/252720, a[6,2] = 0, a[6,3] = 0, a[ 6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a[7,1] = 21400899/35 0000000, a[7,3] = 0, a[7,4] = 3064329829899/27126050000000, a[7,5] = - 21643947/592609375, a[7,6] = 124391943/6756250000, a[15,7] = 105966145 40600968744792362758669/45489031169570058009272287500000-1305842593294 6007192467094947/1451071902055187778276093750000*7^(1/2), a[8,1] = -15 365458811/13609565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -8339128164 608/939060038475, a[8,6] = 341936800488/47951126225, a[8,7] = 19933218 38240/380523459069, a[9,1] = -1840911252282376584438157336464708426954 728061551/2991923615171151921596253813483118262195533733898, a[9,3] = \+ 0, a[9,4] = -14764960804048657303638372252908780219281424435/298169210 2565021975611711269209606363661854518, a[9,5] = -875325048502130441118 613421785266742862694404520560000/170212030428894418395571677575961339 495435011888324169, a[9,6] = 76320519641542909256618497983706456375893 77834346780/1734087257418811583049800347581865260479233950396659, a[9, 7] = 7519834791971137517048532179652347729899303513750000/104567730350 2317596597890707812349832637339039997351, a[9,8] = 1366042683489166351 293315549358278750/144631418224267718165055326464180836641, a[10,1] = \+ -63077736705254280154824845013881/78369357853786633855112190394368, a[ 10,2] = 0, a[9,2] = 0, a[17,11] = 274181904602794756648466714729671493 25313851242007749/1784848804153909612725649386266846447250000000000000 00*7^(1/2)-335375632741370301724806991185800538844107770117192189/6246 97081453868364453977285193396256537500000000000000, a[14,8] = -1684865 8354789365625/225946578416832+22737651361282524171875/8067422582372986 56*7^(1/2), a[16,5] = 0, a[17,1] = 33621176084706347878029792643771803 16187075547467836195538083/8029262041247277939799310295420271216347840 0000000000000000000-29897231422629486601612985772404085799570895498998 35987317/882336488049151421955968164331897935862400000000000000000000* 7^(1/2), a[17,3] = 0, c[15] = 1187/2500, a[10,4] = -319483465108209702 47215/6956009216960026632192, a[10,5] = -3378604805394255292453489375/ 517042670569824692230499952, a[10,6] = 1001587844183325981198091450220 795/184232684207722503701669953872896, a[10,3] = 0, a[18,10] = 2425578 594330712942621238403235096528690693747094280567/187081023881373478418 71676302774430935937500000000000000-1281300529163218452387156728072917 1638148255045076861/21380688443585540390710487203170778212500000000000 0000*7^(1/2), a[15,1] = 130946139152859396534950567713097/281526916178 3203125000000000000000+90670944595916412828478989403/66148981703437500 000000000000000*7^(1/2), a[17,6] = -9331543777192219488090069025678637 265860768694394667197161/204807867773900816582899908914894308064358400 000000000000000*7^(1/2)+1913610790329116866603247637057156481619346745 84562401842333/1433655074417305716080299362404260156450508800000000000 000000, a[18,11] = 372098434581358541752287535215351518628082338823827 /2051550349602194957155918834789478675000000000000000*7^(1/2)-51389535 84119531321031442069620009245122067004757018391/1186924454762349892462 5568418674528874212500000000000000\}: " }{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):\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "63/&%\"aG6$\"#=\"\"\"$\"IQI%*G4Vx$QF?R:[z\"G1#)*> $!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0 /&F%6$F'\"\"'$!G`Hr*HQ)fAsAUW'f@/#)>o'!#S/&F%6$F'\"\"($!Gu/JPgn]*\\Z.s )3k.0)yZ)FD/&F%6$F'\"\")$!H)HAXD-_#)G:w1#e#4!*pk#p#FD/&F%6$F'\"\"*$\"H ]1K-@nNH/02%eoTLp\\J&*F+/&F%6$F'\"#5$!H\")ya_0]@6(\\`Y.<%\\o]+*GFD/&F% 6$F'\"#6$\"Hsm*QH<2b5'ft45]DhO2p%FD/&F%6$F'\"#7$!G`TG\\)Rkh(GC*)o,Wt]; W&)F+/&F%6$F'\"#8$!H!p]vA\"[a0p.A'4X$\\P,W&H!#U/&F%6$F'\"#9$\"I!z'[wD-4e.8#FD/&F%6$F'\"#:$\"I^n([wB$)yDa')yuoSU7#>BXF+/&F%6$F'\"# ;$\"IC+I]79B()eb+*4&GNj*=`y\"FD/&F%6$F'\"# " 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 -1 223 "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_18 := SimpleOrderConditions(7,18,'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]:\nordeqns6 := []:\nfo r ct in whch do\n eqn_group := convert(SO7_18[ct],'polynom_order_con ditions',7):\n ordeqns6 := [op(ordeqns6),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 "eqns6 := []:\nfor ct to nops(ordeqns6) do\n eqns6 := [op(eq ns6),expand(subs(e6,ordeqns6[ct]))];\nend do:\nnops(eqns6);\nnops(inde ts(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 equations." }}{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) min us \{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 wou ld 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$\"#>\"#<\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[19] = 177 3/2500" "6#/&%\"cG6#\"#>*&\"%t<\"\"\"\"%+D!\"\"" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "c_19 := 1773/2500;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c_19G#\"%t<\"%+D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "eq1 := add( subs(\{op(d6),c[19]=c_19\},d[17,i]*c[19]^i),i=1..7)=0:\neq2 := add(sub s(\{op(d6),c[19]=c_19\},d[18,i]*c[19]^i),i=1..7)=0:\ndd := \{d[1,1]=1, seq(d[1,i]=0,i=2..6),seq(d[9,i]=0,i=1..6)\}:\nsol := solve(subs(dd,\{e q1,eq2\}));\ndd_6 := `union`(subs(sol,dd),sol):\nd_6 := `union`(subs(d d_6,d6),dd_6):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<$/&%\"dG6$\" \"*\"\"(,&#\"fp+v=/H[(z&Q)G$[bU2#e\"fp( oPabl]iO#z/BN'Re.Kp!H^5=vHpX-]F\\+m`I)4$znQG*>*Ro\"!\"\"*(\"dp++voH/A9 1OL!*>`b$HP!4%y0/`eVE%fKc#yQ%G,P1nvZk+Gw\"\"\"\"\"ephDWAB_n#3c([QRlp& \\g7)[PFQ(z-Khw1[g3TW\"epam!p40j.#F0F+#F3\"\"#F3/&F(6$F3F+,&#\"fn+]Pf7 F*pqt\"4Vyp0oA5WKK`j)o%zL*>&*\"fn<(>1-:I'oK@X)Hn-J@y(RSL\\V\"y,_b78F0* (\"U++]P;rbcD9\"=%3qF3\"Wzs?-I()p[-ofn'emK*pU`-](\\N0*zBF 0F+F5F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "subs(d_6,matrix([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+$\"'[zd!\"%$ \"'gTqF/$\"',f8!\"$F+$\"'e/*)F/$!'yGAF4$\"'>w\\!\"&$\"'y-#*!\"'$\"'.RE F4$!'+Q@!\"#$!'c1iF/$!'(*R9FC$\"'%*RKFC74F+F+F+F+F+$!'+EGF4$!'F]o$!''*)H'F4$!'l&H*FC$\"'o`FF]o74$!'2vrF;F+F +F+F+$!'](>$F4$!'l%*QF4$!')3[(F4$!'ZFKF;$!'`j[F4$\"'$fA\"FC$!'lVFF/$!' &)ePF;$!';D6F4$\"'*&\"%t<\"\"\"\"%+D!\"\"" }{TEXT -1 83 " to ob tain the linking coefficients in the next stage in the interpolation s cheme." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "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 19110 "e7 := \{a[19,10] = 3 58592747631805583268793478338446049329593175694761/3262434149717031919 96925158739788485908508300781250+1818048548295397550709386176015734990 787971366822/45674078096038446879569522223570388027191162109375*7^(1/2 ), a[19,13] = 2110074255562519597112772669249807784130258601/175219846 041652275588428994064404296875000000000+352558328195921194159226892246 05421615051/52650194123092630885946212158775329589843750*7^(1/2), a[19 ,11] = -132400834676144449170901924810677066147804313106601/6899439957 4634364098688432493225265498352050781250-15311238151448667731596088597 31276680985184261266/1270949465848527759712681651190991732864379882812 5*7^(1/2), a[16,1] = 1013856404118125952763527205161498788180173939254 661359435974309585825318269273/198564667739239660374389171210942338307 45906326832446413024679687500000000000000-1931754243710985131731267585 18148648929759153027347863824763410934840962491/2778932721896691403250 31078817733067056972748113814414737187500000000000000000*7^(1/2), a[18 ,6] = 194471406905941485117086317959160425736682183804756519030733/143 3655074417305716080299362404260156450508800000000000000000-37992144340 2787880967312873271810009475439610374506979309/70623402680655453994103 41686720493381529600000000000000000*7^(1/2), a[16,6] = -19039890149866 49543258492920344376660766478959434824857368133995942073946993/2610549 8804470984484513299908841941049181178531837828644206625000000000000000 +289290358369928611582932131505420731599890246264645126636225544956822 76029/8922777911684418524980131804779960319544348130999257837375311279 29687500000*7^(1/2), c[18] = 4531/10000, a[17,2] = 0, a[7,2] = 0, c[16 ] = 277/2500, a[16,7] = 1471649975107913251117693334592045704450918568 5353579684857180786483042792373/16088988337674671465702156595052532701 6567992369670063678825639487979375000000+35097161131542028791945288573 9900022880241691060058741934303404427219789/74868036898960959337075562 503093457016256710599371634570393697387568750000*7^(1/2), a[14,13] = 3 /392-3/392*7^(1/2), a[18,1] = 6431334448960379167546546351751113222150 988727702037620106614523/151030418995861298047625026656855301579502870 400000000000000000000-431439054490371978946974999326959710988655187178 8377645343/10784112631711850712795166452945419216096000000000000000000 00*7^(1/2), a[15,12] = 12163051029345238407733418657/30168757746093750 00000000000000, a[16,9] = 20790434203496917090062867244491983719450919 6335848531194132841067481793187362526913551902859779165725576921089552 33/1587766758359465966895356458877254530577780731744189320187958035866 71680545234558493166228541026757812500000000000000+1675029191712848711 9349998833576560400111643601891912316132035529723233388789270175212188 77046934971369718057209/7318101931533341081986828123220267795255614895 28186024119552674967375646957459693466445168584156250000000000000000*7 ^(1/2), a[15,13] = -579945478542537523/549316406250000000000+372785656 5327672037/2197265625000000000000*7^(1/2), a[14,2] = 0, a[18,2] = 0, a [17,12] = -113490358552214825884402065259188700089375815491942289/3125 2940536457380106686585123539663360000000000000000000*7^(1/2)+128906890 82071031772471391877143529539151923688844177161/1093852918776008303734 030479323888217600000000000000000000, a[17,5] = 0, a[16,12] = -1248171 12626650602913170312644481143348777609285630839757851632016681433/3983 604783735069658159427207807518356647862336983160384375000000000000000+ 1948856827344727124966973306768815941074244998607323456629155585798032 3/29047118214734882924079156723596488017223996207168877802734375000000 00000*7^(1/2), a[16,11] = 33075872377862145803723943973307330348062898 81443017455732775116590947887/2161276829253336041664932456066917395909 700497105040226143341064453125000-159249481348355430623321425412532136 521666854903188329515308104047100201/530839922974503589180860603244506 027065540472973167774842224121093750000*7^(1/2), a[18,3] = 0, a[16,13] = -46330543544553540874179448395431083082899647522368296822645347/508 7451352083124397391250759096940921758056970520019531250000000+18909196 83186766311626712563992017565381198275789277564189007889/1139589102866 619865015640170037714766473804761396484375000000000000*7^(1/2), a[18,4 ] = 0, a[15,14] = 26094995957293704259/549316406250000000000*7^(1/2), \+ a[18,5] = 0, a[18,7] = 64852855882947391834831101918541943574846248354 66068526129/3958397751944803071621091879785520949375517876777811968000 0-4371944069111595199522286141913363272705812493584890149/671278956712 92956715864398992428451861611685605376000000*7^(1/2), a[14,6] = -14248 9509197785007255/1133707871182130048+996623278509306786175/20973595616 869405888*7^(1/2), a[16,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[19,14] = - 6975558593021648832971791813454390228883838235494737/17497825657641849 0110785321303969544268798828125000000-26886894766313057009210141247088 0702355429907531/6999130263056739604431412852158781770751953125000*7^( 1/2), a[15,2] = 0, a[19,15] = 7309257172110824970392183099608413709717 899/120422243192477433974738629299886453404650000-15346626448658683257 721968231518941311/926324947634441799805681763845280410805*7^(1/2), a[ 17,13] = -966171676795701349259082624752343417581690261/11315349509559 95511243121650862500000000000000000*7^(1/2)+19423380204481918582213207 680232912549352934689479/611934101477002372480280188786440000000000000 0000000, a[10,7] = 187023075231349900768014890274453125/25224698849808 178010752575653374848, a[10,8] = 1908158550070998850625/11708706703918 9929394176, a[10,9] = -52956818288156668227044990077324877908565/29127 79959477433986349822224412353951940608, a[11,1] = -1011610659182690953 4781157993685116703/9562819945036894030442231411871744000, a[11,2] = 0 , a[11,3] = 0, a[11,4] = -9623541317323077848129/386444956497779257344 0, a[11,5] = -4823348333146829406881375/576413233634141239944816, a[11 ,6] = 6566119246514996884067001154977284529/97030548702184632547399086 3582315520, a[11,7] = 2226455130519213549256016892506730559375/3648804 43159675255577435648380047355776, a[11,8] = 39747262782380466933662225 /1756032802431424164410720256, a[11,9] = 48175771419260955335244683805 171548038966866545122229/198978642051381514652888016595206411890385284 3612160000, a[11,10] = -2378292068163246/47768728487211875, a[12,1] = \+ -3218022174758599831659045535578571/1453396753634469525663775847094384 , a[12,2] = 0, a[12,3] = 0, a[12,4] = 26290092604284231996745/57608761 26062860430544, a[12,5] = -697069297560926452045586710000/411079677552 45430594036502319, a[14,1] = -26242/1715+588937/101528*7^(1/2), a[18,8 ] = -1502708592933305431413728452036942690686722169823969/133381846854 80772995898679152668515221504000000000000*7^(1/2)+58589616623242646214 825661347124714930789685430959801/216078591904788522533558602273229946 588364800000000000, a[15,3] = 0, a[16,14] = 37807079247082910935750437 3975352653612596622387047615352035143989/38652465726311665474213406887 371187527810692671413726806640625000000-736694485119203754900544593199 058955707390426548138628337098463663/247375780648394659034965804079175 60017798843309704785156250000000000*7^(1/2), a[19,17] = 0, a[14,12] = \+ 1100613127343/48439922837376-29746300739/1424703612864*7^(1/2), a[17,9 ] = -26470176279648984933316446005698768627318264330801387653782280795 1183236955127319033004381711/26574191783885746772335583276335448051371 609165162367507927820737503487973304320000000000000000*7^(1/2)-1159377 7180292563879425784652912250793223781607737076896497703730077388637121 56809629044640218707/1162620890545001421289681768339675852247507900975 85357847184215726577759883206400000000000000000000, a[12,6] = 18273578 20434213461438077550902273440/139381013914245317709567680839641697, a[ 12,7] = 643504802814241550941949227194107500000/2421246091188365508604 94007545333945331, a[12,8] = 162259938151380266113750/5909108283524418 3497007, a[12,9] = -23028251632873523818545414856857015616678575554130 463402/20013169183191444503443905240405603349978424504151629055, a[12, 10] = 7958341351371843889152/3284467988443203581305, a[12,11] = -50797 4327957860843878400/121555654819179042718967, a[13,1] = 4631674879841/ 103782082379976, a[13,2] = 0, a[19,18] = 0, a[15,8] = 1616191307217293 4315785836819/9932509483065000000000000000-759806684363246716263172375 01/49917227145660000000000000000*7^(1/2), a[15,4] = 0, a[16,15] = 4608 882876307429368775251443876403372924872509408621400283/759222789493608 71648575860105318720457058558153486785997500-8491470298642874191233142 9025972417475312941024930181294/53145595264552610154003102073723104319 94099070744075019825*7^(1/2), a[16,3] = 0, a[17,14] = 1573903182387544 5778785537999179335674270343962861810116243/30961912562078837853331152 4058377597436400000000000000000000+60665205048314621802919589113408546 9196882580940598209/12384765024831535141332460962335103897456000000000 000000*7^(1/2), a[14,7] = 663073659206904988750000000/1371669247574727 9030488259-9227445425051062363134765625/507517621602649324128065583*7^ (1/2), a[19,12] = 116148895768336301414698893782124924558205330714623/ 3179213514857725027128762321323614843750000000000000+31688413394815770 57649164025908284942240477511813/1112724730200203759495066812463265195 312500000000000*7^(1/2), a[8,2] = 0, a[14,4] = 0, a[18,16] = 288266095 365211419064692734565913406048103319/571656346281086243286115746790940 0165793792000+60833408527191271313421144941358828000769/12563875742441 45589639814828111956080394240*7^(1/2), a[15,5] = 0, a[14,9] = 17318895 166382455115817006657422200600438218265372153/554375052657198964316171 246174366066645917060800-274968087654238523490991092528596412316293248 643447707/23287013241323869436363405111359694599520551359840*7^(1/2), \+ a[18,13] = -10435801212172743074622612541578159307187516521/1035144936 6152995973224112880112500000000000000000*7^(1/2)+101126309543475847280 329109976443851552294375363/426356726368281814342696134468000000000000 00000000, a[17,8] = -2643363415762169747333266167756151785333485317345 751/27702383577536990068404949009388454690816000000000000*7^(1/2)+2345 1918187433258820064222913608502147503898444041667/72026197301596174177 852867424409982196121600000000000, a[14,10] = -9081365921021033056/120 504023637725+25400013053688208472/891729774919165*7^(1/2), a[18,14] = \+ 688260040952087965212862705907587124167147667584978421209/114673750229 92162167900426816976948053200000000000000000000+2652859272295366825162 4457086005829986967133546824867/45869500091968648671601707267907792212 8000000000000000*7^(1/2), a[18,9] = 1417729703147854028657911907661502 0298107454902607424914474325122930069419330134278983169782333079/34878 6267163500426386904530501902755674252370292756073541552647179733279649 619200000000000000000000-115752765571029020017132140741067326567200560 39985689215131617039362310394019419472694886093/9842293253291017323087 2530653094252042117070982082842621954891620383288790016000000000000000 0*7^(1/2), a[15,6] = 3353943190402140976803568136084793/19770296302701 765625000000000000000-255748988917794999289135717669191/86495046324320 22460937500000000000*7^(1/2), a[15,11] = -2079418008367563117920929159 07/81839239362061500549316406250+10891006522099224595926000559657/4582 997404275444030761718750000*7^(1/2), a[15,10] = 1446205909971003323550 0874587/10319489926824569702148437500-348155288848452187540396337457/2 52574928278923034667968750000*7^(1/2), a[16,8] = -28591353782492183701 826251126306766396461229874660236919089505718702483/307389544220068686 89408605302697869750592763266032965939343750000000000+3262171864570727 79972278244440069764006022537683484345700377402630409/1743722897253444 403020583118152408439159931363492594965600000000000000*7^(1/2), a[17,1 5] = 14093108260308803339864661009803994368997/66711683261726129441829 8462366640626401280*7^(1/2)-127742518491569943311013221751991606145392 84227/197299803246555027824211770244933965258178560000, a[14,3] = 0, a [18,15] = -40554328659658584237572971829594948977715703/19534633984807 42849744670992524098665922560000+2163159021025400630153842731233252350 39761/8672518824024396827437880010766328143216640*7^(1/2), c[17] = 397 1/10000, a[19,5] = 0, a[15,9] = -3727850143078386229058076561522609367 827766330696417108435354320431359/801634922651385915966816060803238293 7598583484731440429687500000000000+81425134979651212744812417413501615 2994969999173763717229208919303/15898421182640198336198703773925380805 18560957031250000000000000000*7^(1/2), a[17,10] = -7588862103976571500 2449611997758928569635156851427/14951530380129748524972368673545998750 00000000000000*7^(1/2)+38286958923199060861303166062219224463371063912 4838997/1700736580739758894715606936615857357812500000000000000, a[18, 17] = 0, a[16,4] = 0, a[18,12] = -446658928763910751322145433528486036 14252057936533163/1041764684548579336889552837451322112000000000000000 0000*7^(1/2)+11473752696537480050786046301681238985163187194373379961/ 1093852918776008303734030479323888217600000000000000000000, c[19] = 17 73/2500, a[16,10] = -1752032306988852430152559893574903103295734904546 828540573516708228831279/243326283010247711363961558099490763015363068 1641719830799102783203125000+81883999666586172691735974039324906939964 446697501450842550606413648603/733722330307823868112868698269233685400 171713233503210517883300781250000*7^(1/2), a[14,5] = 0, c[2] = 1/16, c [3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, c[7] = 39/2 50, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/ 20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[3,1] = 18928/1134225, a[3,2 ] = 100352/1134225, a[17,16] = 563210386312803219432034499195766363962 9/137337912974056400486943321291590340771840*7^(1/2)+21865481509239331 41497467590323092999998671947/4061768776207718044401348727198784328327 1680000, a[17,4] = 0, a[13,3] = 0, a[13,4] = 0, a[13,5] = 0, a[13,6] = 14327219974204125/40489566827933216, a[13,7] = 2720762324010009765625 000/10917367480696813922225349, a[13,8] = -498533005859375/95352091037 424, a[13,9] = 405932030463777247926705030596175437402459637909765779/ 78803919436321841083201886041201537229769115088303952, a[13,10] = -102 90327637248/1082076946951, a[13,11] = 863264105888000/85814662253313, \+ a[13,12] = -29746300739/247142463456, a[14,11] = 26288142988644492320/ 221311497390123-367631244295366501400/8188525403434551*7^(1/2), a[4,2] = 0, a[4,1] = 14/355, a[4,3] = 42/355, a[5,1] = 94495479/250880000, a [5,2] = 0, a[5,3] = -352806597/250880000, a[5,4] = 178077159/125440000 , a[17,7] = -99976989425049316729418744001270347141667794343723953623/ 1812453183124909831328338772795568200263515511345152000000*7^(1/2)+215 0471462297242216405083705721954726258115913698749425643/13194659173149 343572070306265951736497918392922592706560000, a[6,1] = 12089/252720, \+ a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/520 9191, a[7,1] = 21400899/350000000, a[7,3] = 0, a[7,4] = 3064329829899/ 27126050000000, a[7,5] = -21643947/592609375, a[7,6] = 124391943/67562 50000, a[19,16] = -1438617606921480386578336383098493573/4473222361098 7442664991779565346743585*7^(1/2)-671478610426445930422769177742151257 962163/5815189069428367546448931343495076666050000, a[15,7] = 10596614 540600968744792362758669/45489031169570058009272287500000-130584259329 46007192467094947/1451071902055187778276093750000*7^(1/2), a[8,1] = -1 5365458811/13609565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -833912816 4608/939060038475, a[8,6] = 341936800488/47951126225, a[8,7] = 1993321 838240/380523459069, a[9,1] = -184091125228237658443815733646470842695 4728061551/2991923615171151921596253813483118262195533733898, a[9,3] = 0, a[9,4] = -14764960804048657303638372252908780219281424435/29816921 02565021975611711269209606363661854518, a[9,5] = -87532504850213044111 8613421785266742862694404520560000/17021203042889441839557167757596133 9495435011888324169, a[9,6] = 7632051964154290925661849798370645637589 377834346780/1734087257418811583049800347581865260479233950396659, a[9 ,7] = 7519834791971137517048532179652347729899303513750000/10456773035 02317596597890707812349832637339039997351, a[9,8] = 136604268348916635 1293315549358278750/144631418224267718165055326464180836641, a[10,1] = -63077736705254280154824845013881/78369357853786633855112190394368, a [10,2] = 0, a[9,2] = 0, a[17,11] = 27418190460279475664846671472967149 325313851242007749/178484880415390961272564938626684644725000000000000 000*7^(1/2)-335375632741370301724806991185800538844107770117192189/624 697081453868364453977285193396256537500000000000000, a[14,8] = -168486 58354789365625/225946578416832+22737651361282524171875/806742258237298 656*7^(1/2), a[16,5] = 0, a[17,1] = 3362117608470634787802979264377180 316187075547467836195538083/802926204124727793979931029542027121634784 00000000000000000000-2989723142262948660161298577240408579957089549899 835987317/882336488049151421955968164331897935862400000000000000000000 *7^(1/2), a[17,3] = 0, c[15] = 1187/2500, a[10,4] = -31948346510820970 247215/6956009216960026632192, a[10,5] = -3378604805394255292453489375 /517042670569824692230499952, a[10,6] = 100158784418332598119809145022 0795/184232684207722503701669953872896, a[10,3] = 0, a[19,2] = 0, a[19 ,3] = 0, a[19,9] = -39035035781656022530508280241969035250845924327906 11750125279777950139652909034268868674113/1251512305360287643825181715 0509421290348952100382848437239951021878978575390625000000000000+52540 8232113400333454065597987198106941861647860588141820154051898978741151 535154361/672597784070787985165164837263462180671344834569519685686600 05491863992343750000000000*7^(1/2), a[19,8] = 605307690199379069086337 38073110755509457740271/5427321242464507241169709941678269540000000000 0+5076681881140470540069505760844878092852841139/678415155308063405146 21374270978369250000000000*7^(1/2), a[18,10] = 24255785943307129426212 38403235096528690693747094280567/1870810238813734784187167630277443093 5937500000000000000-12813005291632184523871567280729171638148255045076 861/213806884435855403907104872031707782125000000000000000*7^(1/2), a[ 19,4] = 0, a[15,1] = 130946139152859396534950567713097/281526916178320 3125000000000000000+90670944595916412828478989403/66148981703437500000 000000000000*7^(1/2), a[17,6] = -9331543777192219488090069025678637265 860768694394667197161/204807867773900816582899908914894308064358400000 000000000000*7^(1/2)+1913610790329116866603247637057156481619346745845 62401842333/1433655074417305716080299362404260156450508800000000000000 000, a[19,6] = 3978177873022717803815319922302122241315215052320566257 /12500480210809376012139887018731342044943750000000000000+781657104433 520790811452964723332865309830826665011511/218758403689164080212448022 82779848578651562500000000000*7^(1/2), a[19,1] = 889600145162128348819 614443326379680969740965957107281357/162577869875050483768335334098969 63784031445312500000000000+5874977494478057298078776231893370296631238 27949833/221087811730274500605800768277445947314453125000000000*7^(1/2 ), a[18,11] = 372098434581358541752287535215351518628082338823827/2051 550349602194957155918834789478675000000000000000*7^(1/2)-5138953584119 531321031442069620009245122067004757018391/118692445476234989246255684 18674528874212500000000000000, a[19,7] = 84956449327194365803183807514 4577515571397490531243/21305240325361534113921806320839108256859177560 30000+310169487453029108907565058918136527805795804669099/717003280180 4362442185223281051622971058377063562500*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..18):\nevalf[40](%); " }}{PARA 12 "" 1 "" {XPPMATH 20 "64/&%\"aG6$\"#>\"\"\"$\"IMA$>Hv6H!3Y KP?U7^r*[<'!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6 $F'\"\"&F0/&F%6$F'\"\"'$\"IrvE\"*pqUfoI*yvqN:H(yFT!#S/&F%6$F'\"\"($\"I $*Q[7.CG()ej,v[>fVS6K^FD/&F%6$F'\"\")$\"IdX]2uuN4\\i1$>2eA&GG88!#R/&F% 6$F'\"\"*$!I=YQEr,NB0,M(3Teg7`B\"HFD/&F%6$F'\"#5$\"ICw\\rl)=B\"41'3td& *RqqW?\"FQ/&F%6$F'\"#6$!I!*f0M(G&z:egA'p56HQWxB#FQ/&F%6$F'\"#7$\"IZ]\" )ebeH<9SWs%*Qk#HZoS%F+/&F%6$F'\"#8$\"IxPv+s'p<'[*y:5lbSs49Q\"F+/&F%6$F '\"#9$!I?)pSG$\\)G%eelr7-Vh#3]T\"FD/&F%6$F'\"#:$\"I!e8O%peW$*ph " 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 -1 223 "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_19 := SimpleOrderConditi ons(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$\"#?\"#<\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[20] = 451/500" "6#/&%\"cG6#\"# ?*&\"$^%\"\"\"\"$+&!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c_20 := 451/500;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%c_20G#\"$^%\"$+&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "eq1 := add(subs(\{op(d7),c[20]=c_20\},d[17 ,i]*c[20]^i),i=1..7)=0:\neq2 := add(subs(\{op(d7),c[20]=c_20\},d[18,i] *c[20]^i),i=1..7)=0:\neq3 := add(subs(\{op(d7),c[20]=c_20\},d[19,i]*c[ 20]^i),i=1..7)=0:\ndd := \{d[1,1]=1,seq(d[1,i]=0,i=2..6),seq(d[9,i]=0, i=1..6),seq(d[13,i]=0,i=1..6)\}:\nsol := solve(subs(dd,\{eq1,eq2,eq3\} ));\ndd_7 := `union`(subs(sol,dd),sol):\nd_7 := `union`(subs(dd_7,d7), dd_7):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<%/&%\"dG6$\"\"*\"\"( ,&#\"fp+DcM5NAmNM(p;\\?C1n]hqN[0&>w07, U4!GGR.+0.MM'32@p!zCBNs9*4#Qs!*>l(ephtS&*Q$\"\"\"*(\"bp++]PMfur,P@B@'f 2bYPi?%\\q7Z%yuLm!zgTaBH;We=7xX\"F0\"cp\\4&yNGMZJ1t.fXA6Ds#HG$f3&=I!>2 !G\\v;qiS+&[:Q2Iu\"p!\"\"F+#F0\"\"#F0/&F(6$F0F+,&#\"Z+]i^z![i'[`,y6'eZ O2bds%yR+paz&F0\"U6Od&oL_4n#G&p\")yo\\?AVzDUIdY3)F4F+F5F0/&F(6$\"#8 F+,&#\"N++]ThQLg(pV#o'3D\"z\"y\")zzfB\"O4j:dG`,nz=GMWSme0<:m^BaF4*(\"K +++wHYQ>)QFy&H1E>)yIi`(F0\"N$zU#e<%F4F+F5F0" }}} {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+$\"&E[#!\"$$\"&Y-$F/$\" &&QeF/F+$\"&_#QF/$!&Cd*F/$\"&$Q@!\"%F+$\"&AJ%!\"#$!&B)G!\"\"$!&xU%F/$! &z-#F@$\"&AX%F@$\"&#)H\"F/75F+F+F+F+F+$!&GD\"F=$!&m_\"F=$!&g%HF=F+$!&/ $>F=$\"&9$[F=$!&'y5F/F+$!&$R'F/75F+F+F+F+F+$\"&!=NF=$\"&lG%F=$\"&NF)F=F+$\"&1U&F=$!&lN\"F@$\"&# HIF/F+$\"&cq$F@$!&'>KF*$!&m&RF=$!&z1#F*$\"&%e[F*$\"&xg\"F=75F+F+F+F+F+ $!&,h&F=$!&]$oF=$!&$>8F@F+$!&Qk)F=$\"&I;#F@$!&:$[F/F+$!&,B%F@$\"&*>VF* $\"&fR&F=$\"&Vi#F*$!&4U'F*$!&`H#F=75F+F+F+F+F+$\"&Hv%F=$\"&2z&F=$\"&w6 \"F@F+$\"&EK(F=$!&H$=F@$\"&C4%F/F+$\"&;N#F@$!&B%HF*$!&<.%F=$!&!z;F*$\" &*)H%F*$\"&Tl\"F=75$!&kr\"F:F+F+F+F+$!&\"z;F=$!&![?F=$!&9-%F=$\"&I@'F: $!&$3FF=$\"&Kh'F=$!&RY\"F/$!&J(Q!\"'$!&L_&F=$\"&%*Q)F@$\"&%o8F=$\"&Za% F@$!&w?\"F*$!&h&[F/Q)pprint566\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "Evaluate the \"weight\" polynomials at th e node " }{XPPEDIT 18 0 "c[20] = 451/500;" "6#/&%\"cG6#\"#?*&\"$^%\" \"\"\"$+&!\"\"" }{TEXT -1 82 " to obtain the linking coefficients in t he 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`(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 21890 "e8 := \{a[19,10] = 3585927476318055832687934783384 46049329593175694761/3262434149717031919969251587397884859085083007812 50+1818048548295397550709386176015734990787971366822/45674078096038446 879569522223570388027191162109375*7^(1/2), a[19,13] = 2110074255562519 597112772669249807784130258601/175219846041652275588428994064404296875 000000000+35255832819592119415922689224605421615051/526501941230926308 85946212158775329589843750*7^(1/2), a[19,11] = -1324008346761444491709 01924810677066147804313106601/6899439957463436409868843249322526549835 2050781250-1531123815144866773159608859731276680985184261266/127094946 58485277597126816511909917328643798828125*7^(1/2), a[16,1] = 101385640 4118125952763527205161498788180173939254661359435974309585825318269273 /198564667739239660374389171210942338307459063268324464130246796875000 00000000000-1931754243710985131731267585181486489297591530273478638247 63410934840962491/2778932721896691403250310788177330670569727481138144 14737187500000000000000000*7^(1/2), a[18,6] = 194471406905941485117086 317959160425736682183804756519030733/143365507441730571608029936240426 0156450508800000000000000000-37992144340278788096731287327181000947543 9610374506979309/70623402680655453994103416867204933815296000000000000 00000*7^(1/2), a[16,6] = -19039890149866495432584929203443766607664789 59434824857368133995942073946993/2610549880447098448451329990884194104 9181178531837828644206625000000000000000+28929035836992861158293213150 542073159989024626464512663622554495682276029/892277791168441852498013 180477996031954434813099925783737531127929687500000*7^(1/2), a[20,14] \+ = -39490630384818824403183059789775920215457543414209/7559060684101278 77278592588033148431241210937500000-1522143939087823199874132919971379 275541283867/30236242736405115091143703521325937249648437500*7^(1/2), \+ c[18] = 4531/10000, a[17,2] = 0, a[7,2] = 0, c[16] = 277/2500, a[16,7] = 1471649975107913251117693334592045704450918568535357968485718078648 3042792373/16088988337674671465702156595052532701656799236967006367882 5639487979375000000+35097161131542028791945288573990002288024169106005 8741934303404427219789/74868036898960959337075562503093457016256710599 371634570393697387568750000*7^(1/2), a[14,13] = 3/392-3/392*7^(1/2), a [18,1] = 6431334448960379167546546351751113222150988727702037620106614 523/151030418995861298047625026656855301579502870400000000000000000000 -4314390544903719789469749993269597109886551871788377645343/1078411263 171185071279516645294541921609600000000000000000000*7^(1/2), a[15,12] \+ = 12163051029345238407733418657/3016875774609375000000000000000, a[16, 9] = 20790434203496917090062867244491983719450919633584853119413284106 748179318736252691355190285977916572557692108955233/158776675835946596 6895356458877254530577780731744189320187958035866716805452345584931662 28541026757812500000000000000+1675029191712848711934999883357656040011 1643601891912316132035529723233388789270175212188770469349713697180572 09/7318101931533341081986828123220267795255614895281860241195526749673 75646957459693466445168584156250000000000000000*7^(1/2), a[15,13] = -5 79945478542537523/549316406250000000000+3727856565327672037/2197265625 000000000000*7^(1/2), a[14,2] = 0, a[18,2] = 0, a[17,12] = -1134903585 52214825884402065259188700089375815491942289/3125294053645738010668658 5123539663360000000000000000000*7^(1/2)+128906890820710317724713918771 43529539151923688844177161/1093852918776008303734030479323888217600000 000000000000000, a[20,15] = -337550312755501644844244842597958841915/5 505016831656111267416623053709095012784-574614004344192211053754485728 553750/26466427075269765708733764681293726023*7^(1/2), a[17,5] = 0, a[ 16,12] = -124817112626650602913170312644481143348777609285630839757851 632016681433/398360478373506965815942720780751835664786233698316038437 5000000000000000+19488568273447271249669733067688159410742449986073234 566291555857980323/290471182147348829240791567235964880172239962071688 7780273437500000000000*7^(1/2), a[16,11] = 330758723778621458037239439 7330733034806289881443017455732775116590947887/21612768292533360416649 32456066917395909700497105040226143341064453125000-1592494813483554306 23321425412532136521666854903188329515308104047100201/5308399229745035 89180860603244506027065540472973167774842224121093750000*7^(1/2), a[18 ,3] = 0, a[16,13] = -4633054354455354087417944839543108308289964752236 8296822645347/50874513520831243973912507590969409217580569705200195312 50000000+1890919683186766311626712563992017565381198275789277564189007 889/113958910286661986501564017003771476647380476139648437500000000000 0*7^(1/2), a[18,4] = 0, a[20,16] = -3323161276094946592204852464911653 467445/21532814382797612400565185774770341026288-436308500467491111575 1565268892211250/103523146071142367310409546994088178011*7^(1/2), a[15 ,14] = 26094995957293704259/549316406250000000000*7^(1/2), a[18,5] = 0 , a[18,7] = 6485285588294739183483110191854194357484624835466068526129 /39583977519448030716210918797855209493755178767778119680000-437194406 9111595199522286141913363272705812493584890149/67127895671292956715864 398992428451861611685605376000000*7^(1/2), a[14,6] = -1424895091977850 07255/1133707871182130048+996623278509306786175/20973595616869405888*7 ^(1/2), a[16,2] = 0, a[20,18] = 0, c[14] = 1/2-1/14*7^(1/2), a[19,14] \+ = -6975558593021648832971791813454390228883838235494737/17497825657641 8490110785321303969544268798828125000000-26886894766313057009210141247 0880702355429907531/6999130263056739604431412852158781770751953125000* 7^(1/2), a[15,2] = 0, a[19,15] = 7309257172110824970392183099608413709 717899/120422243192477433974738629299886453404650000-15346626448658683 257721968231518941311/926324947634441799805681763845280410805*7^(1/2), a[17,13] = -966171676795701349259082624752343417581690261/11315349509 55995511243121650862500000000000000000*7^(1/2)+19423380204481918582213 207680232912549352934689479/611934101477002372480280188786440000000000 0000000000, a[20,19] = 0, a[10,7] = 1870230752313499007680148902744531 25/25224698849808178010752575653374848, a[10,8] = 19081585500709988506 25/117087067039189929394176, a[10,9] = -529568182881566682270449900773 24877908565/2912779959477433986349822224412353951940608, a[11,1] = -10 116106591826909534781157993685116703/956281994503689403044223141187174 4000, a[11,2] = 0, a[11,3] = 0, a[11,4] = -9623541317323077848129/3864 449564977792573440, a[11,5] = -4823348333146829406881375/5764132336341 41239944816, a[11,6] = 6566119246514996884067001154977284529/970305487 021846325473990863582315520, a[11,7] = 2226455130519213549256016892506 730559375/364880443159675255577435648380047355776, a[11,8] = 397472627 82380466933662225/1756032802431424164410720256, a[11,9] = 481757714192 60955335244683805171548038966866545122229/1989786420513815146528880165 952064118903852843612160000, a[11,10] = -2378292068163246/477687284872 11875, a[12,1] = -3218022174758599831659045535578571/14533967536344695 25663775847094384, a[12,2] = 0, a[12,3] = 0, a[12,4] = 262900926042842 31996745/5760876126062860430544, a[12,5] = -69706929756092645204558671 0000/41107967755245430594036502319, a[14,1] = -26242/1715+588937/10152 8*7^(1/2), a[18,8] = -150270859293330543141372845203694269068672216982 3969/13338184685480772995898679152668515221504000000000000*7^(1/2)+585 89616623242646214825661347124714930789685430959801/2160785919047885225 33558602273229946588364800000000000, a[15,3] = 0, a[16,14] = 378070792 470829109357504373975352653612596622387047615352035143989/386524657263 11665474213406887371187527810692671413726806640625000000-7366944851192 03754900544593199058955707390426548138628337098463663/2473757806483946 5903496580407917560017798843309704785156250000000000*7^(1/2), a[19,17] = 0, a[14,12] = 1100613127343/48439922837376-29746300739/142470361286 4*7^(1/2), a[17,9] = -264701762796489849333164460056987686273182643308 013876537822807951183236955127319033004381711/265741917838857467723355 8327633544805137160916516236750792782073750348797330432000000000000000 0*7^(1/2)-115937771802925638794257846529122507932237816077370768964977 0373007738863712156809629044640218707/11626208905450014212896817683396 7585224750790097585357847184215726577759883206400000000000000000000, a [12,6] = 1827357820434213461438077550902273440/13938101391424531770956 7680839641697, a[12,7] = 643504802814241550941949227194107500000/24212 4609118836550860494007545333945331, a[12,8] = 162259938151380266113750 /59091082835244183497007, a[12,9] = -230282516328735238185454148568570 15616678575554130463402/2001316918319144450344390524040560334997842450 4151629055, a[12,10] = 7958341351371843889152/3284467988443203581305, \+ a[12,11] = -507974327957860843878400/121555654819179042718967, a[13,1] = 4631674879841/103782082379976, a[13,2] = 0, a[19,18] = 0, a[15,8] = 16161913072172934315785836819/9932509483065000000000000000-7598066843 6324671626317237501/49917227145660000000000000000*7^(1/2), a[15,4] = 0 , a[16,15] = 460888287630742936877525144387640337292487250940862140028 3/75922278949360871648575860105318720457058558153486785997500-84914702 986428741912331429025972417475312941024930181294/531455952645526101540 0310207372310431994099070744075019825*7^(1/2), a[16,3] = 0, a[17,14] = 15739031823875445778785537999179335674270343962861810116243/309619125 620788378533311524058377597436400000000000000000000+606652050483146218 029195891134085469196882580940598209/123847650248315351413324609623351 03897456000000000000000*7^(1/2), a[14,7] = 663073659206904988750000000 /13716692475747279030488259-9227445425051062363134765625/5075176216026 49324128065583*7^(1/2), a[20,17] = 0, a[19,12] = 116148895768336301414 698893782124924558205330714623/317921351485772502712876232132361484375 0000000000000+3168841339481577057649164025908284942240477511813/111272 4730200203759495066812463265195312500000000000*7^(1/2), a[8,2] = 0, a[ 14,4] = 0, a[18,16] = 288266095365211419064692734565913406048103319/57 16563462810862432861157467909400165793792000+6083340852719127131342114 4941358828000769/1256387574244145589639814828111956080394240*7^(1/2), \+ a[15,5] = 0, a[14,9] = 17318895166382455115817006657422200600438218265 372153/554375052657198964316171246174366066645917060800-27496808765423 8523490991092528596412316293248643447707/23287013241323869436363405111 359694599520551359840*7^(1/2), a[18,13] = -104358012121727430746226125 41578159307187516521/1035144936615299597322411288011250000000000000000 0*7^(1/2)+101126309543475847280329109976443851552294375363/42635672636 828181434269613446800000000000000000000, a[17,8] = -264336341576216974 7333266167756151785333485317345751/27702383577536990068404949009388454 690816000000000000*7^(1/2)+2345191818743325882006422291360850214750389 8444041667/72026197301596174177852867424409982196121600000000000, a[14 ,10] = -9081365921021033056/120504023637725+25400013053688208472/89172 9774919165*7^(1/2), a[18,14] = 688260040952087965212862705907587124167 147667584978421209/114673750229921621679004268169769480532000000000000 00000000+26528592722953668251624457086005829986967133546824867/4586950 00919686486716017072679077922128000000000000000*7^(1/2), a[18,9] = 141 7729703147854028657911907661502029810745490260742491447432512293006941 9330134278983169782333079/34878626716350042638690453050190275567425237 0292756073541552647179733279649619200000000000000000000-11575276557102 9020017132140741067326567200560399856892151316170393623103940194194726 94886093/9842293253291017323087253065309425204211707098208284262195489 16203832887900160000000000000000*7^(1/2), a[15,6] = 335394319040214097 6803568136084793/19770296302701765625000000000000000-25574898891779499 9289135717669191/8649504632432022460937500000000000*7^(1/2), a[15,11] \+ = -207941800836756311792092915907/81839239362061500549316406250+108910 06522099224595926000559657/4582997404275444030761718750000*7^(1/2), a[ 15,10] = 14462059099710033235500874587/10319489926824569702148437500-3 48155288848452187540396337457/252574928278923034667968750000*7^(1/2), \+ a[16,8] = -28591353782492183701826251126306766396461229874660236919089 505718702483/307389544220068686894086053026978697505927632660329659393 43750000000000+3262171864570727799722782444400697640060225376834843457 00377402630409/1743722897253444403020583118152408439159931363492594965 600000000000000*7^(1/2), a[17,15] = 1409310826030880333986466100980399 4368997/667116832617261294418298462366640626401280*7^(1/2)-12774251849 156994331101322175199160614539284227/197299803246555027824211770244933 965258178560000, a[14,3] = 0, a[18,15] = -4055432865965858423757297182 9594948977715703/1953463398480742849744670992524098665922560000+216315 902102540063015384273123325235039761/867251882402439682743788001076632 8143216640*7^(1/2), c[17] = 3971/10000, a[19,5] = 0, a[15,9] = -372785 0143078386229058076561522609367827766330696417108435354320431359/80163 49226513859159668160608032382937598583484731440429687500000000000+8142 51349796512127448124174135016152994969999173763717229208919303/1589842 118264019833619870377392538080518560957031250000000000000000*7^(1/2), \+ c[20] = 451/500, a[17,10] = -75888621039765715002449611997758928569635 156851427/1495153038012974852497236867354599875000000000000000*7^(1/2) +382869589231990608613031660622192244633710639124838997/17007365807397 58894715606936615857357812500000000000000, a[18,17] = 0, a[16,4] = 0, \+ a[18,12] = -44665892876391075132214543352848603614252057936533163/1041 7646845485793368895528374513221120000000000000000000*7^(1/2)+114737526 96537480050786046301681238985163187194373379961/1093852918776008303734 030479323888217600000000000000000000, c[19] = 1773/2500, a[16,10] = -1 7520323069888524301525598935749031032957349045468285405735167082288312 79/2433262830102477113639615580994907630153630681641719830799102783203 125000+818839996665861726917359740393249069399644466975014508425506064 13648603/7337223303078238681128686982692336854001717132335032105178833 00781250000*7^(1/2), a[14,5] = 0, c[2] = 1/16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[3,1] = 18928/1134225, a[3,2] = 100352/1134225, a[ 17,16] = 5632103863128032194320344991957663639629/13733791297405640048 6943321291590340771840*7^(1/2)+218654815092393314149746759032309299999 8671947/40617687762077180444013487271987843283271680000, a[17,4] = 0, \+ a[13,3] = 0, a[13,4] = 0, a[13,5] = 0, a[13,6] = 14327219974204125/404 89566827933216, a[13,7] = 2720762324010009765625000/109173674806968139 22225349, a[13,8] = -498533005859375/95352091037424, a[13,9] = 4059320 30463777247926705030596175437402459637909765779/7880391943632184108320 1886041201537229769115088303952, a[13,10] = -10290327637248/1082076946 951, a[13,11] = 863264105888000/85814662253313, a[13,12] = -2974630073 9/247142463456, a[14,11] = 26288142988644492320/221311497390123-367631 244295366501400/8188525403434551*7^(1/2), a[4,2] = 0, a[4,1] = 14/355, a[4,3] = 42/355, a[5,1] = 94495479/250880000, a[5,2] = 0, a[5,3] = -3 52806597/250880000, a[5,4] = 178077159/125440000, a[17,7] = -999769894 25049316729418744001270347141667794343723953623/1812453183124909831328 338772795568200263515511345152000000*7^(1/2)+2150471462297242216405083 705721954726258115913698749425643/131946591731493435720703062659517364 97918392922592706560000, a[6,1] = 12089/252720, a[6,2] = 0, a[6,3] = 0 , a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a[7,1] = 2140089 9/350000000, a[7,3] = 0, a[7,4] = 3064329829899/27126050000000, a[7,5] = -21643947/592609375, a[7,6] = 124391943/6756250000, a[19,16] = -143 8617606921480386578336383098493573/44732223610987442664991779565346743 585*7^(1/2)-671478610426445930422769177742151257962163/581518906942836 7546448931343495076666050000, a[15,7] = 105966145406009687447923627586 69/45489031169570058009272287500000-13058425932946007192467094947/1451 071902055187778276093750000*7^(1/2), a[8,1] = -15365458811/13609565775 , a[8,3] = 0, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8 ,6] = 341936800488/47951126225, a[8,7] = 1993321838240/380523459069, a [9,1] = -1840911252282376584438157336464708426954728061551/29919236151 71151921596253813483118262195533733898, a[9,3] = 0, a[9,4] = -14764960 804048657303638372252908780219281424435/298169210256502197561171126920 9606363661854518, a[9,5] = -875325048502130441118613421785266742862694 404520560000/170212030428894418395571677575961339495435011888324169, a [9,6] = 7632051964154290925661849798370645637589377834346780/173408725 7418811583049800347581865260479233950396659, a[9,7] = 7519834791971137 517048532179652347729899303513750000/104567730350231759659789070781234 9832637339039997351, a[9,8] = 1366042683489166351293315549358278750/14 4631418224267718165055326464180836641, a[10,1] = -63077736705254280154 824845013881/78369357853786633855112190394368, a[10,2] = 0, a[9,2] = 0 , a[17,11] = 27418190460279475664846671472967149325313851242007749/178 484880415390961272564938626684644725000000000000000*7^(1/2)-3353756327 41370301724806991185800538844107770117192189/6246970814538683644539772 85193396256537500000000000000, a[14,8] = -16848658354789365625/2259465 78416832+22737651361282524171875/806742258237298656*7^(1/2), a[16,5] = 0, a[17,1] = 33621176084706347878029792643771803161870755474678361955 38083/80292620412472779397993102954202712163478400000000000000000000-2 989723142262948660161298577240408579957089549899835987317/882336488049 151421955968164331897935862400000000000000000000*7^(1/2), a[17,3] = 0, c[15] = 1187/2500, a[10,4] = -31948346510820970247215/695600921696002 6632192, a[10,5] = -3378604805394255292453489375/517042670569824692230 499952, a[10,6] = 1001587844183325981198091450220795/18423268420772250 3701669953872896, a[10,3] = 0, a[20,8] = -3454743395401760637010669367 685476447099876393/1406761666046800276911188816883007464768000000+8622 1640206701158533615034815831351063336969/87922604127925017306949301055 1879665480000000*7^(1/2), a[20,3] = 0, a[19,2] = 0, a[19,3] = 0, a[20, 10] = -341601321080417106938553052190791919279008005951/66435022685146 831824828395961556928039550781250+141448393038235916181387355191195129 2291436/27116335789855849724419753453696705322265625*7^(1/2), a[19,9] \+ = -3903503578165602253050828024196903525084592432790611750125279777950 139652909034268868674113/125151230536028764382518171505094212903489521 00382848437239951021878978575390625000000000000+5254082321134003334540 65597987198106941861647860588141820154051898978741151535154361/6725977 8407078798516516483726346218067134483456951968568660005491863992343750 000000000*7^(1/2), a[19,8] = 60530769019937906908633738073110755509457 740271/54273212424645072411697099416782695400000000000+507668188114047 0540069505760844878092852841139/67841515530806340514621374270978369250 000000000*7^(1/2), a[20,11] = 2267169993511836812396510162952693484387 133401903/463642365141542926743186266354473784148925781250-27517871172 8623278446792413948735774971947548/17430164103065523561773919787762172 33642578125*7^(1/2), a[18,10] = 24255785943307129426212384032350965286 90693747094280567/1870810238813734784187167630277443093593750000000000 0000-12813005291632184523871567280729171638148255045076861/21380688443 5855403907104872031707782125000000000000000*7^(1/2), a[20,12] = -10393 46366321461511148422012098015726047706061669/2136431481984391218230528 2799294691750000000000000+40679628763326814339683742956800076344180701 /10900160622369342950155756530252393750000000000*7^(1/2), a[20,4] = 0, a[20,5] = 0, a[19,4] = 0, a[15,1] = 130946139152859396534950567713097 /2815269161783203125000000000000000+90670944595916412828478989403/6614 8981703437500000000000000000*7^(1/2), a[17,6] = -933154377719221948809 0069025678637265860768694394667197161/20480786777390081658289990891489 4308064358400000000000000000*7^(1/2)+191361079032911686660324763705715 648161934674584562401842333/143365507441730571608029936240426015645050 8800000000000000000, a[20,13] = -9599705385847962444351258872672433354 9134483/4541698409399626983252079526149359375000000000+598780030683267 309831447210391270733521/682346515835280496281862909577728271484375*7^ (1/2), a[20,1] = 17566897872345846391613972965569453561130934404093320 17/29796089606191070479724003049411162716915812500000000000+1071640173 525649463414934424316330743362810391153/307734545218035512679955414457 274670710937500000000*7^(1/2), a[20,2] = 0, a[19,6] = 3978177873022717 803815319922302122241315215052320566257/125004802108093760121398870187 31342044943750000000000000+7816571044335207908114529647233328653098308 26665011511/21875840368916408021244802282779848578651562500000000000*7 ^(1/2), a[19,1] = 8896001451621283488196144433263796809697409659571072 81357/16257786987505048376833533409896963784031445312500000000000+5874 97749447805729807877623189337029663123827949833/2210878117302745006058 00768277445947314453125000000000*7^(1/2), a[20,7] = 286874742630756193 550077645094469028757436872008145/644270467438932791604995423142174633 687421529423472+35835879510547961853908519812981470449630323307/632133 504159078484698778868860061453774942630910*7^(1/2), a[20,6] = 12660795 150694149245758947395121627071932701286934343/280010756722130022671933 46921958206180674000000000000+3344810444582948978921965249456125361330 745579349/71431315490339291497942211535607668828250000000000*7^(1/2), \+ a[18,11] = 372098434581358541752287535215351518628082338823827/2051550 349602194957155918834789478675000000000000000*7^(1/2)-5138953584119531 321031442069620009245122067004757018391/118692445476234989246255684186 74528874212500000000000000, a[19,7] = 84956449327194365803183807514457 7515571397490531243/21305240325361534113921806320839108256859177560300 00+310169487453029108907565058918136527805795804669099/717003280180436 2442185223281051622971058377063562500*7^(1/2), a[20,9] = 6791979978688 1958166108879101410983034021098405578199468040890142468178502059414851 64496753/2270743926845705900956409703788429398920913869093464020452816 713409721872718875000000000000+948800372201243464997778716741507502515 97084733453730150947384698361284577054335299/9268342558553901636556774 301177262852738423955483526614093129442488660704975000000000*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^ozbyf/E$y=q8PFBp\\q\"o!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F' \"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"I9%***RCP4t(o<'H% egHQE/w&!#S/&F%6$F'\"\"($\"I%f#**fnpkjXCXZFfB#z#f_fFD/&F%6$F'\"\")$!IS EJ!G\\+.lu'y'R&))pUiN'>#!#R/&F%6$F'\"\"*$\"IfeX2>wE3)>rK?=*[%)f;=IFQ/& F%6$F'\"#5$!Ii5&yh0swFU2&)G!4:)=F+/&F%6$F'\"#9$!I;q#)G8+M\\,>/B3Vd-SMa=FD/&F%6$F'\"#: $!ICxv+>owbq$p2.WL<,*e(=\"FD/&F%6$F'\"#;$!I;riF.US0U[S,g>g0*y$eEFD/&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 (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_ 20 := SimpleOrderConditions(7,20,'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]:\nordeqns8 := []:\nfor ct in \+ whch do\n eqn_group := convert(SO7_20[ct],'polynom_order_conditions' ,7):\n ordeqns8 := [op(ordeqns8),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 nops(ordeqns8) do\n eqns8 := [op(eqns8),expa nd(subs(e8,ordeqns8[ct]))];\nend do:\nnops(eqns8);\nnops(indets(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) minus \+ \{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 would \+ like to ensure that " }{XPPEDIT 18 0 "a[21,19] = 0;" "6#/&%\"aG6$\"#@ \"#>\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[21,18] = 0;" "6#/&%\"aG 6$\"#@\"#=\"\"!" }{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] = 63/625" "6#/&%\"cG6#\"#@*&\"#j\"\"\"\"$D'!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "c_21 := 63/625;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c_21G#\"#j\"$D'" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 480 "eq1 := ad d(subs(\{op(d8),c[21]=c_21\},d[17,i]*c[21]^i),i=1..7)=0:\neq2 := add(s ubs(\{op(d8),c[21]=c_21\},d[18,i]*c[21]^i),i=1..7)=0:\neq3 := add(subs (\{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,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)\}:\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$\"\"\"\"\"(,&#\" enD1*GJs.@6ib$>cA#33]ByC\\#GPY$oAz#\"UGjx$R>6Fb/\")G_h*F**( \"cpv$f384;@1;nIghomz8Wi!e&*e?<:Z:*4:qeeC290TG\\Z$)4(F*\"apiy,n03HE:,H Z)R;s??\"F0F+F4F*/&F(6$\"#8F+,&#\"N+++]\\ vhDFS%)HF.BP!)G[x.m$\"J(fuHB6L4\"z*3)f'fq++ymt8F0*(\"K++++@O\\rhuMdkf% fgHqYu$F*\"I$yC&[gq)e5oe$y#yo#pFn'R(F0F+F4F*/&F(6$\"#7F+,&#\"SD1*GO%*4 `726-?psMCF*\"N3LkKeR.0P$39wp')4(zH4738\"F0F+F4F*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 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 9031 "d_8 := \{d[20,2] = \+ -20600996052418012792375/2542130581649162295528+61736707982680046875/2 8887847518740480631*7^(1/2), d[7,1] = 0, d[4,1] = 0, d[2,4] = 0, d[5,1 ] = 0, d[2,2] = 0, d[4,4] = 0, d[2,3] = 0, d[5,7] = 0, d[5,3] = 0, d[3 ,5] = 0, d[11,1] = 0, d[6,4] = 0, d[10,1] = 0, d[3,6] = 0, d[6,6] = 0, d[11,2] = 0, d[2,5] = 0, d[3,1] = 0, d[4,2] = 0, d[6,1] = 0, d[9,1] = 0, d[20,1] = 0, d[8,5] = 0, d[7,4] = 0, d[5,2] = 0, d[12,1] = 0, d[14 ,6] = -1051318486405420129705673046875000000000000/2488555712472690724 43439300557791523679+903465904102648467801171875000000000000000/248855 571247269072443439300557791523679*7^(1/2), d[1,1] = 1, d[2,1] = 0, d[2 ,6] = 0, d[13,3] = 0, d[13,4] = 0, d[13,5] = 0, d[13,6] = 0, d[13,2] = 0, d[3,2] = 0, d[4,3] = 0, d[8,7] = -24190259867134395100312852735623 6760826601409912109375/11634347372303391191909857895396933031479568475 2+82277791751252467004276917779243338908233642578125/14542934215379238 989887322369246166289349460594*7^(1/2), d[11,5] = 0, d[13,1] = 0, d[1, 5] = 0, d[1,4] = 0, d[1,2] = 0, d[1,3] = 0, d[5,5] = 0, d[4,5] = 0, d[ 4,6] = 0, d[16,1] = 0, d[5,6] = 0, d[11,6] = 0, d[12,2] = 0, d[2,7] = \+ 0, d[11,7] = 1166494014177709823840868926041895265397466112500000000/3 14119254049735043688523111755715362082056942597-1505806486898795758095 02994061562968055450000000000/1653259231840710756255384798714291379379 2470663*7^(1/2), d[5,4] = 0, d[8,4] = 0, d[3,3] = 0, d[11,3] = 0, d[7, 6] = 0, d[7,2] = 0, d[19,1] = 0, d[1,7] = -279226834637282492478235008 0822256193556211210372312890625/46261761393856209241320256424571704552 71119393776328+1411055394669868358903790160743262902363747721484375/70 23546770322298973884654315471412634520930253709*7^(1/2), d[9,7] = 1012 6312525310666642583692933784022667332185255807527788923798592152630547 6895814056563671875/96152288119338549706234864645096229378757960134138 537657731187783209221032644536142896+709834749284105140724585870150991 5471517205895580624413796668616030671606211609130859375/12019036014917 3187132793580806370286723447450167673172072163984729011526290805670178 62*7^(1/2), d[13,7] = -366037748288037230327298440272561754950000000/1 3736678000070596598089791093311232974597+37446702960594596457347461714 9362100000000/7396672769268782783586810588706048524783*7^(1/2), d[8,1] = 0, d[9,4] = 0, d[9,5] = 0, d[9,2] = 0, d[9,3] = 0, d[7,5] = 0, d[9, 6] = 0, d[10,4] = 0, d[12,7] = -55726538824572782966617692002110712530 994362890625/904649674383767893580912666964027166661146464+24347219862 229912958790893713089040314939453125/113081209297970986697614083370503 395832643308*7^(1/2), d[4,7] = 0, d[8,3] = 0, d[14,5] = 50836894677451 5707115930200307812500000000/82951857082423024147813100185930507893-43 6874283154411671226199795312500000000000/82951857082423024147813100185 930507893*7^(1/2), d[6,5] = 0, d[16,2] = -31835385463885924469375/1215 011092931332731216-23744887685646171875/11682798970493583954*7^(1/2), \+ d[19,4] = 44029025120822998046875/6758650328656681806-8565555475646972 6562500/37172576807611749933*7^(1/2), d[18,1] = 0, d[20,7] = -42481084 699691412746906280517578125/315362910788309233427469084963*7^(1/2)+126 3867039460241389462947845458984375/2522903286306473867419752679704, d[ 16,7] = 36404054265403955227717588669019233461628234584629535675048828 125/289662237149974340779355632320254494882163414019279349419306*7^(1/ 2)+8775148742002910641879651470479580555659384192067982912063598632812 5/30124872663597331441052985761306467467744995058005052339607824, d[20 ,5] = 4598009683496093750000/35307369189571698549-89928474853515625000 00/317766322706145286941*7^(1/2), d[17,4] = -3508451522825000000000000 0/4315774984633564433289*7^(1/2)-36080953985142100000000000/4315774984 633564433289, d[18,3] = 3033559184729696320000000/42070548396235574721 87-44908515477152200000000000/4207054839623557472187*7^(1/2), d[11,4] \+ = 0, d[10,5] = 0, d[17,3] = 12766898301839882560000000/431577498463356 4433289+44908515477152200000000000/12947324953900693299867*7^(1/2), d[ 15,3] = 43855972145656445312500/5526505228259336139*7^(1/2)-4783897174 2261640625/15745029140339989, d[18,2] = -885137539385394547120000/1402 351613207852490729+1357677232949840000000/547579700588774889*7^(1/2), \+ d[8,2] = 0, d[19,6] = -21133422851562500000000/10137975492985022709*7^ (1/2)+672252551269531250000000/111517730422835249799, d[14,7] = 542235 5613200908620290203505772682593745702919560149649118224086906647000000 00/1859021223078593139914873615054885635480086463411305200599596130020 9099-66582885099706311613896024673382442102666438705266691472654952231 562500000000/265574460439799019987839087864983662211440923344472171514 2280185744157*7^(1/2), d[16,6] = -4226684570312500000000/2278145799246 24887103*7^(1/2)-278063454589843750000000/2505960379170873758133, d[18 ,5] = -92139154600000000000000/8603384130109524483-4604337912500000000 0000000/1402351613207852490729*7^(1/2), d[16,5] = 92972318793945312500 0/5124663352087676397+22482118713378906250000/835320126390291252711*7^ (1/2), d[15,2] = 15293280620121252711875/14737347275358229704-26119376 4542107890625/141705262263059901*7^(1/2), d[18,7] = -21750315366242003 3264160156250000000000/1391744987082474266353028309847*7^(1/2)+7149849 7831104031354675292968750000000/1391744987082474266353028309847, d[1,6 ] = 0, d[17,1] = 0, d[15,7] = 2817634002462947885329102458246328358048 74917922914028167724609375/2425245669905569667777051970810579327523481 340831653948217*7^(1/2)-1848406548570004811129788854162950909818789685 3454477787017822265625/25222554967017924544881340496430025006244205944 6492010614568, d[19,7] = 1062027117492285318672657012939453125/7378284 7242696362379374003238*7^(1/2)-23573555717949801752126216888427734375/ 590262777941570899034992025904, d[16,3] = 32636018887305525546875/3855 32366026288270482+21927986072828222656250/2505960379170873758133*7^(1/ 2), d[7,3] = 0, d[14,2] = -38389984698577741970967104069696181677000/8 2951857082423024147813100185930507893+32990994339665941218873881332513 829000000/82951857082423024147813100185930507893*7^(1/2), d[14,4] = -1 2495844482388965355748423499709375000000/26758663574975169079939709737 39693803+10738486556445883081521282934375000000000/2675866357497516907 993970973739693803*7^(1/2), d[12,3] = 0, d[10,6] = 0, d[3,4] = 0, d[7, 7] = -6294239953663999920667311303343622553411352752208709716796875000 /13320782405460177568954189151542541478313249275933402807027+335127826 8463546478645533706838327067660627136230468750000000/10246755696507828 89919553011657118575254865328917954062079*7^(1/2), d[15,6] = -92987060 546875000000000/5526505228259336139*7^(1/2)-17818364257812500000000/55 26505228259336139, d[20,3] = 11663518684634183828125/31776632270614528 6941-8771194429131289062500/953298968118435860823*7^(1/2), d[17,6] = - 8656250000000000000000000/1177029541263699390897*7^(1/2)-1261795606250 00000000000000/12947324953900693299867, d[10,2] = 0, d[19,2] = 3435124 5211720565153125/54069202629253454448-602662124001171875/2639066899124 046*7^(1/2), d[8,6] = 0, d[6,3] = 0, d[6,7] = -58837713738122452463883 77557345824733447977899072265625/1646772784617035444723231261145356618 2180348426656+5560852894751337061123172170242530505378939892578125/205 8465980771294305904039076431695772772543553332*7^(1/2), d[10,7] = -114 6994512702984249773470057235323567045167700000000/44009729090780619144 0015101144663171480985291+13244347280476214853085643899818686878900000 00000/440097290907806191440015101144663171480985291*7^(1/2), d[16,4] = -1886245630834033203125/11682798970493583954-17131110951293945312500/ 835320126390291252711*7^(1/2), d[18,4] = 149942784621100000000000/4523 7148813156531959+1131758555750000000000000/45237148813156531959*7^(1/2 ), d[17,7] = 217503153662420033264160156250000000000/42831328061156128 07941089279927*7^(1/2)+131520828359960046379699707031250000000/4283132 806115612807941089279927, d[20,4] = -29160124418345166015625/317766322 706145286941+6852444380517578125000/317766322706145286941*7^(1/2), d[1 0,3] = 0, d[15,1] = 0, d[12,4] = 0, d[12,5] = 0, d[12,6] = 0, d[19,3] \+ = -614404107119608542578125/223035460845670499598+10963993036414111328 1250/111517730422835249799*7^(1/2), d[18,6] = 952187500000000000000000 00/4207054839623557472187*7^(1/2)+47857960625000000000000000/420705483 9623557472187, d[19,5] = -321187324432861328125000/3717257680761174993 3+112410593566894531250000/37172576807611749933*7^(1/2), d[14,1] = 0, \+ d[3,7] = 0, d[14,3] = 495838818700665936753060404847037872500000/24885 5571247269072443439300557791523679-42610633449266130907561469147593250 0000000/248855571247269072443439300557791523679*7^(1/2), d[17,2] = -31 6091944871321840000000/392343180421233130299*7^(1/2)-23510526199152774 52880000/4315774984633564433289, d[20,6] = 1690673828125000000000/8666 3542556221441893*7^(1/2)-93909057617187500000000/953298968118435860823 , d[6,2] = 0, d[17,5] = 46043379125000000000000000/4315774984633564433 289*7^(1/2)+6075378534200000000000000/479530553848173825921, d[15,5] = 49230456347656250000/204685378824419857+44964237426757812500000/18421 68409419778713*7^(1/2), d[15,4] = 15789793137646484375/457113749235677 1-1105232964599609375000/59424787400638023*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 0 136 "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%'matr ixG6#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,$!)n[Ig!\"#F+F+F+F+$!)lV,NF0$!)lfQYF0$!)ACk?!\"\"$\")ira5F7$!)sE)f #F7Q)pprint626\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7,$ \"\"!F)F(F(F(F(F(F(F(F(F(7,F(F(F($\")7]%*e!\"&$!)\\(*QQ!\"%$!)B\"z:$! \"'$!)>JwEF0$\")StGfF0$\"(TJ6$F-$!)5b\\C!\"(7,F(F(F($!)tuPDF0$\")#>dz \"!\"$$\")..y5F-$\")H^87FB$!)+7_FFB$!(n``\"F0$\")]9O7F37,F(F(F($\")2!y %fF0$!)0PvXFB$!)<:d@F-$!)I&o)HFB$\")5p]pFB$\"(=&zTF0$!)h=rMF37,F(F(F($ !)Kj0yF0$\")l)=['FB$\")1IEDF-$\")bf*3%FB$!)Nvd(*FB$!(TjR'F0$\")$y_`&F3 7,F(F(F($\")>u!Q&F0$!)e1uZFB$!)$z/g\"F-$!)1L?HFB$\")OsDrFB$\"(:$H^F0$! )v\\*o%F37,$\")4W*o$!\"\"$!)\"[I5'FB$!)*z7l#FB$!)UX;PF)$\")s(4M#\"\"\" $\")lVXKFgp$\")Wh];F`q$!)72@OF`q$!(iX&=F)$\")VgX9!\"#Q)pprint636\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Evaluate the \"weight\" polynomials at the node " }{XPPEDIT 18 0 "c[21] = 63/ 625;" "6#/&%\"cG6#\"#@*&\"#j\"\"\"\"$D'!\"\"" }{TEXT -1 82 " to obtain the linking coefficients in the next stage in the interpolation schem e." }}{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 24670 "e9 := \{a[19,10] = 358592 747631805583268793478338446049329593175694761/326243414971703191996925 158739788485908508300781250+181804854829539755070938617601573499078797 1366822/45674078096038446879569522223570388027191162109375*7^(1/2), a[ 19,13] = 2110074255562519597112772669249807784130258601/17521984604165 2275588428994064404296875000000000+35255832819592119415922689224605421 615051/52650194123092630885946212158775329589843750*7^(1/2), a[19,11] \+ = -132400834676144449170901924810677066147804313106601/689943995746343 64098688432493225265498352050781250-1531123815144866773159608859731276 680985184261266/12709494658485277597126816511909917328643798828125*7^( 1/2), a[16,1] = 101385640411812595276352720516149878818017393925466135 9435974309585825318269273/19856466773923966037438917121094233830745906 326832446413024679687500000000000000-193175424371098513173126758518148 648929759153027347863824763410934840962491/277893272189669140325031078 817733067056972748113814414737187500000000000000000*7^(1/2), a[21,9] = 777700801943859198150427617314612940851111827646018134189347731877322 026062559851954409/698388563259089086956016582059677527363222773458864 3101138365525602108580017089843750000+87224493992030839692237111724153 841314003426044894712796733463953784892697128253/139677712651817817391 2033164119355054726445546917728620227673105120421716003417968750*7^(1/ 2), a[18,6] = 19447140690594148511708631795916042573668218380475651903 0733/1433655074417305716080299362404260156450508800000000000000000-379 921443402787880967312873271810009475439610374506979309/706234026806554 5399410341686720493381529600000000000000000*7^(1/2), a[21,10] = -64217 01118160451978891722278844550761287121305344/2330310106940737085692322 5624270606136322021484375+29660556802459117448907062485917151441054003 2/932124042776294834276929024970824245452880859375*7^(1/2), a[16,6] = \+ -190398901498664954325849292034437666076647895943482485736813399594207 3946993/26105498804470984484513299908841941049181178531837828644206625 000000000000000+289290358369928611582932131505420731599890246264645126 63622554495682276029/8922777911684418524980131804779960319544348130999 25783737531127929687500000*7^(1/2), a[20,14] = -3949063038481882440318 3059789775920215457543414209/75590606841012787727859258803314843124121 0937500000-1522143939087823199874132919971379275541283867/302362427364 05115091143703521325937249648437500*7^(1/2), a[21,11] = 13545515131673 834908011984508226654525037840092928/344971997873171820493442162466126 32749176025390625-1748566585490672015032078287521919035134774272/18156 42094069325371018116644558559618377685546875*7^(1/2), c[18] = 4531/100 00, a[17,2] = 0, a[7,2] = 0, c[16] = 277/2500, a[16,7] = 1471649975107 9132511176933345920457044509185685353579684857180786483042792373/16088 9883376746714657021565950525327016567992369670063678825639487979375000 000+350971611315420287919452885739900022880241691060058741934303404427 219789/748680368989609593370755625030934570162567105993716345703936973 87568750000*7^(1/2), a[21,14] = -8713024202764221725399203625351091305 38767960552/2734035259006538907981020645374524129199981689453125-26867 0858927822983962285521447752757131267008/87489128288209245055392660651 9847721343994140625*7^(1/2), a[14,13] = 3/392-3/392*7^(1/2), a[21,15] \+ = 41976403293032364129548157236863938645504/10751985999328342319173091 90177557619684375-17526079826269964397026946211686912/1323321353763488 28543668823406468630115*7^(1/2), a[18,1] = 643133444896037916754654635 1751113222150988727702037620106614523/15103041899586129804762502665685 5301579502870400000000000000000000-43143905449037197894697499932695971 09886551871788377645343/1078411263171185071279516645294541921609600000 000000000000000*7^(1/2), a[15,12] = 12163051029345238407733418657/3016 875774609375000000000000000, a[16,9] = 2079043420349691709006286724449 1983719450919633584853119413284106748179318736252691355190285977916572 557692108955233/158776675835946596689535645887725453057778073174418932 018795803586671680545234558493166228541026757812500000000000000+167502 9191712848711934999883357656040011164360189191231613203552972323338878 927017521218877046934971369718057209/731810193153334108198682812322026 7795255614895281860241195526749673756469574596934664451685841562500000 00000000000*7^(1/2), a[15,13] = -579945478542537523/549316406250000000 000+3727856565327672037/2197265625000000000000*7^(1/2), a[21,16] = 589 7134866562232986678801419199510559232/51921330977038995950436886995491 755946875-1642923094709652697468208840879616/6390317658712491809284539 937906677655*7^(1/2), a[21,12] = -115554550906634122759578446135576773 50426991089/1774114684630426912460246831095767211914062500000+10097279 02126398950226975944069228679941169/4435286711576067281150617077739418 0297851562500*7^(1/2), a[14,2] = 0, a[21,17] = 0, a[18,2] = 0, a[21,13 ] = -964231278050279193617382643303593077359488/3422262618001021007586 50379032039642333984375+140919432581309585388289967925647945472/263250 97061546315442973106079387664794921875*7^(1/2), a[17,12] = -1134903585 52214825884402065259188700089375815491942289/3125294053645738010668658 5123539663360000000000000000000*7^(1/2)+128906890820710317724713918771 43529539151923688844177161/1093852918776008303734030479323888217600000 000000000000000, a[20,15] = -337550312755501644844244842597958841915/5 505016831656111267416623053709095012784-574614004344192211053754485728 553750/26466427075269765708733764681293726023*7^(1/2), a[17,5] = 0, a[ 21,18] = 0, a[16,12] = -1248171126266506029131703126444811433487776092 85630839757851632016681433/3983604783735069658159427207807518356647862 336983160384375000000000000000+194885682734472712496697330676881594107 42449986073234566291555857980323/2904711821473488292407915672359648801 722399620716887780273437500000000000*7^(1/2), a[16,11] = 3307587237786 214580372394397330733034806289881443017455732775116590947887/216127682 9253336041664932456066917395909700497105040226143341064453125000-15924 9481348355430623321425412532136521666854903188329515308104047100201/53 0839922974503589180860603244506027065540472973167774842224121093750000 *7^(1/2), a[18,3] = 0, a[16,13] = -46330543544553540874179448395431083 082899647522368296822645347/508745135208312439739125075909694092175805 6970520019531250000000+18909196831867663116267125639920175653811982757 89277564189007889/1139589102866619865015640170037714766473804761396484 375000000000000*7^(1/2), a[18,4] = 0, a[20,16] = -33231612760949465922 04852464911653467445/21532814382797612400565185774770341026288-4363085 004674911115751565268892211250/103523146071142367310409546994088178011 *7^(1/2), a[21,20] = 0, a[15,14] = 26094995957293704259/54931640625000 0000000*7^(1/2), a[18,5] = 0, a[18,7] = 648528558829473918348311019185 4194357484624835466068526129/39583977519448030716210918797855209493755 178767778119680000-437194406911159519952228614191336327270581249358489 0149/67127895671292956715864398992428451861611685605376000000*7^(1/2), a[14,6] = -142489509197785007255/1133707871182130048+9966232785093067 86175/20973595616869405888*7^(1/2), a[16,2] = 0, a[20,18] = 0, c[14] = 1/2-1/14*7^(1/2), a[19,14] = -697555859302164883297179181345439022888 3838235494737/174978256576418490110785321303969544268798828125000000-2 68868947663130570092101412470880702355429907531/6999130263056739604431 412852158781770751953125000*7^(1/2), a[15,2] = 0, a[19,15] = 730925717 2110824970392183099608413709717899/12042224319247743397473862929988645 3404650000-15346626448658683257721968231518941311/92632494763444179980 5681763845280410805*7^(1/2), a[17,13] = -96617167679570134925908262475 2343417581690261/1131534950955995511243121650862500000000000000000*7^( 1/2)+19423380204481918582213207680232912549352934689479/61193410147700 23724802801887864400000000000000000000, a[20,19] = 0, a[10,7] = 187023 075231349900768014890274453125/25224698849808178010752575653374848, a[ 10,8] = 1908158550070998850625/117087067039189929394176, a[10,9] = -52 956818288156668227044990077324877908565/291277995947743398634982222441 2353951940608, a[11,1] = -10116106591826909534781157993685116703/95628 19945036894030442231411871744000, a[11,2] = 0, a[11,3] = 0, a[11,4] = \+ -9623541317323077848129/3864449564977792573440, a[11,5] = -48233483331 46829406881375/576413233634141239944816, a[11,6] = 6566119246514996884 067001154977284529/970305487021846325473990863582315520, a[11,7] = 222 6455130519213549256016892506730559375/36488044315967525557743564838004 7355776, a[11,8] = 39747262782380466933662225/175603280243142416441072 0256, a[11,9] = 48175771419260955335244683805171548038966866545122229/ 1989786420513815146528880165952064118903852843612160000, a[11,10] = -2 378292068163246/47768728487211875, a[12,1] = -321802217475859983165904 5535578571/1453396753634469525663775847094384, a[12,2] = 0, a[12,3] = \+ 0, a[12,4] = 26290092604284231996745/5760876126062860430544, a[12,5] = -697069297560926452045586710000/41107967755245430594036502319, a[14,1 ] = -26242/1715+588937/101528*7^(1/2), a[18,8] = -15027085929333054314 13728452036942690686722169823969/1333818468548077299589867915266851522 1504000000000000*7^(1/2)+585896166232426462148256613471247149307896854 30959801/216078591904788522533558602273229946588364800000000000, a[15, 3] = 0, a[16,14] = 378070792470829109357504373975352653612596622387047 615352035143989/386524657263116654742134068873711875278106926714137268 06640625000000-7366944851192037549005445931990589557073904265481386283 37098463663/2473757806483946590349658040791756001779884330970478515625 0000000000*7^(1/2), a[19,17] = 0, a[14,12] = 1100613127343/48439922837 376-29746300739/1424703612864*7^(1/2), a[17,9] = -26470176279648984933 3164460056987686273182643308013876537822807951183236955127319033004381 711/265741917838857467723355832763354480513716091651623675079278207375 03487973304320000000000000000*7^(1/2)-11593777180292563879425784652912 25079322378160773707689649770373007738863712156809629044640218707/1162 6208905450014212896817683396758522475079009758535784718421572657775988 3206400000000000000000000, a[12,6] = 182735782043421346143807755090227 3440/139381013914245317709567680839641697, a[12,7] = 64350480281424155 0941949227194107500000/242124609118836550860494007545333945331, a[12,8 ] = 162259938151380266113750/59091082835244183497007, a[12,9] = -23028 251632873523818545414856857015616678575554130463402/200131691831914445 03443905240405603349978424504151629055, a[12,10] = 7958341351371843889 152/3284467988443203581305, a[12,11] = -507974327957860843878400/12155 5654819179042718967, a[13,1] = 4631674879841/103782082379976, a[13,2] \+ = 0, a[19,18] = 0, a[15,8] = 16161913072172934315785836819/99325094830 65000000000000000-75980668436324671626317237501/4991722714566000000000 0000000*7^(1/2), a[15,4] = 0, a[16,15] = 46088828763074293687752514438 76403372924872509408621400283/7592227894936087164857586010531872045705 8558153486785997500-84914702986428741912331429025972417475312941024930 181294/5314559526455261015400310207372310431994099070744075019825*7^(1 /2), a[16,3] = 0, a[17,14] = 15739031823875445778785537999179335674270 343962861810116243/309619125620788378533311524058377597436400000000000 000000000+606652050483146218029195891134085469196882580940598209/12384 765024831535141332460962335103897456000000000000000*7^(1/2), a[14,7] = 663073659206904988750000000/13716692475747279030488259-92274454250510 62363134765625/507517621602649324128065583*7^(1/2), a[20,17] = 0, a[19 ,12] = 116148895768336301414698893782124924558205330714623/31792135148 57725027128762321323614843750000000000000+3168841339481577057649164025 908284942240477511813/111272473020020375949506681246326519531250000000 0000*7^(1/2), a[8,2] = 0, a[14,4] = 0, a[18,16] = 28826609536521141906 4692734565913406048103319/57165634628108624328611574679094001657937920 00+60833408527191271313421144941358828000769/1256387574244145589639814 828111956080394240*7^(1/2), a[15,5] = 0, a[14,9] = 1731889516638245511 5817006657422200600438218265372153/55437505265719896431617124617436606 6645917060800-274968087654238523490991092528596412316293248643447707/2 3287013241323869436363405111359694599520551359840*7^(1/2), a[18,13] = \+ -10435801212172743074622612541578159307187516521/103514493661529959732 24112880112500000000000000000*7^(1/2)+10112630954347584728032910997644 3851552294375363/42635672636828181434269613446800000000000000000000, a [17,8] = -2643363415762169747333266167756151785333485317345751/2770238 3577536990068404949009388454690816000000000000*7^(1/2)+234519181874332 58820064222913608502147503898444041667/7202619730159617417785286742440 9982196121600000000000, a[14,10] = -9081365921021033056/12050402363772 5+25400013053688208472/891729774919165*7^(1/2), a[18,14] = 68826004095 2087965212862705907587124167147667584978421209/11467375022992162167900 426816976948053200000000000000000000+265285927229536682516244570860058 29986967133546824867/4586950009196864867160170726790779221280000000000 00000*7^(1/2), a[18,9] = 141772970314785402865791190766150202981074549 02607424914474325122930069419330134278983169782333079/3487862671635004 2638690453050190275567425237029275607354155264717973327964961920000000 0000000000000-11575276557102902001713214074106732656720056039985689215 131617039362310394019419472694886093/984229325329101732308725306530942 520421170709820828426219548916203832887900160000000000000000*7^(1/2), \+ a[15,6] = 3353943190402140976803568136084793/1977029630270176562500000 0000000000-255748988917794999289135717669191/8649504632432022460937500 000000000*7^(1/2), a[15,11] = -207941800836756311792092915907/81839239 362061500549316406250+10891006522099224595926000559657/458299740427544 4030761718750000*7^(1/2), a[15,10] = 14462059099710033235500874587/103 19489926824569702148437500-348155288848452187540396337457/252574928278 923034667968750000*7^(1/2), a[16,8] = -2859135378249218370182625112630 6766396461229874660236919089505718702483/30738954422006868689408605302 697869750592763266032965939343750000000000+326217186457072779972278244 440069764006022537683484345700377402630409/174372289725344440302058311 8152408439159931363492594965600000000000000*7^(1/2), a[17,15] = 140931 08260308803339864661009803994368997/6671168326172612944182984623666406 26401280*7^(1/2)-12774251849156994331101322175199160614539284227/19729 9803246555027824211770244933965258178560000, a[14,3] = 0, a[18,15] = - 40554328659658584237572971829594948977715703/1953463398480742849744670 992524098665922560000+216315902102540063015384273123325235039761/86725 18824024396827437880010766328143216640*7^(1/2), c[17] = 3971/10000, a[ 19,5] = 0, a[15,9] = -372785014307838622905807656152260936782776633069 6417108435354320431359/80163492265138591596681606080323829375985834847 31440429687500000000000+8142513497965121274481241741350161529949699991 73763717229208919303/1589842118264019833619870377392538080518560957031 250000000000000000*7^(1/2), c[20] = 451/500, a[17,10] = -7588862103976 5715002449611997758928569635156851427/14951530380129748524972368673545 99875000000000000000*7^(1/2)+38286958923199060861303166062219224463371 0639124838997/1700736580739758894715606936615857357812500000000000000, a[18,17] = 0, a[16,4] = 0, a[18,12] = -446658928763910751322145433528 48603614252057936533163/1041764684548579336889552837451322112000000000 0000000000*7^(1/2)+114737526965374800507860463016812389851631871943733 79961/1093852918776008303734030479323888217600000000000000000000, c[19 ] = 1773/2500, a[16,10] = -1752032306988852430152559893574903103295734 904546828540573516708228831279/243326283010247711363961558099490763015 3630681641719830799102783203125000+81883999666586172691735974039324906 939964446697501450842550606413648603/733722330307823868112868698269233 685400171713233503210517883300781250000*7^(1/2), a[14,5] = 0, c[2] = 1 /16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, c[7] \+ = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[3,1] = 18928/1134225, a[3,2] = 100352/1134225, a[17,16] = 563210386312803219432034499195766 3639629/137337912974056400486943321291590340771840*7^(1/2)+21865481509 23933141497467590323092999998671947/4061768776207718044401348727198784 3283271680000, a[17,4] = 0, a[13,3] = 0, a[13,4] = 0, a[13,5] = 0, a[1 3,6] = 14327219974204125/40489566827933216, a[13,7] = 2720762324010009 765625000/10917367480696813922225349, a[13,8] = -498533005859375/95352 091037424, a[13,9] = 4059320304637772479267050305961754374024596379097 65779/78803919436321841083201886041201537229769115088303952, a[13,10] \+ = -10290327637248/1082076946951, a[13,11] = 863264105888000/8581466225 3313, a[13,12] = -29746300739/247142463456, c[21] = 63/625, a[21,1] = \+ 335495753743289994794018141547401168555952842245322199/907242577427737 0746000855697487144968767547607421875000+97532148879581300967429975910 574331811382242509/459130859022134147064820632463924340524673461914062 5*7^(1/2), a[14,11] = 26288142988644492320/221311497390123-36763124429 5366501400/8188525403434551*7^(1/2), a[4,2] = 0, a[4,1] = 14/355, a[4, 3] = 42/355, a[5,1] = 94495479/250880000, a[5,2] = 0, a[5,3] = -352806 597/250880000, a[5,4] = 178077159/125440000, a[17,7] = -99976989425049 316729418744001270347141667794343723953623/181245318312490983132833877 2795568200263515511345152000000*7^(1/2)+215047146229724221640508370572 1954726258115913698749425643/13194659173149343572070306265951736497918 392922592706560000, a[21,3] = 0, a[21,2] = 0, a[6,1] = 12089/252720, a [6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209 191, a[7,1] = 21400899/350000000, a[7,3] = 0, a[7,4] = 3064329829899/2 7126050000000, a[7,5] = -21643947/592609375, a[7,6] = 124391943/675625 0000, a[21,4] = 0, a[19,16] = -1438617606921480386578336383098493573/4 4732223610987442664991779565346743585*7^(1/2)-671478610426445930422769 177742151257962163/5815189069428367546448931343495076666050000, a[15,7 ] = 10596614540600968744792362758669/45489031169570058009272287500000- 13058425932946007192467094947/1451071902055187778276093750000*7^(1/2), a[8,1] = -15365458811/13609565775, a[8,3] = 0, a[8,4] = -7/5, a[8,5] \+ = -8339128164608/939060038475, a[8,6] = 341936800488/47951126225, a[8, 7] = 1993321838240/380523459069, a[9,1] = -184091125228237658443815733 6464708426954728061551/29919236151711519215962538134831182621955337338 98, a[9,3] = 0, a[9,4] = -14764960804048657303638372252908780219281424 435/2981692102565021975611711269209606363661854518, a[9,5] = -87532504 8502130441118613421785266742862694404520560000/17021203042889441839557 1677575961339495435011888324169, a[9,6] = 7632051964154290925661849798 370645637589377834346780/173408725741881158304980034758186526047923395 0396659, a[9,7] = 7519834791971137517048532179652347729899303513750000 /1045677303502317596597890707812349832637339039997351, a[9,8] = 136604 2683489166351293315549358278750/14463141822426771816505532646418083664 1, a[10,1] = -63077736705254280154824845013881/78369357853786633855112 190394368, a[10,2] = 0, a[9,2] = 0, a[17,11] = 27418190460279475664846 671472967149325313851242007749/178484880415390961272564938626684644725 000000000000000*7^(1/2)-3353756327413703017248069911858005388441077701 17192189/624697081453868364453977285193396256537500000000000000, a[14, 8] = -16848658354789365625/225946578416832+22737651361282524171875/806 742258237298656*7^(1/2), a[16,5] = 0, a[17,1] = 3362117608470634787802 979264377180316187075547467836195538083/802926204124727793979931029542 02712163478400000000000000000000-2989723142262948660161298577240408579 957089549899835987317/882336488049151421955968164331897935862400000000 000000000000*7^(1/2), a[21,5] = 0, a[17,3] = 0, c[15] = 1187/2500, a[1 0,4] = -31948346510820970247215/6956009216960026632192, a[10,5] = -337 8604805394255292453489375/517042670569824692230499952, a[10,6] = 10015 87844183325981198091450220795/184232684207722503701669953872896, a[10, 3] = 0, a[21,6] = -263532707727920749646876297582905726813198902584751 /6975714403353446435345919095274186409008789062500000+2490692665512176 46646812211943842083128601426843/8719643004191808044182398869092733011 26098632812500*7^(1/2), a[20,8] = -34547433954017606370106693676854764 47099876393/1406761666046800276911188816883007464768000000+86221640206 701158533615034815831351063336969/879226041279250173069493010551879665 480000000*7^(1/2), a[20,3] = 0, a[19,2] = 0, a[19,3] = 0, a[20,10] = - 341601321080417106938553052190791919279008005951/664350226851468318248 28395961556928039550781250+1414483930382359161813873551911951292291436 /27116335789855849724419753453696705322265625*7^(1/2), a[19,9] = -3903 5035781656022530508280241969035250845924327906117501252797779501396529 09034268868674113/1251512305360287643825181715050942129034895210038284 8437239951021878978575390625000000000000+52540823211340033345406559798 7198106941861647860588141820154051898978741151535154361/67259778407078 7985165164837263462180671344834569519685686600054918639923437500000000 00*7^(1/2), a[19,8] = 60530769019937906908633738073110755509457740271/ 54273212424645072411697099416782695400000000000+5076681881140470540069 505760844878092852841139/678415155308063405146213742709783692500000000 00*7^(1/2), a[20,11] = 22671699935118368123965101629526934843871334019 03/463642365141542926743186266354473784148925781250-275178711728623278 446792413948735774971947548/174301641030655235617739197877621723364257 8125*7^(1/2), a[18,10] = 242557859433071294262123840323509652869069374 7094280567/18708102388137347841871676302774430935937500000000000000-12 813005291632184523871567280729171638148255045076861/213806884435855403 907104872031707782125000000000000000*7^(1/2), a[21,7] = -9503984093260 58262837165112878937300209484217784/1902253600478708403028732707217777 5229338551393125+12650651861876754164611988947854140315877515136/36581 800009205930827475628984957260056420291140625*7^(1/2), a[20,12] = -103 9346366321461511148422012098015726047706061669/21364314819843912182305 282799294691750000000000000+406796287633268143396837429568000763441807 01/10900160622369342950155756530252393750000000000*7^(1/2), a[20,4] = \+ 0, a[20,5] = 0, a[19,4] = 0, a[15,1] = 1309461391528593965349505677130 97/2815269161783203125000000000000000+90670944595916412828478989403/66 148981703437500000000000000000*7^(1/2), a[17,6] = -9331543777192219488 090069025678637265860768694394667197161/204807867773900816582899908914 894308064358400000000000000000*7^(1/2)+1913610790329116866603247637057 15648161934674584562401842333/1433655074417305716080299362404260156450 508800000000000000000, a[21,8] = -466087863971840796884466316363258971 31144521/212004736033769814108191794596807403906250000+792647131950882 00660606693628633224442743/1325029600211061338176198716230046274414062 50*7^(1/2), a[20,13] = -95997053858479624443512588726724333549134483/4 541698409399626983252079526149359375000000000+598780030683267309831447 210391270733521/682346515835280496281862909577728271484375*7^(1/2), a[ 20,1] = 1756689787234584639161397296556945356113093440409332017/297960 89606191070479724003049411162716915812500000000000+1071640173525649463 414934424316330743362810391153/307734545218035512679955414457274670710 937500000000*7^(1/2), a[20,2] = 0, a[19,6] = 3978177873022717803815319 922302122241315215052320566257/125004802108093760121398870187313420449 43750000000000000+7816571044335207908114529647233328653098308266650115 11/21875840368916408021244802282779848578651562500000000000*7^(1/2), a [19,1] = 889600145162128348819614443326379680969740965957107281357/162 57786987505048376833533409896963784031445312500000000000+5874977494478 05729807877623189337029663123827949833/2210878117302745006058007682774 45947314453125000000000*7^(1/2), a[20,7] = 286874742630756193550077645 094469028757436872008145/644270467438932791604995423142174633687421529 423472+35835879510547961853908519812981470449630323307/632133504159078 484698778868860061453774942630910*7^(1/2), a[20,6] = 12660795150694149 245758947395121627071932701286934343/280010756722130022671933469219582 06180674000000000000+3344810444582948978921965249456125361330745579349 /71431315490339291497942211535607668828250000000000*7^(1/2), a[21,19] \+ = 0, a[18,11] = 372098434581358541752287535215351518628082338823827/20 51550349602194957155918834789478675000000000000000*7^(1/2)-51389535841 19531321031442069620009245122067004757018391/1186924454762349892462556 8418674528874212500000000000000, a[19,7] = 849564493271943658031838075 144577515571397490531243/213052403253615341139218063208391082568591775 6030000+310169487453029108907565058918136527805795804669099/7170032801 804362442185223281051622971058377063562500*7^(1/2), a[20,9] = 67919799 7868819581661088791014109830340210984055781994680408901424681785020594 1485164496753/22707439268457059009564097037884293989209138690934640204 52816713409721872718875000000000000+9488003722012434649977787167415075 0251597084733453730150947384698361284577054335299/92683425585539016365 56774301177262852738423955483526614093129442488660704975000000000*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]I;4[;w74PWz(3*H)=#f.P!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$ F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$!I;*3)))*)4*[/f8# zF$QT5'G-PF+/&F%6$F'\"\"($!IAcY1k]&z5)e0FDqR7gn/\\F+/&F%6$F'\"\")$!ING &*)ylL[Wy!f$pJPVP^E=#!#S/&F%6$F'\"\"*$\"In/_E5i0A$3_\\\"[jG@o@:6FP/&F% 6$F'\"#5$!It96g!eWYD$[-BT>XY#4tu#FP/&F%6$F'\"#6$\"IYuK$>E9GK@8H&yW^\") [2,RFP/&F%6$F'\"#7$!Ik[v!33*QCPcQPzWB+;8`k!#U/&F%6$F'\"#8$!ITWQ)4:='o' =yN(>oBXGO.GFco/&F%6$F'\"#9$!Ib[D[D(H0TH9$)*G&>_gs68\"Fco/&F%6$F'\"#:$ \"IY$H_m+aP\"=jMt6:Y&4?!pQF+/&F%6$F'\"#;$\"I_&y&yvhF:Y5>7:Gxr0)*G6FP/& F%6$F'\"#F0/&F%6$F'\"#?F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "calculation of the interpolation \+ coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 223 "Each standard (simple) order condition gives rise to a group \+ \{list) of equations to be satisfied by the \"d\" coefficients of the \+ weight polynomials for a given stage (corresponding to an \"approximat e\" interpolation scheme)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SO8_21 := SimpleOrderConditions(8,2 1,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 227 "whch := [1,2,3,6,7,8,12,15,16,24,31,48,63,64, 102,117,121,123,125,127,128]:\nordeqns9 := []:\nfor ct in whch do\n \+ eqn_group := convert(SO8_21[ct],'polynom_order_conditions',8):\n ord eqns9 := [op(ordeqns9),op(eqn_group)];\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Substitute for all known \+ coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "eqns9 := [ ]:\nfor ct to nops(ordeqns9) do\n eqns9 := [op(eqns9),expand(subs(e9 ,ordeqns9[ct]))];\nend do:\nnops(eqns9);\nnops(indets(eqns9));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$o\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$o\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Solve the system of equations." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "infolevel[solve]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d9 := solve(\{op(eqns9)\}):\ninfolevel[solve]:=0:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(map(rhs,d9));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can ch eck which of the groups of order conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 285 "nm := NULL:\nfor ct to nops(SO8_21 ) do\n eqn_group := convert(SO8_21[ct],'polynom_order_conditions',8) :\n tt := expand(subs(\{op(e9),op(d9)\},eqn_group));\n tt := map(_ Z->`if`(lhs(_Z)=rhs(_Z),0,1),tt);\n if add(op(i,tt),i=1..nops(tt))=0 then nm := nm,ct end if;\nend do:\nnm;\nnops([%]);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6\\s\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#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 8990 "d9 := \{d[7,1] = 0, d[4,1] = 0, d[2,4] \+ = 0, d[5,1] = 0, d[2,2] = 0, d[4,4] = 0, d[2,3] = 0, d[12,8] = 3631140 2269287109375000000/104044965326975855974897, d[13,3] = -3137719106539 052085627514487187/62250340390487487380380697438, d[5,7] = 0, d[5,3] = 0, d[16,4] = 0, d[3,5] = 0, d[1,5] = 21186629860818988458816633182780 72375000/4010215915503095841633032972051049027, d[11,1] = 0, d[1,7] = \+ 9388894747609968215460540367187500000000/28071511408521670891431230804 357343189, d[19,8] = 860078164760131835937500000000000000/106840453254 0926075379934464499929, d[17,5] = -12249446660371225232915600000000000 000/2213690867197832445116426855041689, d[10,1] = 0, d[3,6] = 0, d[2,5 ] = 0, d[3,1] = 0, d[8,8] = 8519851174354553222656250000000/5619943579 89134905330318407, d[4,2] = 0, d[6,1] = 0, d[9,1] = 0, d[20,1] = 0, d[ 12,2] = 3851719928248196731352373/237817063604516242228336, d[10,3] = \+ -61608702831031228169869590528/3975670571767330087672657, d[5,2] = 0, \+ d[12,1] = 0, d[21,2] = 217058654686563407721751708984375/5445257583257 721410189019668748, d[1,1] = 1, d[2,1] = 0, d[2,6] = 0, d[18,5] = 7325 2540774134787283600000000000000/1987535178317076107007328050495849, d[ 3,2] = 0, d[8,3] = -177334200924459707792125000000/2081460585144944093 8159941, d[9,4] = -397562484985396806815511066887631876870227571942358 05774092455045496125/1061625875307876781720763525336990947818155942221 833704758034190768, d[11,7] = 960974811873198000000000000000000/809250 9434353435846380726569, d[8,6] = 170389381862175762367248535156250/168 5983073967404715990955221, d[4,3] = 0, d[13,6] = 598130068339366973579 83984375000/93375510585731231070571046157, d[20,3] = -6029622724948070 41510135250000000/2533406014554630641654057774329, d[13,1] = 0, d[13,2 ] = 73358409180373102507257577107/17785811540139282108680199268, d[21, 1] = 0, d[14,8] = 0, d[5,5] = 0, d[4,5] = 0, d[14,7] = 0, d[4,6] = 0, \+ d[16,1] = 0, d[5,6] = 0, d[19,6] = 16412576708380742866967773437500000 000/3205213597622778226139803393499787, d[9,2] = -64891832481789846664 6124034587058036789290020132247402066359750613/93617802055368322903065 5666082002599486910001959288981268107752, d[14,6] = 0, d[6,5] = 293151 909787132640811026730468750/51137334124695218389727206301, d[10,8] = 1 205897769990000000000000000000/43732376289440630964399227, d[7,6] = -2 125500241174083759943008422851562500000000/441227792627168594850976342 117325067873, d[5,8] = 0, d[16,3] = 0, d[10,6] = 730814357756731532400 000000000/3975670571767330087672657, d[12,7] = -1034787817309236328125 000000/728314757288830991824279, d[20,2] = 650816378983450106488500000 000/32901376812397800540961789277, d[12,3] = -204064415094291383396888 64/104044965326975855974897, d[8,4] = 34177794322161748191699780273437 5/8991909727826158485285094512, d[2,7] = 0, d[5,4] = 0, d[10,2] = 7267 90938593323216511264256/567952938823904298238951, d[3,3] = 0, d[8,7] = -3853893822639465332031250000000/62443817554348322814479823, d[1,2] = -1165602721447934039669346597095491/91386873028151699547830999722004, d[19,1] = 0, d[9,6] = -1982001103691108737169162644445182233001923277 3674027228704832324218750/19905485162022689657264316100068580271590423 9166593819642131410769, d[7,3] = 2212132485686449696453201250000000000 000/5447256699100846850012053606386729233, d[8,1] = 0, d[12,6] = 72619 5476962818818505859375/312134895980927567924691, d[7,2] = -26096278150 414755183932834472656250000/778179528442978121430293372340961319, d[21 ,3] = -54634182831782148404721175537109375/136520386551675729641167564 552182, d[21,5] = -557231493004750027256052001953125000/15927378431028 8351248028825310879, d[11,4] = -84546524388189435346419196000000/11560 72776336205120911532367, d[13,7] = -37830445780811261453125000000000/9 3375510585731231070571046157, d[10,4] = 302344820296949726493950044800 0/43732376289440630964399227, d[20,6] = 612477462870946448873046875000 00000/22800654130991675774886519968961, d[4,7] = 0, d[4,8] = 0, d[15,8 ] = 0, d[12,5] = -202881528690161447178843750/104044965326975855974897 , d[15,4] = 0, d[14,4] = 0, d[19,3] = -1298602318586779115245257968750 000000/3205213597622778226139803393499787, d[18,1] = 0, d[7,7] = 33652 4289950988082885742187500000000000000/11439239068111778385025312573412 1313893, d[17,8] = 2666184012640625000000000000000000000/2213690867197 832445116426855041689, d[8,2] = 39050510328774558369916015625/55505615 603865175835093176, d[16,6] = 0, d[7,4] = -266466538585844445477870867 767333984375000/147075930875722864950325447372441689291, d[19,4] = 196 5612626586739111259866562500000000/1068404532540926075379934464499929, d[12,4] = 2913298866169434359093066125/3329438890463227391196704, d[1 4,5] = 0, d[9,7] = 313804487612427802205839552949152808835729540399775 9674378906250000000/51606813383021788000314893592770393296715913858005 805092404439829, d[21,4] = 149564238041930800862274083251953125/910135 91034450486427445043034788, d[15,7] = 0, d[11,3] = 6534765059544637151 08552704000/39864578494351900721087323, d[17,1] = 0, d[18,8] = -467684 046390625000000000000000000000/1987535178317076107007328050495849, d[1 0,5] = -6737673780958917488779008000000/43732376289440630964399227, d[ 10,7] = -34365192290067024000000000000000/306126634026084416750794589, d[18,4] = 68025249008090927349728000000000000/28393359690243944385818 9721499407, d[3,4] = 0, d[13,5] = -48698653288136547490137464875000/93 375510585731231070571046157, d[11,6] = -674394837457740114650000000000 000/3468218329008615362734597101, d[14,3] = 0, d[15,1] = 0, d[13,4] = \+ 12204159982374702269960942829125/53357434620417846326040597804, d[19,2 ] = 1535623201637598414687500000000/46353617620761251046897239121, d[1 7,6] = 5302763032231914160000000000000000000/7378969557326108150388089 51680563, d[14,2] = 0, d[20,4] = 10250821296157921075256084500000000/9 771708913282146760665651415269, d[14,1] = 0, d[9,3] = 2062784535709870 0094803875291108609614565218419782669825053775815104/24574673039534184 76205471123465256823653138755143133575828782849, d[3,7] = 0, d[6,7] = \+ 1495207704846970724853515625000000/357961338872866528728090444107, d[9 ,8] = -991044996249456171696057203603943938970848725365639108886718750 000000/663516172067422988575477203335619342386347463888646065473771369 23, d[17,7] = -10349073413752962500000000000000000000/2213690867197832 445116426855041689, d[6,8] = -52467846585220184326171875000000/5113733 4124695218389727206301, d[11,2] = -82364459805231884285455296000/60845 935596642374784817493, d[20,7] = -109383291839734164062500000000000000 /68401962392975027324659559906883, d[13,8] = 9680259277467001953125000 000000/93375510585731231070571046157, d[7,8] = -1062797782816410064697 26562500000000000000/147075930875722864950325447372441689291, d[18,2] \+ = 438275538362730816000000000000/36956084459513138599269780229, d[1,3] = 968320793849692313349204129230762185789/120306477465092875248990989 16153147081, d[15,2] = 0, d[15,3] = 0, d[11,8] = -33721254136250000000 000000000000/1156072776336205120911532367, d[17,2] = 34666457416541829 696000000000000/1064786371908529314630315947591, d[9,5] = 615248199598 937433084042257951670186584008559582302922578442729187500/737240191186 0255428616413370395770470959416265429400727486348547, d[1,8] = -317333 515601090068489243164062500000000/401021591550309584163303297205104902 7, d[16,2] = 0, d[16,8] = 0, d[16,7] = 0, d[15,6] = 0, d[21,6] = 27704 9344903039721904754638671875000/68260193275837864820583782276091, d[20 ,5] = -157281519633298700464322593750000000/68401962392975027324659559 906883, d[3,8] = 0, d[1,6] = -6962024958594838524181325729583789062500 /12030647746509287524899098916153147081, d[17,4] = 7016387410618483556 81312000000000000/316241552456833206445203836434527, d[17,3] = -355368 81022744766695984000000000000/81988550636956757226534327964507, d[6,4] = -4209545994216839073113853785765625/1636394691990246988471270601632 , d[19,7] = -3422515956010727539062500000000000000/1068404532540926075 379934464499929, d[15,5] = 0, d[18,7] = 139879160829496250000000000000 0000000/1987535178317076107007328050495849, d[1,4] = -8779050542684928 997385153298077813720875/32081727324024766733064263776408392216, d[6,2 ] = -5565509389746001724096960186625/116885335142160499176519328688, d [6,6] = -349769957414481490278570556640625/511373341246952183897272063 01, d[8,5] = -5289187792390721190063476562500/624438175543483228144798 23, d[20,8] = 26092757644096679687500000000000000/68401962392975027324 659559906883, d[2,8] = 0, d[18,6] = -128528570970098168000000000000000 0000/1987535178317076107007328050495849, d[21,8] = 9398304351732254028 3203125000000000/159273784310288351248028825310879, d[18,3] = -7294067 7587604374915664000000000000/662511726105692035669109350165283, d[21,7 ] = -387546524205546905517578125000000000/1592737843102883512480288253 10879, d[11,5] = 188409677427920087814816000000000/1156072776336205120 911532367, d[6,3] = 1016762452633897965746731332000/176335634912742132 3783696769, d[19,5] = -4476560877775342997902503906250000000/106840453 2540926075379934464499929, d[7,5] = 6597928700413411021401000976562500 0000000/16341770097302540550036160819160187699, d[16,5] = 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(d9,matrix([seq([seq(d[j,i],j= 1..11)],i=1..8)])):\nevalf[8](%);\nsubs(d9,matrix([seq([seq(d[j,i],j=1 1..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-$!)(faF\"!\"'F+F+F +F+$!)@^hZF/$!)O]`LF/$\")hTNq!\"&$!))p:$pF6$\")vmz7!\"%$!)fl`8F;7-$\") Ny[!)F/F+F+F+F+$\")M1mdF6$\")G+hSF6$!)-q>&)F;$\")]%RR)F;$!)Jk\\:!\"$$ \")5CR;FK7-$!)WYOFF6F+F+F+F+$!):XsDF;$!)Fdo7-$\")YjWLF6F+F+F+F+$ \")*3q<%F;$\");%=%HF;$!)!y<<'FK$\")%z13'FK$!)4eA6Fdo$\")o[(=\"Fdo7-$!) z78zF/F+F+F+F+$!)%=g-\"F;$!)V=EsF6$\")I+;:FK$!)gi$\\\"FK$\")&[uv#FK$!) *zo\"HFKQ)pprint646\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7 *7-$\"\"!F)F(F(F(F(F(F(F(F(F(F(7-$!)fl`8!\"%$\")Yh>;!\"'$\")maCT!\"(F( F(F($\")&>dD$F0$\")h$f=\"F0$\")V%GJ$F0$\")D3y>F0$\")j>')RF07-$\")5CR;! \"$$!))48'>!\"&$!)^[S]F0F(F(F($!)7PMVFD$!)>(45\"FD$!)8`^SFD$!)g/!Q#FD$ !)l!>+%FD7-$!)GD8tFA$\")'>,v)FD$\")kC(G#FDF(F(F($\"))z'=AF-$\")a\"eR#F D$\")WwR=F-$\")1.\\5F-$\")wJV;F-7-$\")*Q(H;!\"#$!)4%*\\>F-$!)gN:_FDF(F (F($!)Z\\LbF-$\")sf&o$F0$!)%\\**=%F-$!):P*H#F-$!)jd)\\$F-7-$!)%*\\W>F_ o$\")PaEBF-$\")j=(F-$!)>tmkFD$\")ve?^F-$\")wA'o#F-$ \")\\seSF-7-$\")o[(=\"F_o$!)vz?9F-$!)7V^SFDF(F(F($!)5.vYF-$\")1#y.(FD$ !)'*Q.KF-$!)^7*f\"F-$!)(4KV#F-7-$!)*zo\"HFA$\")@(**[$FD$\")AqO5FDF(F(F ($\")nS/7F-$!)c3`BFD$\")r6]!)FD$\")7i9QFD$\")Fs+fFDQ)pprint656\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "principal error graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 96 "The interpolation scheme amounts to havin g a Runge-Kutta method for each value of the parameter " }{TEXT 266 1 "u" }{TEXT -1 8 " where " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" } {XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "The nodes and linking coefficients are ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eu9 := map(_U->lhs(_U)=rhs(_U)/u,e9 ):" }}}{PARA 0 "" 0 "" {TEXT -1 43 " ... and the weight polynomials (o f degree " }{XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 33 " and re-using the weight symbol " }{XPPEDIT 18 0 "b[j]" "6#&%\"bG6#%\"jG" }{TEXT -1 10 ") are ... " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 " pols := [seq(b[j]=add(simplify(subs(d9,d[j,i]))*u^(i-1),i=1..8),j=1..2 1)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 " The whole scheme, including the weights, is given by the set of equati ons:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eu := `union`(eu9,\{ op(pols)\}):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can calculate the principal error norm, that is, the \+ root mean square of the residues of the principal error terms. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "errterms8_21 := PrincipalErr orTerms(8,21,'expanded'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "nops(errterms8_21);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$'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#$\"+%H[4N)!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(ssm,u=0 ..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 417 303 303 {PLOTDATA 2 "6%-%'CURV ESG6$7dq7$$\"3`*****\\n5;\"o!#@$\"3QY:*['>1f#*!#G7$$\"3#******\\8ABO\" !#?$\"31'Qs,h!*)oO!#F7$$\"33+++-K[V?F1$\"3?j!fz4gu<)F47$$\"3#)******pU kCFF1$\"3>8jjrD4S9!#E7$$\"3s*****\\Smp3%F1$\"3s&e\"e!4w$zJF?7$$\"3k*** ***R&)G\\aF1$\"3ma\">&zbwXbF?7$$\"3Y******4G$R<)F1$\"3z3#[jH65?\"!#D7$ $\"3%******zqd)*3\"!#>$\"3I.c@`tba?FO7$$\"3*)*****>c'yM;FS$\"3ii:9]L:x UFO7$$\"3')*****fT:(z@FS$\"3#p'f*QR@!GqFO7$$\"3#*******zZ*z7$FS$\"3%Q) ROf[ve7!#C7$$\"33+++XTFwSFS$\"3GS`(\\[1u5Faq$\"3Qka&3B3_F%F_o7$$\"3#******RlD))4\"Faq$\"3)H6r6DgJG%F_o7$$ \"3++++'Q*eB6Faq$\"3+A_Gu6L'G%F_o7$$\"31+++=JN[6Faq$\"3#QyyIQ!*\\G%F_o 7$$\"3)******He!)y>\"Faq$\"3KH1+([3*pUF_o7$$\"3%******z/3uC\"Faq$\"3x! 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g!*Faq$\"3bUt#*3*[m-*F_o7$$\"3%********z\"4)3*Faq$\"3%o:(Hw1uK!*F_o7$$ \"3a*******Q\"y:\"*Faq$\"3UMt25M9L!*F_o7$$\"3D+++!)4ZV\"*Faq$\"3Mmtuaei)F_o7$$\"37+++!oK0e*Faq$\"3g>_qo(3[Y)F_o7$$\"3=+ ++l(z5j*Faq$\"3%yHm@\\,XS)F_o7$$\"3C+++]oi\"o*Faq$\"3;n]jhkpf$)F_o7$$ \"3G+++NRcL$)F_o7$$\"\"\"\"\"!$\"3u%p'o-$[4N)F _o-%'COLOURG6&%$RGBG$\"#5!\"\"$F`\\mF`\\mFj\\m-%+AXESLABELSG6$Q\"u6\"Q !F_]m-%%VIEWG6$;Fj\\mF^\\m%(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 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "abreviated calculation \+ for stages 14 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 4918 "e2 := \{a[7,2] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[14,2] = 0, a[14,6] = -142489509197785007255/1133707871182130048+996 623278509306786175/20973595616869405888*7^(1/2), c[14] = 1/2-1/14*7^(1 /2), a[10,7] = 187023075231349900768014890274453125/252246988498081780 10752575653374848, a[10,8] = 1908158550070998850625/117087067039189929 394176, a[10,9] = -52956818288156668227044990077324877908565/291277995 9477433986349822224412353951940608, a[11,1] = -10116106591826909534781 157993685116703/9562819945036894030442231411871744000, a[11,2] = 0, a[ 11,3] = 0, a[11,4] = -9623541317323077848129/3864449564977792573440, a [11,5] = -4823348333146829406881375/576413233634141239944816, a[11,6] \+ = 6566119246514996884067001154977284529/970305487021846325473990863582 315520, a[11,7] = 2226455130519213549256016892506730559375/36488044315 9675255577435648380047355776, a[11,8] = 39747262782380466933662225/175 6032802431424164410720256, a[11,9] = 481757714192609553352446838051715 48038966866545122229/1989786420513815146528880165952064118903852843612 160000, a[11,10] = -2378292068163246/47768728487211875, a[12,1] = -321 8022174758599831659045535578571/1453396753634469525663775847094384, a[ 12,2] = 0, a[12,3] = 0, a[12,4] = 26290092604284231996745/576087612606 2860430544, a[12,5] = -697069297560926452045586710000/4110796775524543 0594036502319, a[14,1] = -26242/1715+588937/101528*7^(1/2), a[14,12] = 1100613127343/48439922837376-29746300739/1424703612864*7^(1/2), a[12, 6] = 1827357820434213461438077550902273440/139381013914245317709567680 839641697, a[12,7] = 643504802814241550941949227194107500000/242124609 118836550860494007545333945331, a[12,8] = 162259938151380266113750/590 91082835244183497007, a[12,9] = -2302825163287352381854541485685701561 6678575554130463402/20013169183191444503443905240405603349978424504151 629055, a[12,10] = 7958341351371843889152/3284467988443203581305, a[12 ,11] = -507974327957860843878400/121555654819179042718967, a[13,1] = 4 631674879841/103782082379976, a[13,2] = 0, a[14,7] = 66307365920690498 8750000000/13716692475747279030488259-9227445425051062363134765625/507 517621602649324128065583*7^(1/2), a[8,2] = 0, a[14,4] = 0, a[14,9] = 1 7318895166382455115817006657422200600438218265372153/55437505265719896 4316171246174366066645917060800-27496808765423852349099109252859641231 6293248643447707/23287013241323869436363405111359694599520551359840*7^ (1/2), a[14,10] = -9081365921021033056/120504023637725+254000130536882 08472/891729774919165*7^(1/2), a[14,3] = 0, a[14,5] = 0, c[2] = 1/16, \+ c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, c[7] = 39/ 250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19 /20, c[12] = 1, c[13] = 1, a[2,1] = 1/16, a[3,1] = 18928/1134225, a[3, 2] = 100352/1134225, a[13,3] = 0, a[13,4] = 0, a[13,5] = 0, a[13,6] = \+ 14327219974204125/40489566827933216, a[13,7] = 27207623240100097656250 00/10917367480696813922225349, a[13,8] = -498533005859375/953520910374 24, a[13,9] = 405932030463777247926705030596175437402459637909765779/7 8803919436321841083201886041201537229769115088303952, a[13,10] = -1029 0327637248/1082076946951, a[13,11] = 863264105888000/85814662253313, a [13,12] = -29746300739/247142463456, a[14,11] = 26288142988644492320/2 21311497390123-367631244295366501400/8188525403434551*7^(1/2), a[4,2] \+ = 0, a[4,1] = 14/355, a[4,3] = 42/355, a[5,1] = 94495479/250880000, a[ 5,2] = 0, a[5,3] = -352806597/250880000, a[5,4] = 178077159/125440000, a[6,1] = 12089/252720, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685 520, a[6,5] = 960400/5209191, a[7,1] = 21400899/350000000, a[7,3] = 0, a[7,4] = 3064329829899/27126050000000, a[7,5] = -21643947/592609375, \+ a[7,6] = 124391943/6756250000, a[8,1] = -15365458811/13609565775, a[8, 3] = 0, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8,6] = \+ 341936800488/47951126225, a[8,7] = 1993321838240/380523459069, a[9,1] \+ = -1840911252282376584438157336464708426954728061551/29919236151711519 21596253813483118262195533733898, a[9,3] = 0, a[9,4] = -14764960804048 657303638372252908780219281424435/298169210256502197561171126920960636 3661854518, a[9,5] = -875325048502130441118613421785266742862694404520 560000/170212030428894418395571677575961339495435011888324169, a[9,6] \+ = 7632051964154290925661849798370645637589377834346780/173408725741881 1583049800347581865260479233950396659, a[9,7] = 7519834791971137517048 532179652347729899303513750000/104567730350231759659789070781234983263 7339039997351, a[9,8] = 1366042683489166351293315549358278750/14463141 8224267718165055326464180836641, a[10,1] = -63077736705254280154824845 013881/78369357853786633855112190394368, a[10,2] = 0, a[9,2] = 0, a[14 ,8] = -16848658354789365625/225946578416832+22737651361282524171875/80 6742258237298656*7^(1/2), a[10,4] = -31948346510820970247215/695600921 6960026632192, a[10,5] = -3378604805394255292453489375/517042670569824 692230499952, a[10,6] = 1001587844183325981198091450220795/18423268420 7722503701669953872896, a[10,3] = 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 27 "set up order relations e tc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "SO7_14 := SimpleOrderConditions(7,14,'expanded'):\nS O7_15 := SimpleOrderConditions(7,15,'expanded'):\nSO7_16 := SimpleOrde rConditions(7,16,'expanded'):\nSO7_17 := SimpleOrderConditions(7,17,'e xpanded'):\nSO7_18 := SimpleOrderConditions(7,18,'expanded'):\nSO7_19 \+ := SimpleOrderConditions(7,19,'expanded'):\nSO7_20 := SimpleOrderCondi tions(7,20,'expanded'):\nSO8_21 := SimpleOrderConditions(8,21,'expande d'):\nerrterms8_21 := PrincipalErrorTerms(8,21,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1660 " whch := [1,2,3,6,7,8,12,15,16,21,27,31,32,64]:\nordeqns2 := []:\nfor c t in whch do\n eqn_group := convert(SO7_14[ct],'polynom_order_condit ions',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_order_conditi ons',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_order_condi tions',7):\n ordeqns4 := [op(ordeqns4),op(eqn_group)];\nend do:\nwhc h := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nordeqns5 := []:\nfo r ct in whch do\n eqn_group := convert(SO7_17[ct],'polynom_order_con ditions',7):\n ordeqns5 := [op(ordeqns5),op(eqn_group)];\nend do:\nw hch := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nordeqns6 := []:\n for ct in whch do\n eqn_group := convert(SO7_18[ct],'polynom_order_c onditions',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]:\nordeqns7 := [] :\nfor ct in whch do\n eqn_group := convert(SO7_19[ct],'polynom_orde r_conditions',7):\n ordeqns7 := [op(ordeqns7),op(eqn_group)];\nend d o:\nwhch := [1,2,3,6,7,8,12,15,16,21,27,31,32,33,36,64]:\nordeqns8 := \+ []:\nfor ct in whch do\n eqn_group := convert(SO7_20[ct],'polynom_or der_conditions',7):\n ordeqns8 := [op(ordeqns8),op(eqn_group)];\nend do:\nwhch := [1,2,3,6,7,8,12,15,16,24,31,48,63,64,102,117,121,123,125 ,127,128]:\nordeqns9 := []:\nfor ct in whch do\n eqn_group := conver t(SO8_21[ct],'polynom_order_conditions',8):\n ordeqns9 := [op(ordeqn s9),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_coeffs := p roc()\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 ct,eq,e q1,eq2,eq3,eq4,eu9,pols,eu;\n global e2,e3,e4,e5,e6,e7,e8,e9,d9,sm; \n\n eqns2 := []:\n for ct to nops(ordeqns2) do\n eqns2 := [o p(eqns2),expand(subs(e2,ordeqns2[ct]))];\n end do:\n d2 := solve( \{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`(eqs1 5,\{c[15]=c_15\},e2):\n eqns3 := []:\n for ct to nops(ordeqns3) do \n eqns3 := [op(eqns3),expand(subs(e3,ordeqns3[ct]))];\n end do :\n d3 := solve(\{op(eqns3)\}):\n eqs16 := \{seq(a[16,j]=add(expan d(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,ordeqns4[ 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 eqns5 := []:\n for ct to nops(ordeqns5) do\n eqns5 := [op(eqns5),expand (subs(e5,ordeqns5[ct]))];\n end do:\n d5 := solve(\{op(eqns5)\},in dets(eqns5) minus \{seq(d[1,i],i=1..7)\}):\n eq := add(subs(\{op(d5) ,c[18]=c_18\},d[17,i]*c[18]^i),i=1..7)=0:\n dd := \{d[1,1]=1,seq(d[1 ,i]=0,i=2..6)\}:\n sol := \{d[1,7]=expand(rationalize(solve(subs(dd, eq))))\}:\n dd_5 := `union`(subs(sol,dd),sol):\n d_5 := `union`(su bs(dd_5,d5),dd_5):\n 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..17)\}:\n e6 := `union`( eqs18,\{c[18]=c_18\},e5):\n eqns6 := []:\n for ct to nops(ordeqns6 ) do\n eqns6 := [op(eqns6),expand(subs(e6,ordeqns6[ct]))];\n en d 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,d6),d d_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,eq 3\}));\n dd_7 := `union`(subs(sol,dd),sol):\n d_7 := `union`(subs( 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`(eqs 20,\{c[20]=c_20\},e7):\n eqns8 := []:\n for ct to nops(ordeqns8) d o\n eqns8 := [op(eqns8),expand(subs(e8,ordeqns8[ct]))];\n end d o:\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),s eq(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`(subs( sol,dd),sol):\n d_8 := expand(`union`(subs(dd_8,d8),dd_8)):\n eqs2 1 := \{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 := [op(eq ns9),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 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 0 142 "c_15 := 471 /1000:\nc_16 := 11/100:\nc_17 := 37/100:\nc_18 := 9/20:\nc_19 := 7/10: \nc_20 := 9/10:\nc_21 := 1/10:\ncalc_coeffs();\nssmA := sqrt(sm)*u^9: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"#:#\"$r%\"%+5/&F%6#\"#; #\"#6\"$+\"/&F%6#\"#<#\"#PF1/&F%6#\"#=#\"\"*\"#?/&F%6#\"#>#\"\"(\"#5/& F%6#F>#F=FE/&F%6#\"#@#\"\"\"FE" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "c_15 := 59/125: \nc_16 := 1 1/100:\nc_17 := 37/1000: \nc_18 := 4531/10000:\nc_19 := 7091/10000: \n c_20 := 9013/10000: \nc_21 := 1/2:\ncalc_coeffs();\nssmB := sqrt(sm)* u^9:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"#:#\"#f\"$D\"/&F%6# \"#;#\"#6\"$+\"/&F%6#\"#<#\"#P\"%+5/&F%6#\"#=#\"%JX\"&++\"/&F%6#\"#># \"%\"4(F?/&F%6#\"#?#\"%8!*F?/&F%6#\"#@#\"\"\"\"\"#" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "c_15 := 1 187/2500:\nc_16 := 277/2500:\nc_17 := 3971/10000:\nc_18 := 4531/10000: \nc_19 := 1773/2500:\nc_20 := 451/500:\nc_21 := 63/625:\ncalc_coeffs() ;\nssmC := sqrt(sm)*u^9:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6# \"#:#\"%(=\"\"%+D/&F%6#\"#;#\"$x#F*/&F%6#\"#<#\"%rR\"&++\"/&F%6#\"#=# \"%JXF7/&F%6#\"#>#\"%t " 0 "" {MPLTEXT 1 0 95 "plot([ssmA,ssmB,ssmC],u=0..1,color=[COLOR(RGB,0,.6,.6),COLOR(RGB ,.4,0,.9),COLOR(RGB,.9,.3,0)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 549 392 392 {PLOTDATA 2 "6'-%'CURVESG6$7fq7$$\"3`*****\\n5;\"o!#@$\"3'o\\[E%Hj e')!#G7$$\"3#******\\8ABO\"!#?$\"3cvScyE#3V$!#F7$$\"33+++-K[V?F1$\"3]T 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[8 st age scheme] .. (shorter method) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 73 ": This is a slight va riation of the scheme obtained 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 t he nodes are based on the values obtained by trial and error using the longer method, but further improvements 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 c oefficients 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 3978 "e1 := \{c[2]=1/16,c[3]=112/1065,c[4]=56/355,c[5]=39/100,c[6]=7/1 5,c[7]=39/250,c[8]=24/25,\n c[9]=14435868/16178861,c[10]=11/12,c[11]= 19/20,c[12]=1,c[13]=1,\n a[2,1]=1/16,a[3,1]=18928/1134225,a[3,2]=1003 52/1134225,a[4,1]=14/355,a[4,2]=0,\n a[4,3]=42/355,\n a[5,1]=9449547 9/250880000,a[5,2]=0,a[5,3]=-352806597/250880000,a[5,4]=178077159/1254 40000,\n a[6,1]=12089/252720,a[6,2]=0,a[6,3]=0,a[6,4]=2505377/1068552 0,a[6,5]=960400/5209191,\n a[7,1]=21400899/350000000,a[7,2]=0,a[7,3]= 0,a[7,4]=3064329829899/27126050000000,\n a[7,5]=-21643947/592609375,a [7,6]=124391943/6756250000,\n a[8,1]=-15365458811/13609565775,a[8,2]= 0,a[8,3]=0,a[8,4]=-7/5,\n a[8,5]=-8339128164608/939060038475,a[8,6]=3 41936800488/47951126225,\n a[8,7]=1993321838240/380523459069,\n a[9, 1]=-1840911252282376584438157336464708426954728061551/\n \+ 2991923615171151921596253813483118262195533733898,a[9,2]=0,a[9 ,3]=0,\n a[9,4]=-14764960804048657303638372252908780219281424435/\n \+ 2981692102565021975611711269209606363661854518,\n a[9, 5]=-875325048502130441118613421785266742862694404520560000/\n \+ 170212030428894418395571677575961339495435011888324169,\n a[9,6 ]=7632051964154290925661849798370645637589377834346780/\n \+ 1734087257418811583049800347581865260479233950396659,\n a[9,7]=7519 834791971137517048532179652347729899303513750000/\n 1045 677303502317596597890707812349832637339039997351,\n a[9,8]=1366042683 489166351293315549358278750/\n 144631418224267718165055 326464180836641,\n a[10,1]=-63077736705254280154824845013881/78369357 853786633855112190394368,\n a[10,2]=0,a[10,3]=0,a[10,4]=-319483465108 20970247215/6956009216960026632192,\n a[10,5]=-3378604805394255292453 489375/517042670569824692230499952,\n a[10,6]=10015878441833259811980 91450220795/184232684207722503701669953872896,\n a[10,7]=187023075231 349900768014890274453125/25224698849808178010752575653374848,\n a[10, 8]=1908158550070998850625/117087067039189929394176,\n a[10,9]=-529568 18288156668227044990077324877908565/\n 2912779959477 433986349822224412353951940608,\n a[11,1]=-10116106591826909534781157 993685116703/9562819945036894030442231411871744000,\n a[11,2]=0,a[11, 3]=0,a[11,4]=-9623541317323077848129/3864449564977792573440,\n a[11,5 ]=-4823348333146829406881375/576413233634141239944816,\n a[11,6]=6566 119246514996884067001154977284529/970305487021846325473990863582315520 ,\n a[11,7]=2226455130519213549256016892506730559375/3648804431596752 55577435648380047355776,\n a[11,8]=39747262782380466933662225/1756032 802431424164410720256,\n a[11,9]=481757714192609553352446838051715480 38966866545122229/\n 19897864205138151465288801659520641189 03852843612160000,\n a[11,10]=-2378292068163246/47768728487211875,\n \+ a[12,1]=-3218022174758599831659045535578571/1453396753634469525663775 847094384,\n a[12,2]=0,a[12,3]=0,a[12,4]=26290092604284231996745/5760 876126062860430544,\n a[12,5]=-697069297560926452045586710000/4110796 7755245430594036502319,\n a[12,6]=18273578204342134614380775509022734 40/139381013914245317709567680839641697,\n a[12,7]=643504802814241550 941949227194107500000/242124609118836550860494007545333945331,\n a[12 ,8]=162259938151380266113750/59091082835244183497007,\n a[12,9]=-2302 8251632873523818545414856857015616678575554130463402/\n 2 0013169183191444503443905240405603349978424504151629055,\n a[12,10]=7 958341351371843889152/3284467988443203581305,\n a[12,11]=-50797432795 7860843878400/121555654819179042718967,\n a[13,1]=4631674879841/10378 2082379976,a[13,2]=0,a[13,3]=0,a[13,4]=0,a[13,5]=0,\n a[13,6]=1432721 9974204125/40489566827933216,\n a[13,7]=2720762324010009765625000/109 17367480696813922225349,\n a[13,8]=-498533005859375/95352091037424,\n a[13,9]=405932030463777247926705030596175437402459637909765779/\n \+ 78803919436321841083201886041201537229769115088303952,\n \+ a[13,10]=-10290327637248/1082076946951,\n a[13,11]=863264105888000/85 814662253313,\n a[13,12]=-29746300739/247142463456\}:" }}}{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%'matrixG6#7.7/$\" ++++]i!#6F(%!GF+F+F+F+F+F+F+F+F+F+7/$\"+#>V;0\"!#5$\"+!p/)o;F*$\"+MsiZ ))F*F+F+F+F+F+F+F+F+F+F+7/$\"+*ykud\"F/$\"+s>mVRF*$\"\"!F:$\"+#f)4$=\" F/F+F+F+F+F+F+F+F+F+7/$\"+++++RF/$\"+m3cmPF/F9$!+&HwiS\"!\"*$\"+4-i>9F DF+F+F+F+F+F+F+F+7/$\"+nmmmYF/$\"+A\\b$y%F*F9F9$\"+unkWBF/$\"++WmV=F/F +F+F+F+F+F+F+7/$\"++++g:F/$\"+rDa9hF*F9F9$\"+@JmH6F/$!+AEJ_OF*$\"+U)Q6 %=F*F+F+F+F+F+F+7/$\"+++++'*F/$!+M!>!H6FDF9F9$!+++++9FD$!+qGH!)))FD$\" +(4W48(FD$\"+3yOQ_FDF+F+F+F+F+7/$\"+;EnA*)F/$!+K`$H:'F/F9F9$!+jI(=&\\F D$!+#)obU^FD$\"+wD>,WFD$\"+rJN\">(FD$\"+gE*\\W*!#7F+F+F+F+7/$\"+nmmm\" *F/$!+&\\v([!)F/F9F9$!+%>8Hf%FD$!+V'zW`'FD$\"+;s`OaFD$\"+oPG9uFD$\"+)) >pH;F*$!+&4&3==F*F+F+F+7/$\"+++++&*F/$!+1\"ey0\"FDF9F9$!+(yu-\\#FD$!+k `'yO)FD$\"+9R1nnFD$\"+ed(=5'FD$\"+m#pME#F*$\"+!*G:@CF*$!+/Owy\\F*F+F+7 /$\"\"\"F:$!+@)QT@#FDF9F9$\"+Z!eNc%FD$!+@Oq&p\"!\")$\"+z@068F_s$\"+ZAu dEFD$\"+T$Hfu#FD$!+A\\l]6FD$\"+hH-BCFD$!+TW%*yTFDF+7/Fgr$\"+(*[)GY%F*F 9F9F9F9$\"+Gn\\QNF/$\"+y89#\\#F/$!+r%Q$G_FD$\"+#\\l6:&FD$!+sGz4&*FD$\" +9M'f+\"F_s$!+i%4O?\"F/Q)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "convert(ListTools[Enume rate](SimpleOrderConditions(7)),matrix):\nlinalg[augment](linalg[delco ls](%,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(F2F(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(-F 96#*(FF(#F(FTQ )pprint236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "calcul ation 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 \"#5&F%6$F'\"#6&F%6$F'\"#7&F%6$F'\"#8&%\"cG6#F'&F%6$F'\"\"\"&F%6$F'\" \"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$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#<$&%\"cG6#\"#9&%\"aG6$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/,,*&#\"=0hz# pv=l#)QT(H<<\"AOn,'*HU[SHJuXjIe5\"\"\"*$)&%\"cG6#\"#9\"\"#F+F+!\"\"*&# \"=D0)RYy$fKTpq['e)\"@_D,ZsJO0ZtIf(HPzF+*$)F.\"\"$F+F+F+*&#\"=D0)RYy$f KTpq['e)\"@7*Q`m2GokP9>yoFNF+*$)F.\"\"%F+F+F3*&#\">NFd\\)HJcyr*=3@?\" \"@o$3!)\\6U-ZcrG<`\"H&F+*$)F.\"\"&F+F+F+*&#\"?2+AJi-xCf5#z]Pg\"\"A?b7 qC " 0 "" {MPLTEXT 1 0 33 "lhs(eq_14)-rhs(eq_14);\nfactor(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*&#\"=0hz#pv=l#)QT(H<<\"AOn,'*HU[SHJuXjIe5 \"\"\"*$)&%\"cG6#\"#9\"\"#F(F(!\"\"*&#\"=D0)RYy$fKTpq['e)\"@_D,ZsJO0Zt If(HPzF(*$)F+\"\"$F(F(F(*&#\"=D0)RYy$fKTpq['e)\"@7*Q`m2GokP9>yoFNF(*$) F+\"\"%F(F(F0*&#\">NFd\\)HJcyr*=3@?\"\"@o$3!)\\6U-ZcrG<`\"H&F(*$)F+\" \"&F(F(F(*&#\">NFd\\)HJcyr*=3@?\"\"A/^-%\\ME2Tp9'=&fue\"F(*$)F+\"\"'F( F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"=0hz#pv=l#)QT(H<<\"A3-0 ))*o_9#)QHs.>\\<$\"\"\"*()&%\"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]^2-14*c[14]+3;\nsolv e(%);\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$\"+Nw<5JF%" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "We take " }{XPPEDIT 18 0 "c[14]=1/2-sqrt(7)/14" "6#/&%\"cG6#\"#9,&*&\"\"\"F*\"\"#!\"\"F**&-%% sqrtG6#\"\"(F*F'F,F," }{TEXT -1 50 " and obtain values for the linkin g coefficients " }{XPPEDIT 18 0 "a[14,6]" "6#&%\"aG6$\"#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\"#8,&#\"\"$\"$#R\"\"\"*(F.F0F/! \"\"\"\"(#F0\"\"#F2/&F(6$F*\"\"*,(*&#\"JQ%\\(fb[)RN,#))e7%H(\\M^l@%\"G :RE.O>Ar$RB&fE$f/p82#F0&F(6$F*F0F0F2#\"UHqJN5c%=i4=9rdqD6]/$f2*3:K(>\" S!)oa)RKDg+Ih]UH*R]suve7F2F3F4F0/&F(6$F*\"#5,(*&#\".3aMn;z$\"*l*R@xF 0F?F0F0#\"1;g-,nqA&F0\".4b%4;?[F2F3F4F2/&F(6$ F*\"\"),(*&#\"/v=nw&*\\F\"+j&\\&fcF0F?F0F0#\"4DcwoY)))Q*e*\"2w&[Wj*o2O %F2*(\"2D\"G=P&zRJ#F0\"2)GCs\"[%Q!=#F2F3F4F2/&F(6$F*\"#6,(*&#\"1+Of0]n k;\"-([TV2:#F0F?F0F2#\"2+;9s04-v(\"0B,R(\\68AF0*(\"0+a:6@U#RF0FjoF2F3F 4F0/&F(6$F*\"\"',(*&#\"0+C7o,!=@\"/PR#yUAe#F0F?F0F0#\"2DymP\"*pC!H\"3k Gt6tEe>;F2*(\"2D43G&\";f#)[I!zsuvCp;P\"F0*( \"8DJ&p())pBM\\3K#F0FarF2F3F4F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "We choose a value for " }{XPPEDIT 18 0 " a[14,1]" "6#&%\"aG6$\"#9\"\"\"" }{TEXT -1 73 " such that the 2-norm o f the stage 14 linking coefficients is a minimum." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "sm14 := sim plify(subs(`union`(sol,\{seq(a[14,i]=0,i=2..5)\}),add(a[14,i]^2,i=1..1 3)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%sm14G,,#\"cs\\]BL*f?kR]LtU ddt`W\"[y+i?(p4))yrG^FXm+Kz8SZW$HB'QhvGE%G![TB4m=`M#)fpJ$G'GM&=\"]s+o( )oE6#zy=1rZLGgd*)*Q#Rd)ebn37%4.w4+g_()4iBvEV#)zu#)Rk_t3%>;P?<5yhe&)**= '4)\"\"\"*&#\"fqA)[@\"G\"=:x$HfifrSe(*>n!GRq:=)>/Zf+4@iH^K`;%>BPV)***) *H)>D#pxMc\"^qDI40'RO_zEj?\"=1pS+CY$zSZp+gXf+Vgix`'z,_maI%[@**GE&y6/&F )*$)&%\"aG6$\"#9F)\"\"#F)F)F)*&#\"`rVmyXI4/<[%=8+@,aAX/^cf\\5K!*e#pQ&e )GXDk%RZ**pRX9cRN<))H\"4gZsZ!f;,@\"[rg0/%*R&*eOCct%z9P\"))et'p/l2#=LfR ,?0qbK@Q**oz\"Qpu1g@:f6[81oPy:?%F)*&F0F)\"\"(#F)F4F)!\"\"*&#\"dr$[!3'3 cU:_prezAq!**f^5gDs\"]r+_* o)*y@-A9c]5::L)45X%)z0I&4k3tBS)3pMD'\\*GZ0\\zp9@ne0(z\"H/dSKoUrF)F0F)F <*(\"]s\">I,_>j'4/960:!4KDuJTG?HnLX$eqSO\")*[M,;L#efy\"Hz/!HAj[I=(4;JX /y:&Q9$pA(*4\"F)\"ir![o3Fg=$yH&eZ(=E9?>&R*R7FW$fqYE:T#)eqvzX8#[#>Ko5X' Q1jZ!))*yJPFE/N\\aylf[F " 0 "" {MPLTEXT 1 0 26 "plot(sm14,a[14,1]=0..0.1);" }}{PARA 13 "" 1 "" {GLPLOT2D 365 381 381 {PLOTDATA 2 "6%-%'CURVESG6$7S 7$$\"\"!F)$\"35Ho\"[nK\"\\B!#77$$\"3[mmm;arz@!#?$\"3F,7$$\"3-nmm\"z_\"4iF0$\"3gjE_yr)ev\"F,7$ $\"3[mmmT&phN)F0$\"3w46'**f13d\"F,7$$\"3KLLe*=)H\\5!#>$\"3O#R:\"os$oR \"F,7$$\"3omm\"z/3uC\"FE$\"3H!==y,kYC\"F,7$$\"3#****\\7LRDX\"FE$\"3!pM #4lJM'4\"F,7$$\"3emm\"zR'ok;FE$\"3EC)HrQV%G&*!#87$$\"3'****\\i5`h(=FE$ \"3af=jAa=)>)FW7$$\"3gLLL3En$4#FE$\"3!GRym/_T$pFW7$$\"3wmm;/RE&G#FE$\" 39Gao6BS3fFW7$$\"3A+++D.&4]#FE$\"3I]l%y)*Q=&[FW7$$\"3!)*****\\PAvr#FE$ \"3)oMu!)evb*QFW7$$\"3)******\\nHi#HFE$\"3wKBUq)[K2$FW7$$\"3jmm\"z*ev: JFE$\"3$[tVMQg3T#FW7$$\"31LLL347TLFE$\"33z2j@#Gxs\"FW7$$\"3cLLLLY.KNFE $\"3@gg)f(p&yB\"FW7$$\"3!****\\7o7Tv$FE$\"3\\TKouEO0x!#97$$\"3sKLL$Q*o ]RFE$\"3C$*R\\Bem)[%Fdq7$$\"33++D\"=lj;%FE$\"3;X-k&GjK&>Fdq7$$\"33++vV &R3$4_)=Fdq7$$\"3Sm m\"zRQb@&FE$\"3uFZbm\\j_WFdq7$$\"3!****\\(=>Y2aFE$\"3;e<+cEFsvFdq7$$\" 3qmm;zXu9cFE$\"3KX,)Q$)[m=\"FW7$$\"3^******\\y))GeFE$\"3`)z,VEb6t\"FW7 $$\"3!)****\\i_QQgFE$\"3+)4O6q]IO#FW7$$\"3j***\\7y%3TiFE$\"3UY*)=.C$y1 $FW7$$\"3o****\\P![hY'FE$\"3#\\A%QBA)z&RFW7$$\"33KLL$Qx$omFE$\"3'>BMAV 2W&[FW7$$\"3k+++v.I%)oFE$\"3;m]jsGX7fFW7$$\"3Amm\"zpe*zqFE$\"3#f\\&4G4 @hpFW7$$\"37+++D\\'QH(FE$\"3o0F8(y/d?)FW7$$\"3GKLe9S8&\\(FE$\"3+!H]^*y 2q%*FW7$$\"3]++D1#=bq(FE$\"3I\\t5A:&))3\"F,7$$\"3>LLL3s?6zFE$\"3Gfp.*e &4P7F,7$$\"3)*)**\\7`Wl7)FE$\"36$oR\"4sU-9F,7$$\"3[nmmm*RRL)FE$\"37go' 4%3Yr:F,7$$\"3Smm;a<.Y&)FE$\"3'p\"fuTnEaF,7$$\"3u******\\Qk\\*)FE$\"32\"y*y-V#*H@F,7$$\"3!QLL3dg6<*FE$ \"3keV`&yw:N#F,7$$\"3-mmmmxGp$*FE$\"3[@'*Q-(=\"fDF,7$$\"3!3+]7oK0e*FE$ \"3UVd0Ex1!z#F,7$$\"3'****\\(=5s#y*FE$\"3dhfAO)e/-$F,7$$\"3/+++++++5Fj r$\"3Cha<\")GByKF,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q(a[ 14,1]6\"Q!Fh[l-%%VIEWG6$;F($\"\"\"Fc[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "minimize(sm1 4,location):\nevalf[20](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\".3LV .x6(!#9<#7$<#/&%\"aG6$\"#9\"\"\"$\"5j/s)R+;)Q%e%!#@F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "convert (.45843881600398720463e-1,rational,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$!Q\"%*G)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 8 "We set " }{XPPEDIT 18 0 "a[14,1]=380/8289" "6#/&%\"aG6$ \"#9\"\"\"*&\"$!QF(\"%*G)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "e3 := \{c[14 ]=1/2-1/14*7^(1/2),a[14,1] = 380/8289\};\ne4 := subs(e3,sol);\n " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G<$/&%\"aG6$\"#9\"\"\"#\"$!Q\"%*G )/&%\"cG6#F*,&#F+\"\"#F+*&F*!\"\"\"\"(F4F7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e4G<*/&%\"aG6$\"#9\"#8,&#\"\"$\"$#R\"\"\"*(F.F0F/!\" \"\"\"(#F0\"\"#F2/&F(6$F*\"#7,&#\".Vt7815\"\"/wt$G#*R%[F0*(\",R2IY(HF0 \".kGh.ZU\"F2F3F4F2/&F(6$F*\"\"),&#\"5vV)HaG=IIF\"\"4W4de#R-_iWF0*(\"2 D\"G=P&zRJ#F0\"2)GCs\"[%Q!=#F2F3F4F2/&F(6$F*\"#6,&#\"3+[1QJb^]J\"2hxw) pHE%z'F2*(\"0+a:6@U#RF0\"0B,R(\\68AF2F3F4F0/&F(6$F*\"\"*,&#\"Uz>LfkNU* >AtcrEa3*4#oN8ek+8V#\"U?2j%y[\"*)4#4\\Qr;kJJA(zFL;%=)*=#F2*(\"P@k)\\^[ YImGV*3&QL:A1#oG!*R$\\F0\"Qk3_VD8&HW>g+Ih]UH*R]suve7F2F3F4F0/&F(6$F*\" #5,&#\"2[G#H[[2J@\"1:jN^q%*)R(F0*(\".#R/!>qA&F0\".4b%4;?[F2F3F4F2/&F(6 $F*\"\"',&#\"4D(4sW;tc&y*\"5[#*)>qk!)=@(\\F0*(\"2D4 " 0 "" {MPLTEXT 1 0 100 "e5 := `unio n`(e2,e3,e4):\n[c[14]=subs(e5,c[14]),seq(a[14,i]=subs(e5,a[14,i]),i=1. .13)]:\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&%\"cG6#\" #9$\"I@fp*4H)=tTFR'Q'Q&\\jx,6$!#S/&%\"aG6$F(\"\"\"$\"Iw?6RA'phuNL:hz4# \\*)Q%e%!#T/&F.6$F(\"\"#$\"\"!F9/&F.6$F(\"\"$F8/&F.6$F(\"\"%F8/&F.6$F( \"\"&F8/&F.6$F(\"\"'$\"Hf&3Qj$GkW[,x([nrh*)QIOF+/&F.6$F(\"\"($\"IlqMc8 ttHY\"z%)4^jhW#QoBF+/&F.6$F(\"\")$\"Gp4N\"GX4?1$*\\fzKhp\"['*>DiAY3MV&FX/&F.6$F(\"#7$!I54O6%y.XmXq1> dR2HG>D$F3/&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 conditio ns 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 4856 "e5 := \{a[14,1] = 380/8289 , c[2] = 1/16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7 /15, a[7,1] = 21400899/350000000, a[7,2] = 0, a[7,3] = 0, a[7,4] = 306 4329829899/27126050000000, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10 685520, a[6,5] = 960400/5209191, a[5,3] = -352806597/250880000, a[5,4] = 178077159/125440000, a[6,1] = 12089/252720, a[4,2] = 0, a[4,3] = 42 /355, a[5,1] = 94495479/250880000, a[5,2] = 0, c[13] = 1, a[2,1] = 1/1 6, a[3,1] = 18928/1134225, a[3,2] = 100352/1134225, a[4,1] = 14/355, c [7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c [11] = 19/20, c[12] = 1, a[9,6] = 763205196415429092566184979837064563 7589377834346780/1734087257418811583049800347581865260479233950396659, a[9,5] = -875325048502130441118613421785266742862694404520560000/1702 12030428894418395571677575961339495435011888324169, a[9,7] = 751983479 1971137517048532179652347729899303513750000/10456773035023175965978907 07812349832637339039997351, a[9,4] = -14764960804048657303638372252908 780219281424435/2981692102565021975611711269209606363661854518, a[9,1] = -1840911252282376584438157336464708426954728061551/2991923615171151 921596253813483118262195533733898, a[9,2] = 0, a[9,3] = 0, a[8,7] = 19 93321838240/380523459069, a[8,4] = -7/5, a[8,5] = -8339128164608/93906 0038475, a[8,6] = 341936800488/47951126225, a[7,5] = -21643947/5926093 75, a[7,6] = 124391943/6756250000, a[8,1] = -15365458811/13609565775, \+ a[8,2] = 0, a[10,8] = 1908158550070998850625/117087067039189929394176, a[10,9] = -52956818288156668227044990077324877908565/2912779959477433 986349822224412353951940608, a[11,1] = -101161065918269095347811579936 85116703/9562819945036894030442231411871744000, a[11,2] = 0, a[10,5] = -3378604805394255292453489375/517042670569824692230499952, a[10,6] = \+ 1001587844183325981198091450220795/184232684207722503701669953872896, \+ a[10,7] = 187023075231349900768014890274453125/25224698849808178010752 575653374848, a[10,1] = -63077736705254280154824845013881/783693578537 86633855112190394368, a[10,2] = 0, a[10,3] = 0, a[10,4] = -31948346510 820970247215/6956009216960026632192, a[12,10] = 7958341351371843889152 /3284467988443203581305, a[12,6] = 18273578204342134614380775509022734 40/139381013914245317709567680839641697, a[12,7] = 6435048028142415509 41949227194107500000/242124609118836550860494007545333945331, a[12,8] \+ = 162259938151380266113750/59091082835244183497007, a[12,9] = -2302825 1632873523818545414856857015616678575554130463402/20013169183191444503 443905240405603349978424504151629055, a[12,3] = 0, a[12,4] = 262900926 04284231996745/5760876126062860430544, a[12,5] = -69706929756092645204 5586710000/41107967755245430594036502319, a[12,2] = 0, a[12,1] = -3218 022174758599831659045535578571/1453396753634469525663775847094384, a[1 1,8] = 39747262782380466933662225/1756032802431424164410720256, a[11,9 ] = 48175771419260955335244683805171548038966866545122229/198978642051 3815146528880165952064118903852843612160000, a[11,10] = -2378292068163 246/47768728487211875, a[11,7] = 2226455130519213549256016892506730559 375/364880443159675255577435648380047355776, a[11,4] = -96235413173230 77848129/3864449564977792573440, a[11,5] = -4823348333146829406881375/ 576413233634141239944816, a[13,9] = 4059320304637772479267050305961754 37402459637909765779/7880391943632184108320188604120153722976911508830 3952, a[13,10] = -10290327637248/1082076946951, a[13,11] = 86326410588 8000/85814662253313, a[13,6] = 14327219974204125/40489566827933216, a[ 13,7] = 2720762324010009765625000/10917367480696813922225349, a[13,8] \+ = -498533005859375/95352091037424, a[13,5] = 0, a[13,2] = 0, a[13,3] = 0, a[13,4] = 0, a[13,1] = 4631674879841/103782082379976, a[12,11] = - 507974327957860843878400/121555654819179042718967, a[14,4] = 0, c[14] \+ = 1/2-1/14*7^(1/2), a[14,5] = 0, a[14,2] = 0, a[14,3] = 0, a[14,13] = \+ 3/392-3/392*7^(1/2), a[14,12] = 1100613127343/48439922837376-297463007 39/1424703612864*7^(1/2), a[13,12] = -29746300739/247142463456, a[11,6 ] = 6566119246514996884067001154977284529/9703054870218463254739908635 82315520, a[11,3] = 0, a[8,3] = 0, a[9,8] = 13660426834891663512933155 49358278750/144631418224267718165055326464180836641, a[14,8] = 1273030 1828542984375/4462520239258570944-23139795371828125/21803844817224288* 7^(1/2), a[14,11] = -315051553138064800/67942629698767761+392422111155 400/221311497390123*7^(1/2), a[14,9] = -243130064581335682099085426715 6732219942356459331979/21898184163327797223131641671384909209891487846 30720+49339902868206221533385089432866304648514986421/1258757472503992 94250613000601944295132543520864*7^(1/2), a[14,10] = 21310748482922848 /7398947051356315-5227019004392/4820160945509*7^(1/2), a[14,6] = 97855 67316447209725/49721188064701989248-34388281046170925/5668539355910650 24*7^(1/2), a[14,7] = 1016182669829177731250000000/4211024590054414662 359895513-23208493423698876953125/13716692475747279030488259*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 conditi ons can be adapted to give a method of stage by stage construction for an interpolation scheme that avoids dealing with 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 0 219 "whch := [1,2,4,8,16,21,27,31,32,64]:\ninterp_order_eqns15 := \+ []:\nfor ct in whch do\n temp_eqn := convert(SO7_14[ct],'interpolati on_order_condition'):\n interp_order_eqns15 := [op(interp_order_eqns 15),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 0 410 "interp_order_eqns15 := [add(a[15,i],i=1..14)=c[15],seq(op(StageOr derConditions(i,15..15,'expanded')),i=2..7),\n add(a[15,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 s pecify the node for this stage immediately, namely " }{XPPEDIT 18 0 " c[15] = 1187/2500;" "6#/&%\"cG6#\"#:*&\"%(=\"\"\"\"\"%+D!\"\"" }{TEXT -1 80 ", and have enough equations to determine the corresponding link ing coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "e6 := \+ `union`(e5,\{c[15]=1187/2500,seq(a[15,i]=0,i=2..5)\}):\neqs_15 := expa nd(subs(e6,interp_order_eqns15)):\nnops(eqs_15);\nindets(eqs_15);\nnop s(%);" }}{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$F'\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=0:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "e7 := solve(\{op(eqs_15)\} ):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "e8 := `union`(e6,e7):\n[seq(a[15,i] =subs(e8,a[15,i]),i=1..14)]:\nevalf[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70/&%\"aG6$\"#:\"\"\"$\"I$oS.U0Po*eL%zw*)\\\")QQR,&!#T/ &F&6$F(\"\"#$\"\"!F2/&F&6$F(\"\"$F1/&F&6$F(\"\"%F1/&F&6$F(\"\"&F1/&F&6 $F(\"\"'$\"IL,q'f7Z)Q(pTYj&=79ieT\"*F,/&F&6$F(\"\"($\"I3D9h)*\\!3[??sk s@x\\#R\"4#!#S/&F&6$F(\"\")$!Io?-\"pOe%R`Yw=ki!Hr7+S#!#R/&F&6$F(\"\"*$ \"It%R%[N&)))=BivrA+v=W8+*)FK/&F&6$F(\"#5$!Iyj:b'Hs^+4F@*)\\:g&\\`XAFR /&F&6$F(\"#6$\"I3$enY\\2Z-z9;mG(R)***[YPFR/&F&6$F(\"#7$\"I/K-:&\\iY(=q )fkd&*=:r;.%!#U/&F&6$F(\"#8$\"IJ$=eF(G0n!y-uY_Lr*Q*HV$Feo/&F&6$F(\"#9$ \"I6p+:#e_l.q+C$\\b0y1&oD\"FK" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "These linking coefficients agree to 40 di gits with those of the published scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#--------------------------------- ------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "W e can check which of the (adapted) simple order conditions are satisfi ed at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "recd := []:\nfor ct to nops(SO7_14) do\n t t := convert(SO7_14[ct],'interpolation_order_condition'):\n if expan d(subs(e8,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend d o:\nop(recd);\nnops(recd);\nop(\{seq(i,i=1..nops(SO7_14))\} minus \{op (recd)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6J\"\"\"\"\"#\"\"$\"\"%\" \"&\"\"'\"\"(\"\")\"\"*\"#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 6350 "e8 := \{a[15,12] = 12163051029345238407 733418657/3016875774609375000000000000000, a[14,1] = 380/8289, c[2] = \+ 1/16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[7, 1] = 21400899/350000000, a[7,2] = 0, a[7,3] = 0, a[7,4] = 306432982989 9/27126050000000, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a [6,5] = 960400/5209191, a[5,3] = -352806597/250880000, a[5,4] = 178077 159/125440000, a[6,1] = 12089/252720, a[4,2] = 0, a[4,3] = 42/355, a[5 ,1] = 94495479/250880000, a[5,2] = 0, c[13] = 1, a[2,1] = 1/16, a[3,1] = 18928/1134225, a[3,2] = 100352/1134225, a[4,1] = 14/355, c[7] = 39/ 250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19 /20, c[12] = 1, a[9,6] = 763205196415429092566184979837064563758937783 4346780/1734087257418811583049800347581865260479233950396659, a[9,5] = -875325048502130441118613421785266742862694404520560000/1702120304288 94418395571677575961339495435011888324169, a[9,7] = 751983479197113751 7048532179652347729899303513750000/10456773035023175965978907078123498 32637339039997351, a[9,4] = -14764960804048657303638372252908780219281 424435/2981692102565021975611711269209606363661854518, a[9,1] = -18409 11252282376584438157336464708426954728061551/2991923615171151921596253 813483118262195533733898, a[9,2] = 0, a[9,3] = 0, a[8,7] = 19933218382 40/380523459069, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, \+ a[8,6] = 341936800488/47951126225, a[7,5] = -21643947/592609375, a[7,6 ] = 124391943/6756250000, a[8,1] = -15365458811/13609565775, a[8,2] = \+ 0, a[10,8] = 1908158550070998850625/117087067039189929394176, a[10,9] \+ = -52956818288156668227044990077324877908565/2912779959477433986349822 224412353951940608, a[11,1] = -10116106591826909534781157993685116703/ 9562819945036894030442231411871744000, a[11,2] = 0, a[10,5] = -3378604 805394255292453489375/517042670569824692230499952, a[10,6] = 100158784 4183325981198091450220795/184232684207722503701669953872896, a[10,7] = 187023075231349900768014890274453125/25224698849808178010752575653374 848, a[10,1] = -63077736705254280154824845013881/783693578537866338551 12190394368, a[10,2] = 0, a[10,3] = 0, a[10,4] = -31948346510820970247 215/6956009216960026632192, a[12,10] = 7958341351371843889152/32844679 88443203581305, a[12,6] = 1827357820434213461438077550902273440/139381 013914245317709567680839641697, a[12,7] = 6435048028142415509419492271 94107500000/242124609118836550860494007545333945331, a[12,8] = 1622599 38151380266113750/59091082835244183497007, a[12,9] = -2302825163287352 3818545414856857015616678575554130463402/20013169183191444503443905240 405603349978424504151629055, a[12,3] = 0, a[12,4] = 262900926042842319 96745/5760876126062860430544, a[12,5] = -69706929756092645204558671000 0/41107967755245430594036502319, a[12,2] = 0, a[12,1] = -3218022174758 599831659045535578571/1453396753634469525663775847094384, a[11,8] = 39 747262782380466933662225/1756032802431424164410720256, a[11,9] = 48175 771419260955335244683805171548038966866545122229/198978642051381514652 8880165952064118903852843612160000, a[11,10] = -2378292068163246/47768 728487211875, a[11,7] = 2226455130519213549256016892506730559375/36488 0443159675255577435648380047355776, a[11,4] = -9623541317323077848129/ 3864449564977792573440, a[11,5] = -4823348333146829406881375/576413233 634141239944816, a[13,9] = 4059320304637772479267050305961754374024596 37909765779/78803919436321841083201886041201537229769115088303952, a[1 3,10] = -10290327637248/1082076946951, a[13,11] = 863264105888000/8581 4662253313, a[13,6] = 14327219974204125/40489566827933216, a[13,7] = 2 720762324010009765625000/10917367480696813922225349, a[13,8] = -498533 005859375/95352091037424, a[13,5] = 0, a[13,2] = 0, a[13,3] = 0, a[13, 4] = 0, a[13,1] = 4631674879841/103782082379976, a[12,11] = -507974327 957860843878400/121555654819179042718967, a[14,4] = 0, c[14] = 1/2-1/1 4*7^(1/2), a[14,5] = 0, a[14,2] = 0, a[14,3] = 0, a[14,13] = 3/392-3/3 92*7^(1/2), a[14,12] = 1100613127343/48439922837376-29746300739/142470 3612864*7^(1/2), a[15,1] = 130946139152859396534950567713097/281526916 1783203125000000000000000+90670944595916412828478989403/66148981703437 500000000000000000*7^(1/2), a[15,8] = 16161913072172934315785836819/99 32509483065000000000000000-75980668436324671626317237501/4991722714566 0000000000000000*7^(1/2), a[15,6] = 3353943190402140976803568136084793 /19770296302701765625000000000000000-255748988917794999289135717669191 /8649504632432022460937500000000000*7^(1/2), a[15,7] = 105966145406009 68744792362758669/45489031169570058009272287500000-1305842593294600719 2467094947/1451071902055187778276093750000*7^(1/2), a[15,10] = 1446205 9099710033235500874587/10319489926824569702148437500-34815528884845218 7540396337457/252574928278923034667968750000*7^(1/2), a[15,9] = -37278 50143078386229058076561522609367827766330696417108435354320431359/8016 349226513859159668160608032382937598583484731440429687500000000000+814 251349796512127448124174135016152994969999173763717229208919303/158984 2118264019833619870377392538080518560957031250000000000000000*7^(1/2), a[15,11] = -207941800836756311792092915907/81839239362061500549316406 250+10891006522099224595926000559657/4582997404275444030761718750000*7 ^(1/2), a[15,14] = 26094995957293704259/549316406250000000000*7^(1/2), a[15,13] = -579945478542537523/549316406250000000000+3727856565327672 037/2197265625000000000000*7^(1/2), a[13,12] = -29746300739/2471424634 56, a[11,6] = 6566119246514996884067001154977284529/970305487021846325 473990863582315520, a[11,3] = 0, a[8,3] = 0, a[9,8] = 1366042683489166 351293315549358278750/144631418224267718165055326464180836641, a[15,4] = 0, a[15,5] = 0, a[15,2] = 0, a[15,3] = 0, a[14,8] = 127303018285429 84375/4462520239258570944-23139795371828125/21803844817224288*7^(1/2), a[14,11] = -315051553138064800/67942629698767761+392422111155400/2213 11497390123*7^(1/2), a[14,9] = -24313006458133568209908542671567322199 42356459331979/2189818416332779722313164167138490920989148784630720+49 339902868206221533385089432866304648514986421/125875747250399294250613 000601944295132543520864*7^(1/2), a[14,10] = 21310748482922848/7398947 051356315-5227019004392/4820160945509*7^(1/2), a[14,6] = 9785567316447 209725/49721188064701989248-34388281046170925/566853935591065024*7^(1/ 2), a[14,7] = 1016182669829177731250000000/421102459005441466235989551 3-23208493423698876953125/13716692475747279030488259*7^(1/2), c[15] = \+ 1187/2500\}: " }{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 := [1,2,4,8,16,21,27,31,32,63,64]:\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 0 481 "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]*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 ad d(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] = 277/2500; " "6#/&%\"cG6#\"#;*&\"$x#\"\"\"\"%+D!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "e9 := `union`(e8,\{c[16]=277/2500, seq(a[16,i]=0,i=2..5)\}):\neqs_16 := expand(subs(e9,interp_order_eqns1 6)):\nnops(eqs_16);\nindets(eqs_16);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<-&%\"aG6$\"#; \"\"'&F%6$F'\"\"(&F%6$F'\"\")&F%6$F'\"\"*&F%6$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#\"#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)\}):\ninfo level[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$\"#;\"\"\"$\"IFunUTX#\\/M19(Gq@+\"3?#\\!#T/&F%6$F'\"\"#$\"\"! F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"Ieu\\`L Xyf\"3f!*)3S5#R(\\%G\"F+/&F%6$F'\"\"($\"IY3XY`dl;kJ*y+D/PHB(Q5!#S/&F%6 $F'\"\")$!If8!yZQbg)QV)pQ0r+\"zk^VFJ/&F%6$F'\"\"*$\"Ia!f/J>&3/=s>v*p'H >=(*p8FJ/&F%6$F'\"#5$!I/M>:P&o`d1TEhH$*)Qwuh^q!QysmtFJ/&F%6$F'\"#7$!I$>FqH\\+,SG%H!fWTQ*y:e8F+/&F%6$F'\"#8$! I:m&y><&y0/x<4MSCtQt;Z!#U/&F%6$F'\"#9$!I$f_'4j)\\+%>])4Z2em_>5!pF+/&F% 6$F'\"#:$\"I_/tllatnr*>BjsJWH7K%=F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "#-------------------------------------- -------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can check which of the (adapted) simple order conditions are satisfied at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "recd := []:\nfor ct to nops(SO7_15) do\n tt := \+ convert(SO7_15[ct],'interpolation_order_condition'):\n if expand(sub s(e11,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\n op(recd);\nnops(recd);\nop(\{seq(i,i=1..nops(SO7_15))\} minus \{op(rec d)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6R\"\"\"\"\"#\"\"$\"\"%\"\"&\" \"'\"\"(\"\")\"\"*\"#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 9964 "e11 := \{a[15,12] = 12163051029345238407733 418657/3016875774609375000000000000000, a[14,1] = 380/8289, c[2] = 1/1 6, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[7,1] \+ = 21400899/350000000, a[7,2] = 0, a[7,3] = 0, a[7,4] = 3064329829899/2 7126050000000, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6, 5] = 960400/5209191, a[5,3] = -352806597/250880000, a[5,4] = 178077159 /125440000, a[6,1] = 12089/252720, a[4,2] = 0, a[4,3] = 42/355, a[5,1] = 94495479/250880000, a[5,2] = 0, c[13] = 1, a[2,1] = 1/16, a[3,1] = \+ 18928/1134225, a[3,2] = 100352/1134225, a[4,1] = 14/355, c[7] = 39/250 , c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/20 , c[12] = 1, a[9,6] = 763205196415429092566184979837064563758937783434 6780/1734087257418811583049800347581865260479233950396659, a[9,5] = -8 75325048502130441118613421785266742862694404520560000/1702120304288944 18395571677575961339495435011888324169, a[9,7] = 751983479197113751704 8532179652347729899303513750000/10456773035023175965978907078123498326 37339039997351, a[9,4] = -14764960804048657303638372252908780219281424 435/2981692102565021975611711269209606363661854518, a[9,1] = -18409112 52282376584438157336464708426954728061551/2991923615171151921596253813 483118262195533733898, a[9,2] = 0, a[9,3] = 0, a[8,7] = 1993321838240/ 380523459069, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8 ,6] = 341936800488/47951126225, a[7,5] = -21643947/592609375, a[7,6] = 124391943/6756250000, a[8,1] = -15365458811/13609565775, a[8,2] = 0, \+ a[10,8] = 1908158550070998850625/117087067039189929394176, a[10,9] = - 52956818288156668227044990077324877908565/2912779959477433986349822224 412353951940608, a[11,1] = -10116106591826909534781157993685116703/956 2819945036894030442231411871744000, a[11,2] = 0, a[10,5] = -3378604805 394255292453489375/517042670569824692230499952, a[10,6] = 100158784418 3325981198091450220795/184232684207722503701669953872896, a[10,7] = 18 7023075231349900768014890274453125/25224698849808178010752575653374848 , a[10,1] = -63077736705254280154824845013881/783693578537866338551121 90394368, a[10,2] = 0, a[10,3] = 0, a[10,4] = -31948346510820970247215 /6956009216960026632192, a[12,10] = 7958341351371843889152/32844679884 43203581305, a[12,6] = 1827357820434213461438077550902273440/139381013 914245317709567680839641697, a[12,7] = 6435048028142415509419492271941 07500000/242124609118836550860494007545333945331, a[12,8] = 1622599381 51380266113750/59091082835244183497007, a[12,9] = -2302825163287352381 8545414856857015616678575554130463402/20013169183191444503443905240405 603349978424504151629055, a[12,3] = 0, a[12,4] = 262900926042842319967 45/5760876126062860430544, a[12,5] = -697069297560926452045586710000/4 1107967755245430594036502319, a[12,2] = 0, a[12,1] = -3218022174758599 831659045535578571/1453396753634469525663775847094384, a[11,8] = 39747 262782380466933662225/1756032802431424164410720256, a[11,9] = 48175771 419260955335244683805171548038966866545122229/198978642051381514652888 0165952064118903852843612160000, a[11,10] = -2378292068163246/47768728 487211875, a[11,7] = 2226455130519213549256016892506730559375/36488044 3159675255577435648380047355776, a[11,4] = -9623541317323077848129/386 4449564977792573440, a[11,5] = -4823348333146829406881375/576413233634 141239944816, a[13,9] = 4059320304637772479267050305961754374024596379 09765779/78803919436321841083201886041201537229769115088303952, a[13,1 0] = -10290327637248/1082076946951, a[13,11] = 863264105888000/8581466 2253313, a[13,6] = 14327219974204125/40489566827933216, a[13,7] = 2720 762324010009765625000/10917367480696813922225349, a[13,8] = -498533005 859375/95352091037424, a[13,5] = 0, a[13,2] = 0, a[13,3] = 0, a[13,4] \+ = 0, a[13,1] = 4631674879841/103782082379976, a[12,11] = -507974327957 860843878400/121555654819179042718967, a[14,4] = 0, c[14] = 1/2-1/14*7 ^(1/2), a[16,15] = 460888287630742936877525144387640337292487250940862 1400283/75922278949360871648575860105318720457058558153486785997500-84 914702986428741912331429025972417475312941024930181294/531455952645526 1015400310207372310431994099070744075019825*7^(1/2), a[16,7] = 1471649 9751079132511176933345920457044509185685353579684857180786483042792373 /160889883376746714657021565950525327016567992369670063678825639487979 375000000+350971611315420287919452885739900022880241691060058741934303 404427219789/748680368989609593370755625030934570162567105993716345703 93697387568750000*7^(1/2), a[14,5] = 0, a[14,2] = 0, a[14,3] = 0, a[14 ,13] = 3/392-3/392*7^(1/2), a[14,12] = 1100613127343/48439922837376-29 746300739/1424703612864*7^(1/2), a[15,1] = 130946139152859396534950567 713097/2815269161783203125000000000000000+9067094459591641282847898940 3/66148981703437500000000000000000*7^(1/2), a[15,8] = 1616191307217293 4315785836819/9932509483065000000000000000-759806684363246716263172375 01/49917227145660000000000000000*7^(1/2), a[15,6] = 335394319040214097 6803568136084793/19770296302701765625000000000000000-25574898891779499 9289135717669191/8649504632432022460937500000000000*7^(1/2), a[15,7] = 10596614540600968744792362758669/45489031169570058009272287500000-130 58425932946007192467094947/1451071902055187778276093750000*7^(1/2), a[ 15,10] = 14462059099710033235500874587/10319489926824569702148437500-3 48155288848452187540396337457/252574928278923034667968750000*7^(1/2), \+ a[15,9] = -37278501430783862290580765615226093678277663306964171084353 54320431359/8016349226513859159668160608032382937598583484731440429687 500000000000+814251349796512127448124174135016152994969999173763717229 208919303/158984211826401983361987037739253808051856095703125000000000 0000000*7^(1/2), a[15,11] = -207941800836756311792092915907/8183923936 2061500549316406250+10891006522099224595926000559657/45829974042754440 30761718750000*7^(1/2), a[15,14] = 26094995957293704259/54931640625000 0000000*7^(1/2), a[15,13] = -579945478542537523/549316406250000000000+ 3727856565327672037/2197265625000000000000*7^(1/2), a[13,12] = -297463 00739/247142463456, a[11,6] = 6566119246514996884067001154977284529/97 0305487021846325473990863582315520, a[11,3] = 0, a[8,3] = 0, a[9,8] = \+ 1366042683489166351293315549358278750/14463141822426771816505532646418 0836641, a[15,4] = 0, a[15,5] = 0, a[15,2] = 0, a[15,3] = 0, a[16,2] = 0, a[16,4] = 0, a[16,3] = 0, a[16,5] = 0, c[16] = 277/2500, a[14,8] = 12730301828542984375/4462520239258570944-23139795371828125/2180384481 7224288*7^(1/2), a[14,11] = -315051553138064800/67942629698767761+3924 22111155400/221311497390123*7^(1/2), a[14,9] = -2431300645813356820990 854267156732219942356459331979/218981841633277972231316416713849092098 9148784630720+49339902868206221533385089432866304648514986421/12587574 7250399294250613000601944295132543520864*7^(1/2), a[14,10] = 213107484 82922848/7398947051356315-5227019004392/4820160945509*7^(1/2), a[14,6] = 9785567316447209725/49721188064701989248-34388281046170925/56685393 5591065024*7^(1/2), a[14,7] = 1016182669829177731250000000/42110245900 54414662359895513-23208493423698876953125/13716692475747279030488259*7 ^(1/2), a[16,14] = 378070792470829109357504373975352653612596622387047 615352035143989/386524657263116654742134068873711875278106926714137268 06640625000000-7366944851192037549005445931990589557073904265481386283 37098463663/2473757806483946590349658040791756001779884330970478515625 0000000000*7^(1/2), a[16,6] = -190398901498664954325849292034437666076 6478959434824857368133995942073946993/26105498804470984484513299908841 941049181178531837828644206625000000000000000+289290358369928611582932 13150542073159989024626464512663622554495682276029/8922777911684418524 98013180477996031954434813099925783737531127929687500000*7^(1/2), a[16 ,9] = 2079043420349691709006286724449198371945091963358485311941328410 6748179318736252691355190285977916572557692108955233/15877667583594659 6689535645887725453057778073174418932018795803586671680545234558493166 228541026757812500000000000000+167502919171284871193499988335765604001 1164360189191231613203552972323338878927017521218877046934971369718057 209/731810193153334108198682812322026779525561489528186024119552674967 375646957459693466445168584156250000000000000000*7^(1/2), a[16,13] = - 46330543544553540874179448395431083082899647522368296822645347/5087451 352083124397391250759096940921758056970520019531250000000+189091968318 6766311626712563992017565381198275789277564189007889/11395891028666198 65015640170037714766473804761396484375000000000000*7^(1/2), a[16,1] = \+ 1013856404118125952763527205161498788180173939254661359435974309585825 318269273/198564667739239660374389171210942338307459063268324464130246 79687500000000000000-1931754243710985131731267585181486489297591530273 47863824763410934840962491/2778932721896691403250310788177330670569727 48113814414737187500000000000000000*7^(1/2), a[16,10] = -1752032306988 852430152559893574903103295734904546828540573516708228831279/243326283 0102477113639615580994907630153630681641719830799102783203125000+81883 999666586172691735974039324906939964446697501450842550606413648603/733 722330307823868112868698269233685400171713233503210517883300781250000* 7^(1/2), a[16,12] = -1248171126266506029131703126444811433487776092856 30839757851632016681433/3983604783735069658159427207807518356647862336 983160384375000000000000000+194885682734472712496697330676881594107424 49986073234566291555857980323/2904711821473488292407915672359648801722 399620716887780273437500000000000*7^(1/2), a[16,11] = 3307587237786214 580372394397330733034806289881443017455732775116590947887/216127682925 3336041664932456066917395909700497105040226143341064453125000-15924948 1348355430623321425412532136521666854903188329515308104047100201/53083 9922974503589180860603244506027065540472973167774842224121093750000*7^ (1/2), c[15] = 1187/2500, a[16,8] = -285913537824921837018262511263067 66396461229874660236919089505718702483/3073895442200686868940860530269 7869750592763266032965939343750000000000+32621718645707277997227824444 0069764006022537683484345700377402630409/17437228972534444030205831181 52408439159931363492594965600000000000000*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 "calculat ion for stage 17" }}{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_16 := SimpleOrderConditions(7,16,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1, 2,4,8,16,21,27,31,32,61,63,64]:\ninterp_order_eqns17 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_16[ct],'interpolation_order_condi tion'):\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 specified explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 555 "interp_orde r_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]*ad d(a[k,l]*c[l]^2,l=2..k-1),k=2..j-1),\n j=2..i -1),i=2..16)=c[17]^6/360, #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 specif y the node " }{XPPEDIT 18 0 " c[17]=993/2500" "6#/&%\"cG6#\"#<*&\"$$ **\"\"\"\"%+D!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "e12 := `union`(e11,\{c[17]=993/2500,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$\"#<\"\"\"&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]:=0:" }}}{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$\"#<\"\"\"$\"I3BySnoo&[\"*\\16A(QJ#o1H$!#T/&F%6$F'\"\"#$\"\"!F1/&F%6 $F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"Hz76_@G>l%3X!f u!RTS$*)G\"!#S/&F%6$F'\"\"($\"HgX!32RC)R(RE4Vo*yPd$)p\"FD/&F%6$F'\"\") $\"H;ocv4a#4&[iWz/5j/kGH(FD/&F%6$F'\"\"*$!IvXv#fB[/nZ&e[WFxh!RDi$F+/&F %6$F'\"#5$\"HyrG;%\\Y\"4%H0)RG**HVVq0*FD/&F%6$F'\"#6$!Iim9Lk*)zU)p\\YG \"4Q)[q/I\"FD/&F%6$F'\"#7$\"I\"=+GWR(Rz\"*eV$=FVBU)[q@!#U/&F%6$F'\"#8$ \"HxM(H3!G`jsF$3&3Mc!ejB\"*Fco/&F%6$F'\"#9$\"I]D94(piaN&fm45C*[g[^!=FD /&F%6$F'\"#:$!H;!>NU(yhA*3s^=]E1`fp()F+/&F%6$F'\"#;$\"Iqc_Ttdbo)zGxj%4 q4llB;FD" }}}{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 0 244 "recd : = []:\nfor ct to nops(SO7_16) do\n tt := convert(SO7_16[ct],'interpo lation_order_condition'):\n if expand(subs(e14,lhs(tt)=rhs(tt))) the n recd := [op(recd),ct] end if: \nend do:\nop(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 12831 "e14 := \{a[15,12] = 1216305102934523840773 3418657/3016875774609375000000000000000, a[14,1] = 380/8289, c[2] = 1/ 16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[7,1] = 21400899/350000000, a[7,2] = 0, a[7,3] = 0, a[7,4] = 3064329829899/ 27126050000000, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6 ,5] = 960400/5209191, a[5,3] = -352806597/250880000, a[5,4] = 17807715 9/125440000, a[6,1] = 12089/252720, a[4,2] = 0, a[4,3] = 42/355, a[5,1 ] = 94495479/250880000, a[5,2] = 0, c[13] = 1, a[2,1] = 1/16, a[3,1] = 18928/1134225, a[3,2] = 100352/1134225, a[4,1] = 14/355, c[7] = 39/25 0, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/2 0, c[12] = 1, a[9,6] = 76320519641542909256618497983706456375893778343 46780/1734087257418811583049800347581865260479233950396659, a[9,5] = - 875325048502130441118613421785266742862694404520560000/170212030428894 418395571677575961339495435011888324169, a[9,7] = 75198347919711375170 48532179652347729899303513750000/1045677303502317596597890707812349832 637339039997351, a[9,4] = -1476496080404865730363837225290878021928142 4435/2981692102565021975611711269209606363661854518, a[9,1] = -1840911 252282376584438157336464708426954728061551/299192361517115192159625381 3483118262195533733898, a[9,2] = 0, a[9,3] = 0, a[8,7] = 1993321838240 /380523459069, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[ 8,6] = 341936800488/47951126225, a[7,5] = -21643947/592609375, a[7,6] \+ = 124391943/6756250000, a[8,1] = -15365458811/13609565775, a[8,2] = 0, a[10,8] = 1908158550070998850625/117087067039189929394176, a[10,9] = \+ -52956818288156668227044990077324877908565/291277995947743398634982222 4412353951940608, a[11,1] = -10116106591826909534781157993685116703/95 62819945036894030442231411871744000, a[11,2] = 0, a[10,5] = -337860480 5394255292453489375/517042670569824692230499952, a[10,6] = 10015878441 83325981198091450220795/184232684207722503701669953872896, a[10,7] = 1 87023075231349900768014890274453125/2522469884980817801075257565337484 8, a[10,1] = -63077736705254280154824845013881/78369357853786633855112 190394368, a[10,2] = 0, a[10,3] = 0, a[10,4] = -3194834651082097024721 5/6956009216960026632192, a[12,10] = 7958341351371843889152/3284467988 443203581305, a[12,6] = 1827357820434213461438077550902273440/13938101 3914245317709567680839641697, a[12,7] = 643504802814241550941949227194 107500000/242124609118836550860494007545333945331, a[12,8] = 162259938 151380266113750/59091082835244183497007, a[12,9] = -230282516328735238 18545414856857015616678575554130463402/2001316918319144450344390524040 5603349978424504151629055, a[12,3] = 0, a[12,4] = 26290092604284231996 745/5760876126062860430544, a[12,5] = -697069297560926452045586710000/ 41107967755245430594036502319, a[12,2] = 0, a[12,1] = -321802217475859 9831659045535578571/1453396753634469525663775847094384, a[11,8] = 3974 7262782380466933662225/1756032802431424164410720256, a[11,9] = 4817577 1419260955335244683805171548038966866545122229/19897864205138151465288 80165952064118903852843612160000, a[11,10] = -2378292068163246/4776872 8487211875, a[11,7] = 2226455130519213549256016892506730559375/3648804 43159675255577435648380047355776, a[11,4] = -9623541317323077848129/38 64449564977792573440, a[11,5] = -4823348333146829406881375/57641323363 4141239944816, a[13,9] = 405932030463777247926705030596175437402459637 909765779/78803919436321841083201886041201537229769115088303952, a[13, 10] = -10290327637248/1082076946951, a[13,11] = 863264105888000/858146 62253313, a[13,6] = 14327219974204125/40489566827933216, a[13,7] = 272 0762324010009765625000/10917367480696813922225349, a[13,8] = -49853300 5859375/95352091037424, a[13,5] = 0, a[13,2] = 0, a[13,3] = 0, a[13,4] = 0, a[13,1] = 4631674879841/103782082379976, a[12,11] = -50797432795 7860843878400/121555654819179042718967, a[17,1] = 14296927109732800365 803280238811548992426955660898056576097/341413526737606015913504201607 836239464660351562500000000000-354981099311588698585708772578884720992 15653607797737/1047152271922481952869292729750448532280273437500000000 0*7^(1/2), a[17,9] = -777365420445936318089732417738641004514876429071 6888867400388432148819282908940948159254319/78845275237698121560986448 0482093541291983982324119451546116914378375650249609375000000000000-34 9211535189884795491496750852447695276558730568496460079234407077243879 427767491465619/350423445500880540271050880214263796129770658810719756 24271862861261140011093750000000000*7^(1/2), a[17,14] = 88988313395506 75631062765096204752299374508560508163/1749782565764184901107853213039 69544268798828125000000+3430004043103091387006765566734797526578597451 69/6999130263056739604431412852158781770751953125000*7^(1/2), a[17,6] \+ = 11680941707325157677955382237279696768268247094386992999/87503361475 665632084979209131119394314606250000000000000-997172433570276305925254 457325512017866796579074407189/218758403689164080212448022827798485786 51562500000000000*7^(1/2), a[17,7] = 729247118061923611585935970034673 8458665766464861503/44741004683259221639235793273762127339404272876630 000-395688161558937580577353827374940476509802536449201/71700328018043 62442185223281051622971058377063562500*7^(1/2), a[17,16] = 31295100384 6638093776403168468573978963277/58151890694283675464489313434950766660 50000+1835267423444683727710449934904130527/44732223610987442664991779 565346743585*7^(1/2), a[17,11] = -259187471300326360195328568552656084 453334237831807/482960797022440548690819027452576858488464355468750+19 53279068514198003642387066372792263574848990934/1270949465848527759712 6816511909917328643798828125*7^(1/2), a[17,15] = -77127127250912159388 465163390223095790721/1192299437549281524502362666335509439650000+1957 7936100246207024301582455131137389/92632494763444179980568176384528041 0805*7^(1/2), a[17,8] = 1589943165386807608189280067146301169342437436 39/488458911821805651705273894751044258600000000000-647640403594581213 3836473370020724114834745161/67841515530806340514621374270978369250000 000000*7^(1/2), a[17,10] = 5136523946898556275020265594155602860649294 95444927/2283703904801922343978476111178519401359558105468750-21084668 4210425267635251476041683325933811084598/41521889178216769890517747475 97308002471923828125*7^(1/2), a[17,12] = 26221430874444225133059937920 1914827178216725701161/22254494604004075189901336249265303906250000000 000000-4042541431743211698640695349601479330547680120287/1112724730200 203759495066812463265195312500000000000*7^(1/2), a[17,13] = 5503805654 154447779082385289578389472169301/173484996080843837216266330756835937 5000000000-44976428168893445098944962904416522821649/52650194123092630 885946212158775329589843750*7^(1/2), a[14,4] = 0, c[14] = 1/2-1/14*7^( 1/2), a[17,5] = 0, a[17,4] = 0, a[17,3] = 0, a[17,2] = 0, a[16,15] = 4 608882876307429368775251443876403372924872509408621400283/759222789493 60871648575860105318720457058558153486785997500-8491470298642874191233 1429025972417475312941024930181294/53145595264552610154003102073723104 31994099070744075019825*7^(1/2), a[16,7] = 147164997510791325111769333 45920457044509185685353579684857180786483042792373/1608898833767467146 57021565950525327016567992369670063678825639487979375000000+3509716113 15420287919452885739900022880241691060058741934303404427219789/7486803 6898960959337075562503093457016256710599371634570393697387568750000*7^ (1/2), a[14,5] = 0, a[14,2] = 0, a[14,3] = 0, a[14,13] = 3/392-3/392*7 ^(1/2), a[14,12] = 1100613127343/48439922837376-29746300739/1424703612 864*7^(1/2), a[15,1] = 130946139152859396534950567713097/2815269161783 203125000000000000000+90670944595916412828478989403/661489817034375000 00000000000000*7^(1/2), a[15,8] = 16161913072172934315785836819/993250 9483065000000000000000-75980668436324671626317237501/49917227145660000 000000000000*7^(1/2), a[15,6] = 3353943190402140976803568136084793/197 70296302701765625000000000000000-255748988917794999289135717669191/864 9504632432022460937500000000000*7^(1/2), a[15,7] = 1059661454060096874 4792362758669/45489031169570058009272287500000-13058425932946007192467 094947/1451071902055187778276093750000*7^(1/2), a[15,10] = 14462059099 710033235500874587/10319489926824569702148437500-348155288848452187540 396337457/252574928278923034667968750000*7^(1/2), a[15,9] = -372785014 3078386229058076561522609367827766330696417108435354320431359/80163492 26513859159668160608032382937598583484731440429687500000000000+8142513 49796512127448124174135016152994969999173763717229208919303/1589842118 264019833619870377392538080518560957031250000000000000000*7^(1/2), a[1 5,11] = -207941800836756311792092915907/81839239362061500549316406250+ 10891006522099224595926000559657/4582997404275444030761718750000*7^(1/ 2), a[15,14] = 26094995957293704259/549316406250000000000*7^(1/2), a[1 5,13] = -579945478542537523/549316406250000000000+3727856565327672037/ 2197265625000000000000*7^(1/2), a[13,12] = -29746300739/247142463456, \+ a[11,6] = 6566119246514996884067001154977284529/9703054870218463254739 90863582315520, a[11,3] = 0, a[8,3] = 0, a[9,8] = 13660426834891663512 93315549358278750/144631418224267718165055326464180836641, a[15,4] = 0 , a[15,5] = 0, a[15,2] = 0, a[15,3] = 0, a[16,2] = 0, a[16,4] = 0, a[1 6,3] = 0, c[17] = 993/2500, a[16,5] = 0, c[16] = 277/2500, a[14,8] = 1 2730301828542984375/4462520239258570944-23139795371828125/218038448172 24288*7^(1/2), a[14,11] = -315051553138064800/67942629698767761+392422 111155400/221311497390123*7^(1/2), a[14,9] = -243130064581335682099085 4267156732219942356459331979/21898184163327797223131641671384909209891 48784630720+49339902868206221533385089432866304648514986421/1258757472 50399294250613000601944295132543520864*7^(1/2), a[14,10] = 21310748482 922848/7398947051356315-5227019004392/4820160945509*7^(1/2), a[14,6] = 9785567316447209725/49721188064701989248-34388281046170925/5668539355 91065024*7^(1/2), a[14,7] = 1016182669829177731250000000/4211024590054 414662359895513-23208493423698876953125/13716692475747279030488259*7^( 1/2), a[16,14] = 37807079247082910935750437397535265361259662238704761 5352035143989/38652465726311665474213406887371187527810692671413726806 640625000000-736694485119203754900544593199058955707390426548138628337 098463663/247375780648394659034965804079175600177988433097047851562500 00000000*7^(1/2), a[16,6] = -19039890149866495432584929203443766607664 78959434824857368133995942073946993/2610549880447098448451329990884194 1049181178531837828644206625000000000000000+28929035836992861158293213 150542073159989024626464512663622554495682276029/892277791168441852498 013180477996031954434813099925783737531127929687500000*7^(1/2), a[16,9 ] = 207904342034969170900628672444919837194509196335848531194132841067 48179318736252691355190285977916572557692108955233/1587766758359465966 8953564588772545305777807317441893201879580358667168054523455849316622 8541026757812500000000000000+16750291917128487119349998833576560400111 6436018919123161320355297232333887892701752121887704693497136971805720 9/73181019315333410819868281232202677952556148952818602411955267496737 5646957459693466445168584156250000000000000000*7^(1/2), a[16,13] = -46 330543544553540874179448395431083082899647522368296822645347/508745135 2083124397391250759096940921758056970520019531250000000+18909196831867 66311626712563992017565381198275789277564189007889/1139589102866619865 015640170037714766473804761396484375000000000000*7^(1/2), a[16,1] = 10 1385640411812595276352720516149878818017393925466135943597430958582531 8269273/19856466773923966037438917121094233830745906326832446413024679 687500000000000000-193175424371098513173126758518148648929759153027347 863824763410934840962491/277893272189669140325031078817733067056972748 113814414737187500000000000000000*7^(1/2), a[16,10] = -175203230698885 2430152559893574903103295734904546828540573516708228831279/24332628301 02477113639615580994907630153630681641719830799102783203125000+8188399 9666586172691735974039324906939964446697501450842550606413648603/73372 2330307823868112868698269233685400171713233503210517883300781250000*7^ (1/2), a[16,12] = -124817112626650602913170312644481143348777609285630 839757851632016681433/398360478373506965815942720780751835664786233698 3160384375000000000000000+19488568273447271249669733067688159410742449 986073234566291555857980323/290471182147348829240791567235964880172239 9620716887780273437500000000000*7^(1/2), a[16,11] = 330758723778621458 0372394397330733034806289881443017455732775116590947887/21612768292533 36041664932456066917395909700497105040226143341064453125000-1592494813 48355430623321425412532136521666854903188329515308104047100201/5308399 22974503589180860603244506027065540472973167774842224121093750000*7^(1 /2), c[15] = 1187/2500, a[16,8] = -28591353782492183701826251126306766 396461229874660236919089505718702483/307389544220068686894086053026978 69750592763266032965939343750000000000+3262171864570727799722782444400 69764006022537683484345700377402630409/1743722897253444403020583118152 408439159931363492594965600000000000000*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 "calculatio n for stage 18" }}{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_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_condi tion'):\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 explicitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 555 "interp_orde r_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]*ad d(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 specif y " }{XPPEDIT 18 0 "c[18] = 453/1000;" "6#/&%\"cG6#\"#=*&\"$`%\"\"\" \"%+5!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[18,17]=0" "6#/&%\" aG6$\"#=\"#<\"\"!" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "e15 := `union`(e14,\{c[18]=453/1000,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'\"\"(&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 "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$\"#=\"\"\"$\"ISXy5W\"GDZ2#pBr(elYG**>$!#T/&F%6 $F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F' \"\"'$!GEVV%R0(*4(Gk'G+,\\)pChm!#S/&F%6$F'\"\"($!G#yq-k2*\\h>qi5Jh5n)4 X)FD/&F%6$F'\"\")$!HGS*f98S55d6e[t]y'RBo#FD/&F%6$F'\"\"*$\"Hi$=k!=:bIm fd:l(**z2L&[*F+/&F%6$F'\"#5$!H0`PLs\"HW-UN\"fv;;SWy(GFD/&F%6$F'\"#6$\" HQ!4iBiiCqx%HkWR.i,Dn%FD/&F%6$F'\"#7$!G_Ngo$>!GMZLA(*o(G>%z7&)F+/&F%6$ F'\"#8$!H#=Y&\\)[C#znGtQ6GA[`?%H!#U/&F%6$F'\"#9$\"IgOgB@YNr!HG#3p5EU;2 I@FD/&F%6$F'\"#:$\"I*zWrSEa#o`L#3u.v\"[1&G^%F+/&F%6$F'\"#;$\"I*y#= " 0 " " {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_17) do\n tt := c onvert(SO7_17[ct],'interpolation_order_condition'):\n if expand(subs (e17,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\no p(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 15628 "e17 := \{c[18] = 453/1000, a[15,12] = 12163051029345238407733 418657/3016875774609375000000000000000, a[14,1] = 380/8289, c[2] = 1/1 6, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[7,1] \+ = 21400899/350000000, a[7,2] = 0, a[7,3] = 0, a[7,4] = 3064329829899/2 7126050000000, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6, 5] = 960400/5209191, a[5,3] = -352806597/250880000, a[5,4] = 178077159 /125440000, a[6,1] = 12089/252720, a[4,2] = 0, a[4,3] = 42/355, a[5,1] = 94495479/250880000, a[5,2] = 0, c[13] = 1, a[2,1] = 1/16, a[3,1] = \+ 18928/1134225, a[3,2] = 100352/1134225, a[4,1] = 14/355, c[7] = 39/250 , c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/20 , c[12] = 1, a[9,6] = 763205196415429092566184979837064563758937783434 6780/1734087257418811583049800347581865260479233950396659, a[9,5] = -8 75325048502130441118613421785266742862694404520560000/1702120304288944 18395571677575961339495435011888324169, a[9,7] = 751983479197113751704 8532179652347729899303513750000/10456773035023175965978907078123498326 37339039997351, a[9,4] = -14764960804048657303638372252908780219281424 435/2981692102565021975611711269209606363661854518, a[9,1] = -18409112 52282376584438157336464708426954728061551/2991923615171151921596253813 483118262195533733898, a[9,2] = 0, a[9,3] = 0, a[8,7] = 1993321838240/ 380523459069, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8 ,6] = 341936800488/47951126225, a[7,5] = -21643947/592609375, a[7,6] = 124391943/6756250000, a[8,1] = -15365458811/13609565775, a[8,2] = 0, \+ a[10,8] = 1908158550070998850625/117087067039189929394176, a[10,9] = - 52956818288156668227044990077324877908565/2912779959477433986349822224 412353951940608, a[11,1] = -10116106591826909534781157993685116703/956 2819945036894030442231411871744000, a[11,2] = 0, a[10,5] = -3378604805 394255292453489375/517042670569824692230499952, a[10,6] = 100158784418 3325981198091450220795/184232684207722503701669953872896, a[10,7] = 18 7023075231349900768014890274453125/25224698849808178010752575653374848 , a[10,1] = -63077736705254280154824845013881/783693578537866338551121 90394368, a[10,2] = 0, a[10,3] = 0, a[10,4] = -31948346510820970247215 /6956009216960026632192, a[12,10] = 7958341351371843889152/32844679884 43203581305, a[12,6] = 1827357820434213461438077550902273440/139381013 914245317709567680839641697, a[12,7] = 6435048028142415509419492271941 07500000/242124609118836550860494007545333945331, a[12,8] = 1622599381 51380266113750/59091082835244183497007, a[12,9] = -2302825163287352381 8545414856857015616678575554130463402/20013169183191444503443905240405 603349978424504151629055, a[12,3] = 0, a[12,4] = 262900926042842319967 45/5760876126062860430544, a[12,5] = -697069297560926452045586710000/4 1107967755245430594036502319, a[12,2] = 0, a[12,1] = -3218022174758599 831659045535578571/1453396753634469525663775847094384, a[11,8] = 39747 262782380466933662225/1756032802431424164410720256, a[11,9] = 48175771 419260955335244683805171548038966866545122229/198978642051381514652888 0165952064118903852843612160000, a[11,10] = -2378292068163246/47768728 487211875, a[11,7] = 2226455130519213549256016892506730559375/36488044 3159675255577435648380047355776, a[11,4] = -9623541317323077848129/386 4449564977792573440, a[11,5] = -4823348333146829406881375/576413233634 141239944816, a[13,9] = 4059320304637772479267050305961754374024596379 09765779/78803919436321841083201886041201537229769115088303952, a[13,1 0] = -10290327637248/1082076946951, a[13,11] = 863264105888000/8581466 2253313, a[13,6] = 14327219974204125/40489566827933216, a[13,7] = 2720 762324010009765625000/10917367480696813922225349, a[13,8] = -498533005 859375/95352091037424, a[13,5] = 0, a[13,2] = 0, a[13,3] = 0, a[13,4] \+ = 0, a[13,1] = 4631674879841/103782082379976, a[12,11] = -507974327957 860843878400/121555654819179042718967, a[17,1] = 142969271097328003658 03280238811548992426955660898056576097/3414135267376060159135042016078 36239464660351562500000000000-3549810993115886985857087725788847209921 5653607797737/10471522719224819528692927297504485322802734375000000000 *7^(1/2), a[17,9] = -7773654204459363180897324177386410045148764290716 888867400388432148819282908940948159254319/788452752376981215609864480 482093541291983982324119451546116914378375650249609375000000000000-349 2115351898847954914967508524476952765587305684964600792344070772438794 27767491465619/3504234455008805402710508802142637961297706588107197562 4271862861261140011093750000000000*7^(1/2), a[17,14] = 889883133955067 5631062765096204752299374508560508163/17497825657641849011078532130396 9544268798828125000000+34300040431030913870067655667347975265785974516 9/6999130263056739604431412852158781770751953125000*7^(1/2), a[17,6] = 11680941707325157677955382237279696768268247094386992999/875033614756 65632084979209131119394314606250000000000000-9971724335702763059252544 57325512017866796579074407189/2187584036891640802124480228277984857865 1562500000000000*7^(1/2), a[17,7] = 7292471180619236115859359700346738 458665766464861503/447410046832592216392357932737621273394042728766300 00-395688161558937580577353827374940476509802536449201/717003280180436 2442185223281051622971058377063562500*7^(1/2), a[17,16] = 312951003846 638093776403168468573978963277/581518906942836754644893134349507666605 0000+1835267423444683727710449934904130527/447322236109874426649917795 65346743585*7^(1/2), a[17,11] = -2591874713003263601953285685526560844 53334237831807/482960797022440548690819027452576858488464355468750+195 3279068514198003642387066372792263574848990934/12709494658485277597126 816511909917328643798828125*7^(1/2), a[17,15] = -771271272509121593884 65163390223095790721/1192299437549281524502362666335509439650000+19577 936100246207024301582455131137389/926324947634441799805681763845280410 805*7^(1/2), a[17,8] = 15899431653868076081892800671463011693424374363 9/488458911821805651705273894751044258600000000000-6476404035945812133 836473370020724114834745161/678415155308063405146213742709783692500000 00000*7^(1/2), a[17,10] = 51365239468985562750202655941556028606492949 5444927/2283703904801922343978476111178519401359558105468750-210846684 210425267635251476041683325933811084598/415218891782167698905177474759 7308002471923828125*7^(1/2), a[17,12] = 262214308744442251330599379201 914827178216725701161/222544946040040751899013362492653039062500000000 00000-4042541431743211698640695349601479330547680120287/11127247302002 03759495066812463265195312500000000000*7^(1/2), a[17,13] = 55038056541 54447779082385289578389472169301/1734849960808438372162663307568359375 000000000-44976428168893445098944962904416522821649/526501941230926308 85946212158775329589843750*7^(1/2), a[14,4] = 0, a[18,3] = 0, a[18,4] \+ = 0, a[18,5] = 0, a[18,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[17,5] = 0, \+ a[17,4] = 0, a[18,17] = 0, a[17,3] = 0, a[17,2] = 0, a[16,15] = 460888 2876307429368775251443876403372924872509408621400283/75922278949360871 648575860105318720457058558153486785997500-849147029864287419123314290 25972417475312941024930181294/5314559526455261015400310207372310431994 099070744075019825*7^(1/2), a[16,7] = 14716499751079132511176933345920 457044509185685353579684857180786483042792373/160889883376746714657021 565950525327016567992369670063678825639487979375000000+350971611315420 287919452885739900022880241691060058741934303404427219789/748680368989 60959337075562503093457016256710599371634570393697387568750000*7^(1/2) , a[14,5] = 0, a[14,2] = 0, a[14,3] = 0, a[14,13] = 3/392-3/392*7^(1/2 ), a[14,12] = 1100613127343/48439922837376-29746300739/1424703612864*7 ^(1/2), a[15,1] = 130946139152859396534950567713097/281526916178320312 5000000000000000+90670944595916412828478989403/66148981703437500000000 000000000*7^(1/2), a[15,8] = 16161913072172934315785836819/99325094830 65000000000000000-75980668436324671626317237501/4991722714566000000000 0000000*7^(1/2), a[15,6] = 3353943190402140976803568136084793/19770296 302701765625000000000000000-255748988917794999289135717669191/86495046 32432022460937500000000000*7^(1/2), a[15,7] = 105966145406009687447923 62758669/45489031169570058009272287500000-1305842593294600719246709494 7/1451071902055187778276093750000*7^(1/2), a[15,10] = 1446205909971003 3235500874587/10319489926824569702148437500-34815528884845218754039633 7457/252574928278923034667968750000*7^(1/2), a[15,9] = -37278501430783 86229058076561522609367827766330696417108435354320431359/8016349226513 859159668160608032382937598583484731440429687500000000000+814251349796 512127448124174135016152994969999173763717229208919303/158984211826401 9833619870377392538080518560957031250000000000000000*7^(1/2), a[15,11] = -207941800836756311792092915907/81839239362061500549316406250+10891 006522099224595926000559657/4582997404275444030761718750000*7^(1/2), a [15,14] = 26094995957293704259/549316406250000000000*7^(1/2), a[15,13] = -579945478542537523/549316406250000000000+3727856565327672037/21972 65625000000000000*7^(1/2), a[18,10] = 78813744026468418092863662132343 791904150881627/607405921692771033827002477362806199218750000000-17245 957150753093018755976031205721533364483773/287816959817497659105718096 965760475937500000000*7^(1/2), a[18,14] = 1720418743754626973815631730 013574951008671541792083/286684375574804054197510670424423701330000000 00000000+66312564220445944370265939039199371363080688129/1146737502299 216216790042681697694805320000000000*7^(1/2), a[18,9] = 12403847049624 5645770949475501124572057558878067871133108876781024786308880699933251 5507/30553476572716320332431456121614523605789653657517438727841957249 705077232000000000000000-447108394135293641224551688365209147458637766 147319719227389653577377825685260302229/380221041793803097470258120624 53629376093791218243923750203324577410762777600000000000*7^(1/2), a[18 ,7] = 428930755082588455942724722283447907984915278705435/261798793118 0425311918711560704709622602855738609664-76498733805890742729378834290 483857132585978507041/117473817424762674252762698236749790757820449809 4080*7^(1/2), a[18,6] = 6945489182798665490581179457354700952470236849 3567653/512019669434752041457249772287235770160896000000000000-1927842 07275057942900067020977105865032917681968026949/3584137686043264290200 748406010650391126272000000000000*7^(1/2), a[18,1] = 77336247783224311 32379739805371154872928342814315555219/1816142604567836676859367804916 48991798344000000000000000-7549168255858056864842589434727287526200394 3330267987/18872197105495738747391541292654483628168000000000000000*7^ (1/2), a[18,16] = 1501373575980631550465262783336687863485/29773768035 473241837818528478694792530176+17058357454582951976373885014055586875/ 352352284443470317607319863653192811008*7^(1/2), a[18,8] = 54262248367 54120276540228725914955825217338781/2000727702822115949384801872900277 2832256000000-96314521713121628743334127199055813316326877/85501183881 2870063839658920042853539840000000*7^(1/2), a[18,15] = -25461882459643 0425581156792109862619835/12209146240504642810904193703275616662016+18 1972080991961001018895701030477170625/72965903567512954077001394321351 31851264*7^(1/2), a[18,13] = 84303053727623559901892597740221920795377 /35529727197356817861891344539000000000000000-668871667581022998424770 230088072148293/663554446548268972642571338468750000000000*7^(1/2), a[ 18,11] = -611946677966621853095211488458472723282243226327/14130053032 88511776741139097461253437406250000000+1888145641282084603097985496601 71819794928530747/1041161802423113940756628808655660427562500000000*7^ (1/2), a[18,12] = 9047237292228637467633523053986950065115624317/86238 7983897830576895325196565664000000000000000-51758147301442652275148489 19003486791055011817/1207343177456962807653455275191929600000000000000 *7^(1/2), a[13,12] = -29746300739/247142463456, a[11,6] = 656611924651 4996884067001154977284529/970305487021846325473990863582315520, a[11,3 ] = 0, a[8,3] = 0, a[9,8] = 1366042683489166351293315549358278750/1446 31418224267718165055326464180836641, a[15,4] = 0, a[15,5] = 0, a[15,2] = 0, a[15,3] = 0, a[16,2] = 0, a[16,4] = 0, a[16,3] = 0, c[17] = 993/ 2500, a[16,5] = 0, c[16] = 277/2500, a[14,8] = 12730301828542984375/44 62520239258570944-23139795371828125/21803844817224288*7^(1/2), a[14,11 ] = -315051553138064800/67942629698767761+392422111155400/221311497390 123*7^(1/2), a[14,9] = -2431300645813356820990854267156732219942356459 331979/2189818416332779722313164167138490920989148784630720+4933990286 8206221533385089432866304648514986421/12587574725039929425061300060194 4295132543520864*7^(1/2), a[14,10] = 21310748482922848/739894705135631 5-5227019004392/4820160945509*7^(1/2), a[14,6] = 9785567316447209725/4 9721188064701989248-34388281046170925/566853935591065024*7^(1/2), a[14 ,7] = 1016182669829177731250000000/4211024590054414662359895513-232084 93423698876953125/13716692475747279030488259*7^(1/2), a[16,14] = 37807 0792470829109357504373975352653612596622387047615352035143989/38652465 726311665474213406887371187527810692671413726806640625000000-736694485 119203754900544593199058955707390426548138628337098463663/247375780648 39465903496580407917560017798843309704785156250000000000*7^(1/2), a[16 ,6] = -190398901498664954325849292034437666076647895943482485736813399 5942073946993/26105498804470984484513299908841941049181178531837828644 206625000000000000000+289290358369928611582932131505420731599890246264 64512663622554495682276029/8922777911684418524980131804779960319544348 13099925783737531127929687500000*7^(1/2), a[16,9] = 207904342034969170 9006286724449198371945091963358485311941328410674817931873625269135519 0285977916572557692108955233/15877667583594659668953564588772545305777 8073174418932018795803586671680545234558493166228541026757812500000000 000000+167502919171284871193499988335765604001116436018919123161320355 2972323338878927017521218877046934971369718057209/73181019315333410819 8682812322026779525561489528186024119552674967375646957459693466445168 584156250000000000000000*7^(1/2), a[16,13] = -463305435445535408741794 48395431083082899647522368296822645347/5087451352083124397391250759096 940921758056970520019531250000000+189091968318676631162671256399201756 5381198275789277564189007889/11395891028666198650156401700377147664738 04761396484375000000000000*7^(1/2), a[16,1] = 101385640411812595276352 7205161498788180173939254661359435974309585825318269273/19856466773923 966037438917121094233830745906326832446413024679687500000000000000-193 1754243710985131731267585181486489297591530273478638247634109348409624 91/2778932721896691403250310788177330670569727481138144147371875000000 00000000000*7^(1/2), a[16,10] = -1752032306988852430152559893574903103 295734904546828540573516708228831279/243326283010247711363961558099490 7630153630681641719830799102783203125000+81883999666586172691735974039 324906939964446697501450842550606413648603/733722330307823868112868698 269233685400171713233503210517883300781250000*7^(1/2), a[16,12] = -124 817112626650602913170312644481143348777609285630839757851632016681433/ 3983604783735069658159427207807518356647862336983160384375000000000000 000+194885682734472712496697330676881594107424499860732345662915558579 80323/2904711821473488292407915672359648801722399620716887780273437500 000000000*7^(1/2), a[16,11] = 3307587237786214580372394397330733034806 289881443017455732775116590947887/216127682925333604166493245606691739 5909700497105040226143341064453125000-15924948134835543062332142541253 2136521666854903188329515308104047100201/53083992297450358918086060324 4506027065540472973167774842224121093750000*7^(1/2), c[15] = 1187/2500 , a[16,8] = -285913537824921837018262511263067663964612298746602369190 89505718702483/3073895442200686868940860530269786975059276326603296593 9343750000000000+32621718645707277997227824444006976400602253768348434 5700377402630409/17437228972534444030205831181524084391599313634925949 65600000000000000*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 19" }} {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_18 := SimpleOrderCon ditions(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_orde r_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 0 558 "interp_order_eqns19 := [add(a[19,i ],i=1..18)=c[19],seq(op(StageOrderConditions(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]^6/36 0, #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 specify " }{XPPEDIT 18 0 "c[19 ] = 1779/2500;" "6#/&%\"cG6#\"#>*&\"%z<\"\"\"\"%+D!\"\"" }{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 0 170 "e18 : = `union`(e17,\{c[19]=1779/2500,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$\"#>\"\"\"&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 "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$\"#>\"\"\"$\"IN!\\jL'z`dz)eSP.)=@(=#Hi!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$ F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\"\"'$\"IhyiF!e(>hZN_Z` \\5pQ]+U!#S/&F%6$F'\"\"($\"I!H:P>x1x*)HMXr(GWk9U=_FD/&F%6$F'\"\")$\"I3 31qR?z[RpK&4E[LF`FK\"!#R/&F%6$F'\"\"*$!I*\\,*>FWFR#G7$ew@) *piS^AFQ/&F%6$F'\"#7$\"I%)f?H1Z..*>0Z]z8'3;Z]WF+/&F%6$F'\"#8$\"IV`*H,i aQ'>v3s'\\&*>\"p$yQ\"F+/&F%6$F'\"#9$!IOlYbR3SA1W_8%4ORh\\pY\"FD/&F%6$F '\"#:$\"I^SSrM(z8>$HimEns?Ti88F+/&F%6$F'\"#;$!IvSg=H))eBZ/EyIy&yrB-2#F D/&F%6$F'\"# " 0 " " {MPLTEXT 1 0 205 "recd := []:\nfor ct to nops(SO7_18) do\n tt := c onvert(SO7_18[ct],'interpolation_order_condition'):\n if expand(subs (e20,lhs(tt)=rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\no p(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 18510 "e20 := \{a[19,3] = 0, a[19,4] = 0, c[18] = 453/1000, a[15,12] = 12163051029345238407733418657/3016875774609375000000000000000, a[14 ,1] = 380/8289, c[2] = 1/16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39 /100, c[6] = 7/15, a[7,1] = 21400899/350000000, a[7,2] = 0, a[7,3] = 0 , a[7,4] = 3064329829899/27126050000000, a[6,2] = 0, a[6,3] = 0, a[6,4 ] = 2505377/10685520, a[6,5] = 960400/5209191, a[5,3] = -352806597/250 880000, a[5,4] = 178077159/125440000, a[6,1] = 12089/252720, a[4,2] = \+ 0, a[4,3] = 42/355, a[5,1] = 94495479/250880000, a[5,2] = 0, c[13] = 1 , a[2,1] = 1/16, a[3,1] = 18928/1134225, a[3,2] = 100352/1134225, a[4, 1] = 14/355, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[ 10] = 11/12, c[11] = 19/20, c[12] = 1, a[9,6] = 7632051964154290925661 849798370645637589377834346780/173408725741881158304980034758186526047 9233950396659, a[9,5] = -875325048502130441118613421785266742862694404 520560000/170212030428894418395571677575961339495435011888324169, a[9, 7] = 7519834791971137517048532179652347729899303513750000/104567730350 2317596597890707812349832637339039997351, a[9,4] = -147649608040486573 03638372252908780219281424435/2981692102565021975611711269209606363661 854518, a[9,1] = -1840911252282376584438157336464708426954728061551/29 91923615171151921596253813483118262195533733898, a[9,2] = 0, a[9,3] = \+ 0, a[8,7] = 1993321838240/380523459069, a[8,4] = -7/5, a[8,5] = -83391 28164608/939060038475, a[8,6] = 341936800488/47951126225, a[19,5] = 0, a[7,5] = -21643947/592609375, a[7,6] = 124391943/6756250000, a[8,1] = -15365458811/13609565775, a[8,2] = 0, a[10,8] = 190815855007099885062 5/117087067039189929394176, a[10,9] = -5295681828815666822704499007732 4877908565/2912779959477433986349822224412353951940608, a[11,1] = -101 16106591826909534781157993685116703/9562819945036894030442231411871744 000, a[11,2] = 0, a[10,5] = -3378604805394255292453489375/517042670569 824692230499952, a[10,6] = 1001587844183325981198091450220795/18423268 4207722503701669953872896, a[10,7] = 187023075231349900768014890274453 125/25224698849808178010752575653374848, a[10,1] = -630777367052542801 54824845013881/78369357853786633855112190394368, a[10,2] = 0, a[10,3] \+ = 0, a[10,4] = -31948346510820970247215/6956009216960026632192, a[12,1 0] = 7958341351371843889152/3284467988443203581305, a[12,6] = 18273578 20434213461438077550902273440/139381013914245317709567680839641697, a[ 12,7] = 643504802814241550941949227194107500000/2421246091188365508604 94007545333945331, a[12,8] = 162259938151380266113750/5909108283524418 3497007, a[12,9] = -23028251632873523818545414856857015616678575554130 463402/20013169183191444503443905240405603349978424504151629055, a[12, 3] = 0, a[12,4] = 26290092604284231996745/5760876126062860430544, a[12 ,5] = -697069297560926452045586710000/41107967755245430594036502319, a [12,2] = 0, a[12,1] = -3218022174758599831659045535578571/145339675363 4469525663775847094384, a[11,8] = 39747262782380466933662225/175603280 2431424164410720256, a[11,9] = 481757714192609553352446838051715480389 66866545122229/1989786420513815146528880165952064118903852843612160000 , a[11,10] = -2378292068163246/47768728487211875, a[11,7] = 2226455130 519213549256016892506730559375/364880443159675255577435648380047355776 , a[11,4] = -9623541317323077848129/3864449564977792573440, a[11,5] = \+ -4823348333146829406881375/576413233634141239944816, a[13,9] = 4059320 30463777247926705030596175437402459637909765779/7880391943632184108320 1886041201537229769115088303952, a[13,10] = -10290327637248/1082076946 951, a[13,11] = 863264105888000/85814662253313, a[13,6] = 143272199742 04125/40489566827933216, a[13,7] = 2720762324010009765625000/109173674 80696813922225349, a[13,8] = -498533005859375/95352091037424, a[13,5] \+ = 0, a[13,2] = 0, a[13,3] = 0, a[13,4] = 0, a[13,1] = 4631674879841/10 3782082379976, a[12,11] = -507974327957860843878400/121555654819179042 718967, a[19,2] = 0, a[17,1] = 142969271097328003658032802388115489924 26955660898056576097/3414135267376060159135042016078362394646603515625 00000000000-35498109931158869858570877257888472099215653607797737/1047 1522719224819528692927297504485322802734375000000000*7^(1/2), a[17,9] \+ = -7773654204459363180897324177386410045148764290716888867400388432148 819282908940948159254319/788452752376981215609864480482093541291983982 324119451546116914378375650249609375000000000000-349211535189884795491 496750852447695276558730568496460079234407077243879427767491465619/350 4234455008805402710508802142637961297706588107197562427186286126114001 1093750000000000*7^(1/2), a[17,14] = 889883133955067563106276509620475 2299374508560508163/17497825657641849011078532130396954426879882812500 0000+343000404310309138700676556673479752657859745169/6999130263056739 604431412852158781770751953125000*7^(1/2), a[17,6] = 11680941707325157 677955382237279696768268247094386992999/875033614756656320849792091311 19394314606250000000000000-9971724335702763059252544573255120178667965 79074407189/21875840368916408021244802282779848578651562500000000000*7 ^(1/2), a[17,7] = 7292471180619236115859359700346738458665766464861503 /44741004683259221639235793273762127339404272876630000-395688161558937 580577353827374940476509802536449201/717003280180436244218522328105162 2971058377063562500*7^(1/2), a[17,16] = 312951003846638093776403168468 573978963277/5815189069428367546448931343495076666050000+1835267423444 683727710449934904130527/44732223610987442664991779565346743585*7^(1/2 ), a[17,11] = -259187471300326360195328568552656084453334237831807/482 960797022440548690819027452576858488464355468750+195327906851419800364 2387066372792263574848990934/12709494658485277597126816511909917328643 798828125*7^(1/2), a[17,15] = -771271272509121593884651633902230957907 21/1192299437549281524502362666335509439650000+19577936100246207024301 582455131137389/926324947634441799805681763845280410805*7^(1/2), a[17, 8] = 158994316538680760818928006714630116934243743639/4884589118218056 51705273894751044258600000000000-6476404035945812133836473370020724114 834745161/67841515530806340514621374270978369250000000000*7^(1/2), a[1 7,10] = 513652394689855627502026559415560286064929495444927/2283703904 801922343978476111178519401359558105468750-210846684210425267635251476 041683325933811084598/415218891782167698905177474759730800247192382812 5*7^(1/2), a[17,12] = 262214308744442251330599379201914827178216725701 161/22254494604004075189901336249265303906250000000000000-404254143174 3211698640695349601479330547680120287/11127247302002037594950668124632 65195312500000000000*7^(1/2), a[17,13] = 55038056541544477790823852895 78389472169301/1734849960808438372162663307568359375000000000-44976428 168893445098944962904416522821649/526501941230926308859462121587753295 89843750*7^(1/2), a[14,4] = 0, a[18,3] = 0, a[18,4] = 0, a[18,5] = 0, \+ a[18,2] = 0, c[14] = 1/2-1/14*7^(1/2), a[17,5] = 0, a[17,4] = 0, a[19, 17] = 0, a[19,18] = 0, a[18,17] = 0, a[17,3] = 0, a[17,2] = 0, a[16,15 ] = 4608882876307429368775251443876403372924872509408621400283/7592227 8949360871648575860105318720457058558153486785997500-84914702986428741 912331429025972417475312941024930181294/531455952645526101540031020737 2310431994099070744075019825*7^(1/2), a[16,7] = 1471649975107913251117 6933345920457044509185685353579684857180786483042792373/16088988337674 6714657021565950525327016567992369670063678825639487979375000000+35097 1611315420287919452885739900022880241691060058741934303404427219789/74 8680368989609593370755625030934570162567105993716345703936973875687500 00*7^(1/2), a[14,5] = 0, a[14,2] = 0, a[14,3] = 0, a[14,13] = 3/392-3/ 392*7^(1/2), a[14,12] = 1100613127343/48439922837376-29746300739/14247 03612864*7^(1/2), a[15,1] = 130946139152859396534950567713097/28152691 61783203125000000000000000+90670944595916412828478989403/6614898170343 7500000000000000000*7^(1/2), a[15,8] = 16161913072172934315785836819/9 932509483065000000000000000-75980668436324671626317237501/499172271456 60000000000000000*7^(1/2), a[15,6] = 335394319040214097680356813608479 3/19770296302701765625000000000000000-25574898891779499928913571766919 1/8649504632432022460937500000000000*7^(1/2), a[15,7] = 10596614540600 968744792362758669/45489031169570058009272287500000-130584259329460071 92467094947/1451071902055187778276093750000*7^(1/2), a[15,10] = 144620 59099710033235500874587/10319489926824569702148437500-3481552888484521 87540396337457/252574928278923034667968750000*7^(1/2), a[15,9] = -3727 850143078386229058076561522609367827766330696417108435354320431359/801 6349226513859159668160608032382937598583484731440429687500000000000+81 4251349796512127448124174135016152994969999173763717229208919303/15898 42118264019833619870377392538080518560957031250000000000000000*7^(1/2) , a[15,11] = -207941800836756311792092915907/8183923936206150054931640 6250+10891006522099224595926000559657/4582997404275444030761718750000* 7^(1/2), a[15,14] = 26094995957293704259/549316406250000000000*7^(1/2) , a[15,13] = -579945478542537523/549316406250000000000+372785656532767 2037/2197265625000000000000*7^(1/2), a[18,10] = 7881374402646841809286 3662132343791904150881627/60740592169277103382700247736280619921875000 0000-17245957150753093018755976031205721533364483773/28781695981749765 9105718096965760475937500000000*7^(1/2), a[18,14] = 172041874375462697 3815631730013574951008671541792083/28668437557480405419751067042442370 133000000000000000+66312564220445944370265939039199371363080688129/114 6737502299216216790042681697694805320000000000*7^(1/2), a[18,9] = 1240 3847049624564577094947550112457205755887806787113310887678102478630888 06999332515507/3055347657271632033243145612161452360578965365751743872 7841957249705077232000000000000000-44710839413529364122455168836520914 7458637766147319719227389653577377825685260302229/38022104179380309747 025812062453629376093791218243923750203324577410762777600000000000*7^( 1/2), a[18,7] = 428930755082588455942724722283447907984915278705435/26 17987931180425311918711560704709622602855738609664-7649873380589074272 9378834290483857132585978507041/11747381742476267425276269823674979075 78204498094080*7^(1/2), a[18,6] = 694548918279866549058117945735470095 24702368493567653/5120196694347520414572497722872357701608960000000000 00-192784207275057942900067020977105865032917681968026949/358413768604 3264290200748406010650391126272000000000000*7^(1/2), a[18,1] = 7733624 778322431132379739805371154872928342814315555219/181614260456783667685 936780491648991798344000000000000000-754916825585805686484258943472728 75262003943330267987/1887219710549573874739154129265448362816800000000 0000000*7^(1/2), a[18,16] = 1501373575980631550465262783336687863485/2 9773768035473241837818528478694792530176+17058357454582951976373885014 055586875/352352284443470317607319863653192811008*7^(1/2), a[18,8] = 5 426224836754120276540228725914955825217338781/200072770282211594938480 18729002772832256000000-96314521713121628743334127199055813316326877/8 55011838812870063839658920042853539840000000*7^(1/2), a[18,15] = -2546 18824596430425581156792109862619835/1220914624050464281090419370327561 6662016+181972080991961001018895701030477170625/7296590356751295407700 139432135131851264*7^(1/2), a[18,13] = 8430305372762355990189259774022 1920795377/35529727197356817861891344539000000000000000-66887166758102 2998424770230088072148293/663554446548268972642571338468750000000000*7 ^(1/2), a[18,11] = -611946677966621853095211488458472723282243226327/1 413005303288511776741139097461253437406250000000+188814564128208460309 798549660171819794928530747/104116180242311394075662880865566042756250 0000000*7^(1/2), a[18,12] = 904723729222863746763352305398695006511562 4317/862387983897830576895325196565664000000000000000-5175814730144265 227514848919003486791055011817/120734317745696280765345527519192960000 0000000000*7^(1/2), a[13,12] = -29746300739/247142463456, a[11,6] = 65 66119246514996884067001154977284529/9703054870218463254739908635823155 20, a[11,3] = 0, a[8,3] = 0, a[9,8] = 13660426834891663512933155493582 78750/144631418224267718165055326464180836641, a[19,16] = -59220160570 843125612523224803799201068911/498444777379574361124194115156720857090 000-639182399323564197784537438328437546/19170952976137475427853619813 720032965*7^(1/2), a[19,8] = 54585293581684876354958260975120352371260 4801307/488458911821805651705273894751044258600000000000+7894550931548 752199808312441931046629102173173/101762273296209510771932061406467553 875000000000*7^(1/2), a[19,12] = 8165955601839090547398038247963403997 73958361526093/22254494604004075189901336249265303906250000000000000+3 3522055523510853377725329217930880599252333953/11354333981634732239745 579719012910156250000000000*7^(1/2), a[19,13] = 6329822636137785234855 809795487253116923767139/525659538124956826765286982193212890625000000 000+54824977090333610968656925721307374512157/789752911846389463289193 18238162994384765625*7^(1/2), a[19,15] = 20154542909143405317604887167 0250775086069/3440635519785069542135389408568184382990000-227285172186 1698000685159054540758474/132332135376348828543668823406468630115*7^(1 /2), a[19,7] = 18039033114928576144292936355618923845713121206510289/4 4741004683259221639235793273762127339404272876630000+32811736739714398 20995171767975524126665743950319/7316360001841186165495125796991452011 2840582281250*7^(1/2), a[19,1] = 1877895419043249864139736722550177731 0616443565686827961261/34141352673760601591350420160783623946466035156 2500000000000+3237979363730976751083677681250337532341535825485533/117 5374999096663416485940819107646311743164062500000000*7^(1/2), a[19,6] \+ = 28179885214710124253594866402805577138221840971048257987/87503361475 665632084979209131119394314606250000000000000+826887497606742913768075 6848115769916895601168366091/22322286090731028593106941104877396508828 1250000000000*7^(1/2), a[19,14] = -36158067991345032910431367229095843 52046302608498253/8748912828820924505539266065198477213439941406250000 0-139369221270545100600958791779240838739971407839/3499565131528369802 215706426079390885375976562500*7^(1/2), a[19,10] = 2498001978259689024 640690501707586024679437753648551/228370390480192234397847611117851940 1359558105468750+38464989471746279636532325535691058627774079964/93212 4042776294834276929024970824245452880859375*7^(1/2), a[19,11] = -92775 3364230868120383187286221818715585159819007991/48296079702244054869081 9027452576858488464355468750-22676106773548404552804801717321237135105 3066444/1815642094069325371018116644558559618377685546875*7^(1/2), a[1 9,9] = -24140640626611869758219261752350837961521246348971485476648481 6260099015472852370732495500147/78845275237698121560986448048209354129 1983982324119451546116914378375650249609375000000000000+86873235585153 9238451654312133160482449873052828422043968217269298030501500884642378 3/10727248331659608375644414700436646820299101800328155803348529447324 83877890625000000000*7^(1/2), a[15,4] = 0, a[15,5] = 0, a[15,2] = 0, a [15,3] = 0, a[16,2] = 0, a[16,4] = 0, a[16,3] = 0, c[17] = 993/2500, a [16,5] = 0, c[16] = 277/2500, a[14,8] = 12730301828542984375/446252023 9258570944-23139795371828125/21803844817224288*7^(1/2), a[14,11] = -31 5051553138064800/67942629698767761+392422111155400/221311497390123*7^( 1/2), a[14,9] = -2431300645813356820990854267156732219942356459331979/ 2189818416332779722313164167138490920989148784630720+49339902868206221 533385089432866304648514986421/125875747250399294250613000601944295132 543520864*7^(1/2), a[14,10] = 21310748482922848/7398947051356315-52270 19004392/4820160945509*7^(1/2), a[14,6] = 9785567316447209725/49721188 064701989248-34388281046170925/566853935591065024*7^(1/2), a[14,7] = 1 016182669829177731250000000/4211024590054414662359895513-2320849342369 8876953125/13716692475747279030488259*7^(1/2), a[16,14] = 378070792470 829109357504373975352653612596622387047615352035143989/386524657263116 65474213406887371187527810692671413726806640625000000-7366944851192037 54900544593199058955707390426548138628337098463663/2473757806483946590 3496580407917560017798843309704785156250000000000*7^(1/2), a[16,6] = - 1903989014986649543258492920344376660766478959434824857368133995942073 946993/261054988044709844845132999088419410491811785318378286442066250 00000000000000+2892903583699286115829321315054207315998902462646451266 3622554495682276029/89227779116844185249801318047799603195443481309992 5783737531127929687500000*7^(1/2), a[16,9] = 2079043420349691709006286 7244491983719450919633584853119413284106748179318736252691355190285977 916572557692108955233/158776675835946596689535645887725453057778073174 418932018795803586671680545234558493166228541026757812500000000000000+ 1675029191712848711934999883357656040011164360189191231613203552972323 338878927017521218877046934971369718057209/731810193153334108198682812 3220267795255614895281860241195526749673756469574596934664451685841562 50000000000000000*7^(1/2), a[16,13] = -4633054354455354087417944839543 1083082899647522368296822645347/50874513520831243973912507590969409217 58056970520019531250000000+1890919683186766311626712563992017565381198 275789277564189007889/113958910286661986501564017003771476647380476139 6484375000000000000*7^(1/2), a[16,1] = 1013856404118125952763527205161 498788180173939254661359435974309585825318269273/198564667739239660374 38917121094233830745906326832446413024679687500000000000000-1931754243 71098513173126758518148648929759153027347863824763410934840962491/2778 9327218966914032503107881773306705697274811381441473718750000000000000 0000*7^(1/2), a[16,10] = -17520323069888524301525598935749031032957349 04546828540573516708228831279/2433262830102477113639615580994907630153 630681641719830799102783203125000+818839996665861726917359740393249069 39964446697501450842550606413648603/7337223303078238681128686982692336 85400171713233503210517883300781250000*7^(1/2), a[16,12] = -1248171126 26650602913170312644481143348777609285630839757851632016681433/3983604 783735069658159427207807518356647862336983160384375000000000000000+194 88568273447271249669733067688159410742449986073234566291555857980323/2 9047118214734882924079156723596488017223996207168877802734375000000000 00*7^(1/2), a[16,11] = 33075872377862145803723943973307330348062898814 43017455732775116590947887/2161276829253336041664932456066917395909700 497105040226143341064453125000-159249481348355430623321425412532136521 666854903188329515308104047100201/530839922974503589180860603244506027 065540472973167774842224121093750000*7^(1/2), c[15] = 1187/2500, c[19] = 1779/2500, a[16,8] = -285913537824921837018262511263067663964612298 74660236919089505718702483/3073895442200686868940860530269786975059276 3266032965939343750000000000+32621718645707277997227824444006976400602 2537683484345700377402630409/17437228972534444030205831181524084391599 31363492594965600000000000000*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 "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_19 := SimpleOr derConditions(7,19,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "whch := [1,2,4,8,16,21,27,3 1,32,61,63,64]:\ninterp_order_eqns20 := []:\nfor ct in whch do\n tem p_eqn := convert(SO7_19[ct],'interpolation_order_condition'):\n inte rp_order_eqns20 := [op(interp_order_eqns20),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 0 558 "interp_order_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]*add(a[k,l]*c[l]^2, l=2..k-1),k=2..j-1),\n j=2..i-1),i=2..19)=c[2 0]^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] = 9029/10000;" "6#/&%\"cG6#\"#?*&\"%H!*\"\"\"\"&++\"!\"\" " }{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 0 173 "e21 := `union`(e20,\{c[20]=9029/10000,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#<.&%\"a G6$\"#?\"\"'&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[40](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "65/&%\"aG6$\"#?\"\"\"$\"I3s&=DxLN,v6V_y+(34`$z'!#T/&F%6$F '\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F%6$F'\"\"&F0/&F%6$F'\" \"'$\"IPm6&pwH2<*[_5x'GM=W!Rd!#S/&F%6$F'\"\"($\"I&pMR9g(z!R*e)QYn!\\7m *o\"fFD/&F%6$F'\"\")$!I'ee7K#o'fo'=Ga$3&Q(*Q^LA!#R/&F%6$F'\"\"*$\"II25 '4U[['\\&[[`a\"=T_ZXIFQ/&F%6$F'\"#5$!I5$>07!=V4(>%pb2%z<+w/1&FQ/&F%6$F '\"#6$\"IwRFT'Rss;n%zMGpOx`-TXFQ/&F%6$F'\"#7$!I)fm\\\"[Ym(R'=2loA\"G2W e(RF+/&F%6$F'\"#8$!I(yuBpm4?zQy?2NvD6uL!>F+/&F%6$F'\"#9$!IJ9)pNwATHu_e 9!**3;zjM=FD/&F%6$F'\"#:$!I`7NY.\"3QwJ)))H4'HHhIw<\"FD/&F%6$F'\"#;$!IM $QhmP/K&[u0a$)42L2@JEFD/&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) simpl e 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],'interpolatio n_order_condition'):\n if expand(subs(e23,lhs(tt)=rhs(tt))) then rec d := [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 21703 "e23 := \{a[19,3] = 0 , a[19,4] = 0, c[18] = 453/1000, a[15,12] = 12163051029345238407733418 657/3016875774609375000000000000000, a[20,6] = 64705474043359104750113 9839926183281833250508652625223013947/14336550744173057160802993624042 60156450508800000000000000000+6641827613172956904051390551401057910139 7635988537328052807/14336550744173057160802993624042601564505088000000 00000000000*7^(1/2), a[20,1] = 897330932052009060320705191604115082179 14220092982292430255743/1525559787836982808561868956129851531106089600 000000000000000000+234076470257947541762894493668275072391709451556335 985408969/679399095797846594906095486535561410614048000000000000000000 00*7^(1/2), a[20,16] = -5895996755822171065621140032186175996965052639 37/3858680337397332142181281290838845111910809600000-99015163742825465 63618749633979325112870981/2374572515321435164419250025131596991945113 60*7^(1/2), a[20,7] = 584912615672238396354166551337437211351694198276 3467945837/13194659173149343572070306265951736497918392922592706560000 +711597078575774626289414881120414051622031127652898916601/12687172281 874368819298371409568977401844608579416064000000*7^(1/2), a[20,12] = - 54175810754967243498377218440527251354798928847932480401/1093852918776 008303734030479323888217600000000000000000000+807779950733732671058695 914872912518198545978092589343/218770583755201660746806095864777643520 000000000000000000*7^(1/2), a[20,10] = -883876388558690332747751484238 8396732758843333309547477/17007365807397588947156069366158573578125000 00000000000+77240806625230137147359751322705182070781822501923107/1496 648191050987827349734104221954474875000000000000000*7^(1/2), a[20,11] \+ = 58803614160031168862776673639561218650451486878573351031/11869244547 623498924625568418674528874212500000000000000-195151947898929484657927 742905600534538199744564712363/124939416290773672890795457038679251307 5000000000000000*7^(1/2), a[20,8] = -179360592777609544171797254107830 289034341017967853947/720261973015961741778528674244099821961216000000 00000+34941077136336931914073500569509004206732646116349683/3601309865 07980870889264337122049910980608000000000000*7^(1/2), a[20,15] = -6010 8815292626420255491361488348250991962662149/98649901623277513912105885 1224669826290892800000-1304019969083779956900885316764115910875407/607 07631768170777792065160075364297002516480*7^(1/2), a[20,14] = -1600346 6868997045499992466728944319268578868436842768854363/30961912562078837 8533311524058377597436400000000000000000000-61684454924277584852709070 8712335253938511951736215769/12384765024831535141332460962335103897456 000000000000000*7^(1/2), a[14,1] = 380/8289, c[2] = 1/16, c[3] = 112/1 065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[7,1] = 21400899/3500 00000, a[7,2] = 0, a[7,3] = 0, a[7,4] = 3064329829899/27126050000000, \+ a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/520 9191, a[5,3] = -352806597/250880000, a[5,4] = 178077159/125440000, a[6 ,1] = 12089/252720, a[4,2] = 0, a[4,3] = 42/355, a[5,1] = 94495479/250 880000, a[5,2] = 0, c[13] = 1, a[2,1] = 1/16, a[3,1] = 18928/1134225, \+ a[3,2] = 100352/1134225, a[4,1] = 14/355, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/20, c[12] = 1, a[ 9,6] = 7632051964154290925661849798370645637589377834346780/1734087257 418811583049800347581865260479233950396659, a[9,5] = -8753250485021304 41118613421785266742862694404520560000/1702120304288944183955716775759 61339495435011888324169, a[9,7] = 751983479197113751704853217965234772 9899303513750000/1045677303502317596597890707812349832637339039997351, a[9,4] = -14764960804048657303638372252908780219281424435/29816921025 65021975611711269209606363661854518, a[9,1] = -18409112522823765844381 57336464708426954728061551/2991923615171151921596253813483118262195533 733898, a[9,2] = 0, a[9,3] = 0, a[8,7] = 1993321838240/380523459069, a [8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8,6] = 341936800 488/47951126225, a[19,5] = 0, a[7,5] = -21643947/592609375, a[7,6] = 1 24391943/6756250000, a[8,1] = -15365458811/13609565775, a[8,2] = 0, a[ 10,8] = 1908158550070998850625/117087067039189929394176, a[10,9] = -52 956818288156668227044990077324877908565/291277995947743398634982222441 2353951940608, a[11,1] = -10116106591826909534781157993685116703/95628 19945036894030442231411871744000, a[11,2] = 0, a[20,19] = 0, a[10,5] = -3378604805394255292453489375/517042670569824692230499952, a[10,6] = \+ 1001587844183325981198091450220795/184232684207722503701669953872896, \+ a[10,7] = 187023075231349900768014890274453125/25224698849808178010752 575653374848, a[10,1] = -63077736705254280154824845013881/783693578537 86633855112190394368, a[10,2] = 0, a[10,3] = 0, a[10,4] = -31948346510 820970247215/6956009216960026632192, a[12,10] = 7958341351371843889152 /3284467988443203581305, a[12,6] = 18273578204342134614380775509022734 40/139381013914245317709567680839641697, a[12,7] = 6435048028142415509 41949227194107500000/242124609118836550860494007545333945331, a[12,8] \+ = 162259938151380266113750/59091082835244183497007, a[12,9] = -2302825 1632873523818545414856857015616678575554130463402/20013169183191444503 443905240405603349978424504151629055, a[12,3] = 0, a[12,4] = 262900926 04284231996745/5760876126062860430544, a[12,5] = -69706929756092645204 5586710000/41107967755245430594036502319, a[12,2] = 0, a[12,1] = -3218 022174758599831659045535578571/1453396753634469525663775847094384, a[1 1,8] = 39747262782380466933662225/1756032802431424164410720256, a[11,9 ] = 48175771419260955335244683805171548038966866545122229/198978642051 3815146528880165952064118903852843612160000, a[11,10] = -2378292068163 246/47768728487211875, a[11,7] = 2226455130519213549256016892506730559 375/364880443159675255577435648380047355776, a[11,4] = -96235413173230 77848129/3864449564977792573440, a[11,5] = -4823348333146829406881375/ 576413233634141239944816, a[13,9] = 4059320304637772479267050305961754 37402459637909765779/7880391943632184108320188604120153722976911508830 3952, a[13,10] = -10290327637248/1082076946951, a[13,11] = 86326410588 8000/85814662253313, a[13,6] = 14327219974204125/40489566827933216, a[ 13,7] = 2720762324010009765625000/10917367480696813922225349, a[13,8] \+ = -498533005859375/95352091037424, a[13,5] = 0, a[13,2] = 0, a[13,3] = 0, a[13,4] = 0, a[13,1] = 4631674879841/103782082379976, a[12,11] = - 507974327957860843878400/121555654819179042718967, a[19,2] = 0, a[17,1 ] = 14296927109732800365803280238811548992426955660898056576097/341413 526737606015913504201607836239464660351562500000000000-354981099311588 69858570877257888472099215653607797737/1047152271922481952869292729750 4485322802734375000000000*7^(1/2), a[17,9] = -777365420445936318089732 4177386410045148764290716888867400388432148819282908940948159254319/78 8452752376981215609864480482093541291983982324119451546116914378375650 249609375000000000000-349211535189884795491496750852447695276558730568 496460079234407077243879427767491465619/350423445500880540271050880214 26379612977065881071975624271862861261140011093750000000000*7^(1/2), a [17,14] = 8898831339550675631062765096204752299374508560508163/1749782 56576418490110785321303969544268798828125000000+3430004043103091387006 76556673479752657859745169/6999130263056739604431412852158781770751953 125000*7^(1/2), a[17,6] = 11680941707325157677955382237279696768268247 094386992999/87503361475665632084979209131119394314606250000000000000- 997172433570276305925254457325512017866796579074407189/218758403689164 08021244802282779848578651562500000000000*7^(1/2), a[17,7] = 729247118 0619236115859359700346738458665766464861503/44741004683259221639235793 273762127339404272876630000-395688161558937580577353827374940476509802 536449201/7170032801804362442185223281051622971058377063562500*7^(1/2) , a[17,16] = 312951003846638093776403168468573978963277/58151890694283 67546448931343495076666050000+1835267423444683727710449934904130527/44 732223610987442664991779565346743585*7^(1/2), a[17,11] = -259187471300 326360195328568552656084453334237831807/482960797022440548690819027452 576858488464355468750+195327906851419800364238706637279226357484899093 4/12709494658485277597126816511909917328643798828125*7^(1/2), a[17,15] = -77127127250912159388465163390223095790721/119229943754928152450236 2666335509439650000+19577936100246207024301582455131137389/92632494763 4441799805681763845280410805*7^(1/2), a[17,8] = 1589943165386807608189 28006714630116934243743639/4884589118218056517052738947510442586000000 00000-6476404035945812133836473370020724114834745161/67841515530806340 514621374270978369250000000000*7^(1/2), a[17,10] = 5136523946898556275 02026559415560286064929495444927/2283703904801922343978476111178519401 359558105468750-210846684210425267635251476041683325933811084598/41521 88917821676989051774747597308002471923828125*7^(1/2), a[17,12] = 26221 4308744442251330599379201914827178216725701161/22254494604004075189901 336249265303906250000000000000-404254143174321169864069534960147933054 7680120287/1112724730200203759495066812463265195312500000000000*7^(1/2 ), a[17,13] = 5503805654154447779082385289578389472169301/173484996080 8438372162663307568359375000000000-44976428168893445098944962904416522 821649/52650194123092630885946212158775329589843750*7^(1/2), a[14,4] = 0, a[18,3] = 0, a[18,4] = 0, a[18,5] = 0, a[18,2] = 0, c[14] = 1/2-1/ 14*7^(1/2), a[17,5] = 0, a[17,4] = 0, a[19,17] = 0, a[19,18] = 0, a[18 ,17] = 0, a[17,3] = 0, a[17,2] = 0, a[16,15] = 46088828763074293687752 51443876403372924872509408621400283/7592227894936087164857586010531872 0457058558153486785997500-84914702986428741912331429025972417475312941 024930181294/531455952645526101540031020737231043199409907074407501982 5*7^(1/2), a[16,7] = 1471649975107913251117693334592045704450918568535 3579684857180786483042792373/16088988337674671465702156595052532701656 7992369670063678825639487979375000000+35097161131542028791945288573990 0022880241691060058741934303404427219789/74868036898960959337075562503 093457016256710599371634570393697387568750000*7^(1/2), a[14,5] = 0, a[ 14,2] = 0, a[14,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[14,12] = 110 0613127343/48439922837376-29746300739/1424703612864*7^(1/2), a[15,1] = 130946139152859396534950567713097/2815269161783203125000000000000000+ 90670944595916412828478989403/66148981703437500000000000000000*7^(1/2) , a[15,8] = 16161913072172934315785836819/9932509483065000000000000000 -75980668436324671626317237501/49917227145660000000000000000*7^(1/2), \+ a[15,6] = 3353943190402140976803568136084793/1977029630270176562500000 0000000000-255748988917794999289135717669191/8649504632432022460937500 000000000*7^(1/2), a[15,7] = 10596614540600968744792362758669/45489031 169570058009272287500000-13058425932946007192467094947/145107190205518 7778276093750000*7^(1/2), a[15,10] = 14462059099710033235500874587/103 19489926824569702148437500-348155288848452187540396337457/252574928278 923034667968750000*7^(1/2), a[15,9] = -3727850143078386229058076561522 609367827766330696417108435354320431359/801634922651385915966816060803 2382937598583484731440429687500000000000+81425134979651212744812417413 5016152994969999173763717229208919303/15898421182640198336198703773925 38080518560957031250000000000000000*7^(1/2), a[15,11] = -2079418008367 56311792092915907/81839239362061500549316406250+1089100652209922459592 6000559657/4582997404275444030761718750000*7^(1/2), a[15,14] = 2609499 5957293704259/549316406250000000000*7^(1/2), a[15,13] = -5799454785425 37523/549316406250000000000+3727856565327672037/2197265625000000000000 *7^(1/2), a[18,10] = 78813744026468418092863662132343791904150881627/6 07405921692771033827002477362806199218750000000-1724595715075309301875 5976031205721533364483773/28781695981749765910571809696576047593750000 0000*7^(1/2), a[18,14] = 172041874375462697381563173001357495100867154 1792083/28668437557480405419751067042442370133000000000000000+66312564 220445944370265939039199371363080688129/114673750229921621679004268169 7694805320000000000*7^(1/2), a[18,9] = 1240384704962456457709494755011 245720575588780678711331088767810247863088806999332515507/305534765727 1632033243145612161452360578965365751743872784195724970507723200000000 0000000-44710839413529364122455168836520914745863776614731971922738965 3577377825685260302229/38022104179380309747025812062453629376093791218 243923750203324577410762777600000000000*7^(1/2), a[18,7] = 42893075508 2588455942724722283447907984915278705435/26179879311804253119187115607 04709622602855738609664-7649873380589074272937883429048385713258597850 7041/1174738174247626742527626982367497907578204498094080*7^(1/2), a[1 8,6] = 69454891827986654905811794573547009524702368493567653/512019669 434752041457249772287235770160896000000000000-192784207275057942900067 020977105865032917681968026949/358413768604326429020074840601065039112 6272000000000000*7^(1/2), a[18,1] = 7733624778322431132379739805371154 872928342814315555219/181614260456783667685936780491648991798344000000 000000000-75491682558580568648425894347272875262003943330267987/188721 97105495738747391541292654483628168000000000000000*7^(1/2), a[18,16] = 1501373575980631550465262783336687863485/2977376803547324183781852847 8694792530176+17058357454582951976373885014055586875/35235228444347031 7607319863653192811008*7^(1/2), a[18,8] = 5426224836754120276540228725 914955825217338781/20007277028221159493848018729002772832256000000-963 14521713121628743334127199055813316326877/8550118388128700638396589200 42853539840000000*7^(1/2), a[18,15] = -2546188245964304255811567921098 62619835/12209146240504642810904193703275616662016+1819720809919610010 18895701030477170625/7296590356751295407700139432135131851264*7^(1/2), a[18,13] = 84303053727623559901892597740221920795377/3552972719735681 7861891344539000000000000000-668871667581022998424770230088072148293/6 63554446548268972642571338468750000000000*7^(1/2), a[18,11] = -6119466 77966621853095211488458472723282243226327/1413005303288511776741139097 461253437406250000000+188814564128208460309798549660171819794928530747 /1041161802423113940756628808655660427562500000000*7^(1/2), a[18,12] = 9047237292228637467633523053986950065115624317/8623879838978305768953 25196565664000000000000000-5175814730144265227514848919003486791055011 817/1207343177456962807653455275191929600000000000000*7^(1/2), a[13,12 ] = -29746300739/247142463456, a[11,6] = 65661192465149968840670011549 77284529/970305487021846325473990863582315520, a[11,3] = 0, a[8,3] = 0 , a[9,8] = 1366042683489166351293315549358278750/144631418224267718165 055326464180836641, a[19,16] = -59220160570843125612523224803799201068 911/498444777379574361124194115156720857090000-63918239932356419778453 7438328437546/19170952976137475427853619813720032965*7^(1/2), a[19,8] \+ = 545852935816848763549582609751203523712604801307/4884589118218056517 05273894751044258600000000000+7894550931548752199808312441931046629102 173173/101762273296209510771932061406467553875000000000*7^(1/2), a[19, 12] = 816595560183909054739803824796340399773958361526093/222544946040 04075189901336249265303906250000000000000+3352205552351085337772532921 7930880599252333953/11354333981634732239745579719012910156250000000000 *7^(1/2), a[19,13] = 6329822636137785234855809795487253116923767139/52 5659538124956826765286982193212890625000000000+54824977090333610968656 925721307374512157/78975291184638946328919318238162994384765625*7^(1/2 ), a[19,15] = 201545429091434053176048871670250775086069/3440635519785 069542135389408568184382990000-2272851721861698000685159054540758474/1 32332135376348828543668823406468630115*7^(1/2), a[19,7] = 180390331149 28576144292936355618923845713121206510289/4474100468325922163923579327 3762127339404272876630000+32811736739714398209951717679755241266657439 50319/73163600018411861654951257969914520112840582281250*7^(1/2), a[19 ,1] = 18778954190432498641397367225501777310616443565686827961261/3414 13526737606015913504201607836239464660351562500000000000+3237979363730 976751083677681250337532341535825485533/117537499909666341648594081910 7646311743164062500000000*7^(1/2), a[19,6] = 2817988521471012425359486 6402805577138221840971048257987/87503361475665632084979209131119394314 606250000000000000+826887497606742913768075684811576991689560116836609 1/223222860907310285931069411048773965088281250000000000*7^(1/2), a[19 ,14] = -3615806799134503291043136722909584352046302608498253/874891282 88209245055392660651984772134399414062500000-1393692212705451006009587 91779240838739971407839/3499565131528369802215706426079390885375976562 500*7^(1/2), a[19,10] = 2498001978259689024640690501707586024679437753 648551/2283703904801922343978476111178519401359558105468750+3846498947 1746279636532325535691058627774079964/93212404277629483427692902497082 4245452880859375*7^(1/2), a[19,11] = -92775336423086812038318728622181 8715585159819007991/48296079702244054869081902745257685848846435546875 0-226761067735484045528048017173212371351053066444/1815642094069325371 018116644558559618377685546875*7^(1/2), a[19,9] = -2414064062661186975 8219261752350837961521246348971485476648481626009901547285237073249550 0147/78845275237698121560986448048209354129198398232411945154611691437 8375650249609375000000000000+86873235585153923845165431213316048244987 30528284220439682172692980305015008846423783/1072724833165960837564441 470043664682029910180032815580334852944732483877890625000000000*7^(1/2 ), a[20,17] = 0, a[20,18] = 0, a[20,3] = 0, a[20,4] = 0, a[20,5] = 0, \+ a[20,2] = 0, a[15,4] = 0, a[15,5] = 0, a[15,2] = 0, a[15,3] = 0, a[16, 2] = 0, a[16,4] = 0, a[16,3] = 0, c[17] = 993/2500, a[16,5] = 0, c[16] = 277/2500, a[14,8] = 12730301828542984375/4462520239258570944-231397 95371828125/21803844817224288*7^(1/2), a[14,11] = -315051553138064800/ 67942629698767761+392422111155400/221311497390123*7^(1/2), a[14,9] = - 2431300645813356820990854267156732219942356459331979/21898184163327797 22313164167138490920989148784630720+4933990286820622153338508943286630 4648514986421/125875747250399294250613000601944295132543520864*7^(1/2) , a[14,10] = 21310748482922848/7398947051356315-5227019004392/48201609 45509*7^(1/2), a[14,6] = 9785567316447209725/49721188064701989248-3438 8281046170925/566853935591065024*7^(1/2), a[14,7] = 101618266982917773 1250000000/4211024590054414662359895513-23208493423698876953125/137166 92475747279030488259*7^(1/2), a[16,14] = 37807079247082910935750437397 5352653612596622387047615352035143989/38652465726311665474213406887371 187527810692671413726806640625000000-736694485119203754900544593199058 955707390426548138628337098463663/247375780648394659034965804079175600 17798843309704785156250000000000*7^(1/2), a[16,6] = -19039890149866495 43258492920344376660766478959434824857368133995942073946993/2610549880 4470984484513299908841941049181178531837828644206625000000000000000+28 9290358369928611582932131505420731599890246264645126636225544956822760 29/8922777911684418524980131804779960319544348130999257837375311279296 87500000*7^(1/2), a[16,9] = 207904342034969170900628672444919837194509 1963358485311941328410674817931873625269135519028597791657255769210895 5233/15877667583594659668953564588772545305777807317441893201879580358 6671680545234558493166228541026757812500000000000000+16750291917128487 1193499988335765604001116436018919123161320355297232333887892701752121 8877046934971369718057209/73181019315333410819868281232202677952556148 9528186024119552674967375646957459693466445168584156250000000000000000 *7^(1/2), a[16,13] = -463305435445535408741794483954310830828996475223 68296822645347/5087451352083124397391250759096940921758056970520019531 250000000+189091968318676631162671256399201756538119827578927756418900 7889/11395891028666198650156401700377147664738047613964843750000000000 00*7^(1/2), a[16,1] = 101385640411812595276352720516149878818017393925 4661359435974309585825318269273/19856466773923966037438917121094233830 745906326832446413024679687500000000000000-193175424371098513173126758 518148648929759153027347863824763410934840962491/277893272189669140325 031078817733067056972748113814414737187500000000000000000*7^(1/2), a[1 6,10] = -1752032306988852430152559893574903103295734904546828540573516 708228831279/243326283010247711363961558099490763015363068164171983079 9102783203125000+81883999666586172691735974039324906939964446697501450 842550606413648603/733722330307823868112868698269233685400171713233503 210517883300781250000*7^(1/2), a[16,12] = -124817112626650602913170312 644481143348777609285630839757851632016681433/398360478373506965815942 7207807518356647862336983160384375000000000000000+19488568273447271249 669733067688159410742449986073234566291555857980323/290471182147348829 2407915672359648801722399620716887780273437500000000000*7^(1/2), a[16, 11] = 3307587237786214580372394397330733034806289881443017455732775116 590947887/216127682925333604166493245606691739590970049710504022614334 1064453125000-15924948134835543062332142541253213652166685490318832951 5308104047100201/53083992297450358918086060324450602706554047297316777 4842224121093750000*7^(1/2), c[20] = 9029/10000, c[15] = 1187/2500, c[ 19] = 1779/2500, a[16,8] = -285913537824921837018262511263067663964612 29874660236919089505718702483/3073895442200686868940860530269786975059 2763266032965939343750000000000+32621718645707277997227824444006976400 6022537683484345700377402630409/17437228972534444030205831181524084391 59931363492594965600000000000000*7^(1/2), a[20,9] = 350957869536825723 7736586575771915705978852862387597398280776123742867347028861250535310 02333688587/1162620890545001421289681768339675852247507900975853578471 84215726577759883206400000000000000000000+1884043540249331018790584629 707285984800654018527960079175320009299480163532738452671310798657/186 0193424872002274063490829343481363596012641561365725554947451625244158 13130240000000000000000*7^(1/2), a[20,13] = -2480077906845787040851366 835821277889558088410267941/116267479280630450771253235869423600000000 000000000000+242653922955367016465634203061511070435162932747/27948913 2886130891277051047763037500000000000000000*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 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_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],'interpolation_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 specified expl icitly as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 558 "inter p_order_eqns21 := [add(a[21,i],i=1..20)=c[21],seq(op(StageOrderConditi ons(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 s pecify " }{XPPEDIT 18 0 "c[21] = 253/2500;" "6#/&%\"cG6#\"#@*&\"$`#\" \"\"\"%+D!\"\"" }{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[2 1,20] = 0;" "6#/&%\"aG6$\"#@\"#?\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "e24 := `union`(e23,\{c[21]=253/250 0,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(eqs_21);\nnops( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%\"aG6$\"#@\"\"\"&F%6$F'\"\")&F%6$F'\"#5&F%6$F'\"#6& F%6$F'\"\"(&F%6$F'\"\"'&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 "e25 := solve(\{op(eqs_21)\}):\ninfolevel[solve]:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "e26 := `u nion`(e24,e25):\nseq(a[21,i]=subs(e26,a[21,i]),i=1..20):\nevalf[40](%) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "66/&%\"aG6$\"#@\"\"\"$\"IQ7UtJ0QEi, 7EPp%*)\\@Xq$!#T/&F%6$F'\"\"#$\"\"!F1/&F%6$F'\"\"$F0/&F%6$F'\"\"%F0/&F %6$F'\"\"&F0/&F%6$F'\"\"'$!I*fx0$p)3)>P\\Ns\"zz+0Dlq$F+/&F%6$F'\"\"($! IL.iWHY:TN\\H;b\"Qe)QG5\\F+/&F%6$F'\"\")$!IQa'*zT+:>dUsD(3i$zX7&=#!#S/ &F%6$F'\"\"*$\"IZ`.>sU)p15Y&pqQXQT]FFP/&F%6$F'\"#6$\"I^,V,^zK+$Q0Fvmy5u#\\0RFP/&F%6$F'\"#7$!I`Z5A^1; 5cFZwQhY`eXgk!#U/&F%6$F'\"#8$!I4vFrKr&>cBvv[=/l+Cl!GFco/&F%6$F'\"#9$!I +fcP6Fco/&F%6$F'\"#:$\"IVICJe@w-+ddZ\"H_fhuN(QF+/&F%6$F '\"#;$\"IQqH9$f:q#GJ'ppADJx)[L6FP/&F%6$F'\"#F0/&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 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_20) do\n tt := convert(SO7_20 [ct],'interpolation_order_condition'):\n if expand(subs(e26,lhs(tt)= rhs(tt))) then recd := [op(recd),ct] end if: \nend do:\nop(recd);\nnop s(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 24599 "e26 := \{a[21,3] = 0, a[21,4] = 0, a[21,5] = 0, a[21,2] = 0, a[ 19,3] = 0, a[19,4] = 0, c[18] = 453/1000, a[15,12] = 12163051029345238 407733418657/3016875774609375000000000000000, a[20,6] = 64705474043359 1047501139839926183281833250508652625223013947/14336550744173057160802 99362404260156450508800000000000000000+6641827613172956904051390551401 0579101397635988537328052807/14336550744173057160802993624042601564505 08800000000000000000*7^(1/2), a[20,1] = 897330932052009060320705191604 11508217914220092982292430255743/1525559787836982808561868956129851531 106089600000000000000000000+234076470257947541762894493668275072391709 451556335985408969/679399095797846594906095486535561410614048000000000 00000000000*7^(1/2), a[20,16] = -5895996755822171065621140032186175996 96505263937/3858680337397332142181281290838845111910809600000-99015163 74282546563618749633979325112870981/2374572515321435164419250025131596 99194511360*7^(1/2), a[20,7] = 584912615672238396354166551337437211351 6941982763467945837/13194659173149343572070306265951736497918392922592 706560000+711597078575774626289414881120414051622031127652898916601/12 687172281874368819298371409568977401844608579416064000000*7^(1/2), a[2 0,12] = -54175810754967243498377218440527251354798928847932480401/1093 852918776008303734030479323888217600000000000000000000+807779950733732 671058695914872912518198545978092589343/218770583755201660746806095864 777643520000000000000000000*7^(1/2), a[20,10] = -883876388558690332747 7514842388396732758843333309547477/17007365807397588947156069366158573 57812500000000000000+7724080662523013714735975132270518207078182250192 3107/1496648191050987827349734104221954474875000000000000000*7^(1/2), \+ a[20,11] = 58803614160031168862776673639561218650451486878573351031/11 869244547623498924625568418674528874212500000000000000-195151947898929 484657927742905600534538199744564712363/124939416290773672890795457038 6792513075000000000000000*7^(1/2), a[20,8] = -179360592777609544171797 254107830289034341017967853947/720261973015961741778528674244099821961 21600000000000+34941077136336931914073500569509004206732646116349683/3 60130986507980870889264337122049910980608000000000000*7^(1/2), a[21,17 ] = 0, a[21,18] = 0, a[21,19] = 0, a[21,20] = 0, c[21] = 253/2500, a[2 0,15] = -60108815292626420255491361488348250991962662149/9864990162327 75139121058851224669826290892800000-1304019969083779956900885316764115 910875407/60707631768170777792065160075364297002516480*7^(1/2), a[20,1 4] = -16003466868997045499992466728944319268578868436842768854363/3096 19125620788378533311524058377597436400000000000000000000-6168445492427 75848527090708712335253938511951736215769/1238476502483153514133246096 2335103897456000000000000000*7^(1/2), a[14,1] = 380/8289, c[2] = 1/16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[7,1] = \+ 21400899/350000000, a[7,2] = 0, a[7,3] = 0, a[7,4] = 3064329829899/271 26050000000, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a[5,3] = -352806597/250880000, a[5,4] = 178077159/1 25440000, a[6,1] = 12089/252720, a[4,2] = 0, a[4,3] = 42/355, a[5,1] = 94495479/250880000, a[5,2] = 0, c[13] = 1, a[2,1] = 1/16, a[3,1] = 18 928/1134225, a[3,2] = 100352/1134225, a[4,1] = 14/355, c[7] = 39/250, \+ c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/20, \+ c[12] = 1, a[9,6] = 76320519641542909256618497983706456375893778343467 80/1734087257418811583049800347581865260479233950396659, a[9,5] = -875 325048502130441118613421785266742862694404520560000/170212030428894418 395571677575961339495435011888324169, a[9,7] = 75198347919711375170485 32179652347729899303513750000/1045677303502317596597890707812349832637 339039997351, a[9,4] = -1476496080404865730363837225290878021928142443 5/2981692102565021975611711269209606363661854518, a[9,1] = -1840911252 282376584438157336464708426954728061551/299192361517115192159625381348 3118262195533733898, a[9,2] = 0, a[9,3] = 0, a[8,7] = 1993321838240/38 0523459069, a[8,4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8,6 ] = 341936800488/47951126225, a[19,5] = 0, a[7,5] = -21643947/59260937 5, a[7,6] = 124391943/6756250000, a[8,1] = -15365458811/13609565775, a [8,2] = 0, a[10,8] = 1908158550070998850625/117087067039189929394176, \+ a[10,9] = -52956818288156668227044990077324877908565/29127799594774339 86349822224412353951940608, a[11,1] = -1011610659182690953478115799368 5116703/9562819945036894030442231411871744000, a[11,2] = 0, a[20,19] = 0, a[10,5] = -3378604805394255292453489375/51704267056982469223049995 2, a[10,6] = 1001587844183325981198091450220795/1842326842077225037016 69953872896, a[10,7] = 187023075231349900768014890274453125/2522469884 9808178010752575653374848, a[10,1] = -63077736705254280154824845013881 /78369357853786633855112190394368, a[10,2] = 0, a[10,3] = 0, a[10,4] = -31948346510820970247215/6956009216960026632192, a[12,10] = 795834135 1371843889152/3284467988443203581305, a[12,6] = 1827357820434213461438 077550902273440/139381013914245317709567680839641697, a[12,7] = 643504 802814241550941949227194107500000/242124609118836550860494007545333945 331, a[12,8] = 162259938151380266113750/59091082835244183497007, a[12, 9] = -23028251632873523818545414856857015616678575554130463402/2001316 9183191444503443905240405603349978424504151629055, a[12,3] = 0, a[12,4 ] = 26290092604284231996745/5760876126062860430544, a[12,5] = -6970692 97560926452045586710000/41107967755245430594036502319, a[12,2] = 0, a[ 12,1] = -3218022174758599831659045535578571/14533967536344695256637758 47094384, a[11,8] = 39747262782380466933662225/17560328024314241644107 20256, a[11,9] = 48175771419260955335244683805171548038966866545122229 /1989786420513815146528880165952064118903852843612160000, a[11,10] = - 2378292068163246/47768728487211875, a[11,7] = 222645513051921354925601 6892506730559375/364880443159675255577435648380047355776, a[11,4] = -9 623541317323077848129/3864449564977792573440, a[11,5] = -4823348333146 829406881375/576413233634141239944816, a[13,9] = 405932030463777247926 705030596175437402459637909765779/788039194363218410832018860412015372 29769115088303952, a[13,10] = -10290327637248/1082076946951, a[13,11] \+ = 863264105888000/85814662253313, a[13,6] = 14327219974204125/40489566 827933216, a[13,7] = 2720762324010009765625000/10917367480696813922225 349, a[13,8] = -498533005859375/95352091037424, a[13,5] = 0, a[13,2] = 0, a[13,3] = 0, a[13,4] = 0, a[13,1] = 4631674879841/103782082379976, a[12,11] = -507974327957860843878400/121555654819179042718967, a[19,2 ] = 0, a[17,1] = 14296927109732800365803280238811548992426955660898056 576097/341413526737606015913504201607836239464660351562500000000000-35 498109931158869858570877257888472099215653607797737/104715227192248195 28692927297504485322802734375000000000*7^(1/2), a[17,9] = -77736542044 5936318089732417738641004514876429071688886740038843214881928290894094 8159254319/78845275237698121560986448048209354129198398232411945154611 6914378375650249609375000000000000-34921153518988479549149675085244769 5276558730568496460079234407077243879427767491465619/35042344550088054 0271050880214263796129770658810719756242718628612611400110937500000000 00*7^(1/2), a[17,14] = 88988313395506756310627650962047522993745085605 08163/174978256576418490110785321303969544268798828125000000+343000404 310309138700676556673479752657859745169/699913026305673960443141285215 8781770751953125000*7^(1/2), a[17,6] = 1168094170732515767795538223727 9696768268247094386992999/87503361475665632084979209131119394314606250 000000000000-997172433570276305925254457325512017866796579074407189/21 875840368916408021244802282779848578651562500000000000*7^(1/2), a[17,7 ] = 7292471180619236115859359700346738458665766464861503/4474100468325 9221639235793273762127339404272876630000-39568816155893758057735382737 4940476509802536449201/71700328018043624421852232810516229710583770635 62500*7^(1/2), a[17,16] = 312951003846638093776403168468573978963277/5 815189069428367546448931343495076666050000+183526742344468372771044993 4904130527/44732223610987442664991779565346743585*7^(1/2), a[17,11] = \+ -259187471300326360195328568552656084453334237831807/48296079702244054 8690819027452576858488464355468750+19532790685141980036423870663727922 63574848990934/12709494658485277597126816511909917328643798828125*7^(1 /2), a[17,15] = -77127127250912159388465163390223095790721/11922994375 49281524502362666335509439650000+1957793610024620702430158245513113738 9/926324947634441799805681763845280410805*7^(1/2), a[17,8] = 158994316 538680760818928006714630116934243743639/488458911821805651705273894751 044258600000000000-6476404035945812133836473370020724114834745161/6784 1515530806340514621374270978369250000000000*7^(1/2), a[17,10] = 513652 394689855627502026559415560286064929495444927/228370390480192234397847 6111178519401359558105468750-21084668421042526763525147604168332593381 1084598/4152188917821676989051774747597308002471923828125*7^(1/2), a[1 7,12] = 262214308744442251330599379201914827178216725701161/2225449460 4004075189901336249265303906250000000000000-40425414317432116986406953 49601479330547680120287/1112724730200203759495066812463265195312500000 000000*7^(1/2), a[17,13] = 5503805654154447779082385289578389472169301 /1734849960808438372162663307568359375000000000-4497642816889344509894 4962904416522821649/52650194123092630885946212158775329589843750*7^(1/ 2), a[14,4] = 0, a[18,3] = 0, a[18,4] = 0, a[18,5] = 0, a[18,2] = 0, c [14] = 1/2-1/14*7^(1/2), a[17,5] = 0, a[17,4] = 0, a[19,17] = 0, a[19, 18] = 0, a[18,17] = 0, a[17,3] = 0, a[17,2] = 0, a[16,15] = 4608882876 307429368775251443876403372924872509408621400283/759222789493608716485 75860105318720457058558153486785997500-8491470298642874191233142902597 2417475312941024930181294/53145595264552610154003102073723104319940990 70744075019825*7^(1/2), a[16,7] = 147164997510791325111769333459204570 44509185685353579684857180786483042792373/1608898833767467146570215659 50525327016567992369670063678825639487979375000000+3509716113154202879 19452885739900022880241691060058741934303404427219789/7486803689896095 9337075562503093457016256710599371634570393697387568750000*7^(1/2), a[ 14,5] = 0, a[14,2] = 0, a[14,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a [14,12] = 1100613127343/48439922837376-29746300739/1424703612864*7^(1/ 2), a[15,1] = 130946139152859396534950567713097/2815269161783203125000 000000000000+90670944595916412828478989403/661489817034375000000000000 00000*7^(1/2), a[15,8] = 16161913072172934315785836819/993250948306500 0000000000000-75980668436324671626317237501/49917227145660000000000000 000*7^(1/2), a[15,6] = 3353943190402140976803568136084793/197702963027 01765625000000000000000-255748988917794999289135717669191/864950463243 2022460937500000000000*7^(1/2), a[15,7] = 1059661454060096874479236275 8669/45489031169570058009272287500000-13058425932946007192467094947/14 51071902055187778276093750000*7^(1/2), a[15,10] = 14462059099710033235 500874587/10319489926824569702148437500-348155288848452187540396337457 /252574928278923034667968750000*7^(1/2), a[15,9] = -372785014307838622 9058076561522609367827766330696417108435354320431359/80163492265138591 59668160608032382937598583484731440429687500000000000+8142513497965121 27448124174135016152994969999173763717229208919303/1589842118264019833 619870377392538080518560957031250000000000000000*7^(1/2), a[15,11] = - 207941800836756311792092915907/81839239362061500549316406250+108910065 22099224595926000559657/4582997404275444030761718750000*7^(1/2), a[15, 14] = 26094995957293704259/549316406250000000000*7^(1/2), a[15,13] = - 579945478542537523/549316406250000000000+3727856565327672037/219726562 5000000000000*7^(1/2), a[21,1] = 3099708133843084297023805030555436368 5394154010924527196929/83801502017412385724223758576468895141325722656 2500000000000+4565336092001834973218894721195940768038738096387/213704 545290302439361080148928662965771484375000000000*7^(1/2), a[21,11] = 5 69572596978601761554246747325447992832658077510179/1448882391067321646 072457082357730575465393066406250-175844934070379499464053180779527598 7505836638/1815642094069325371018116644558559618377685546875*7^(1/2), \+ a[21,13] = -1482763015196564094793258170458172355589102597/52565953812 4956826765286982193212890625000000000+85029538607620379403603914020417 3286753/157950582369277892657838636476325988769531250*7^(1/2), a[21,8] = -2902819118865630209141335036817092955423296512547/1318839061918875 2596042395158278194982200000000000+12243872389001035990165802669519097 5148385817/203524546592419021543864122812935107750000000000*7^(1/2), a [21,10] = -57276921633460666879267544795444795665922000283843/20760944 5891083849452588737379865400123596191406250+27116541674232832942490198 536154073869071698/84738549343299530388811729542802204132080078125*7^( 1/2), a[21,6] = -33098372644113630504863314574774471907135574827406634 01/87503361475665632084979209131119394314606250000000000000+1282442166 57060373956178084993533317618563230091239/4464457218146205718621388220 97547930176562500000000000*7^(1/2), a[21,7] = -60428090962983762028102 222659035092334443807259518019/120800712644799898425936641839157743816 3915367669010000+50888609242748421551480430147730676944005130651/14632 7200036823723309902515939829040225681164562500*7^(1/2), a[21,15] = 672 440045006965423277162352231808676922717/172031775989253477106769470428 40921914950000-17625135794859273212919783295280073/1323321353763488285 43668823406468630115*7^(1/2), a[21,12] = -4353666069118873002783354278 07133982215783539255317/6676348381201222556970400874779591171875000000 0000000+519902618408155831782878999596105234295116437/2270866796326946 4479491159438025820312500000000000*7^(1/2), a[21,14] = -56078524815090 382874736120910671348469312408461137/174978256576418490110785321303969 544268798828125000000-2161514917044480886135528855393203658546310731/6 999130263056739604431412852158781770751953125000*7^(1/2), a[21,9] = 23 7323297964284489468432451754522328128037640343850828667723380707628841 0234603005442258460187/21288224314178492821466340973016525614883567522 751225191745156688216142556739453125000000000000+134734048807457466338 865854369659975898277761095422843616676876929084259549554219507/214544 9666331921675128882940087329364059820360065631160669705889464967755781 250000000000*7^(1/2), a[21,16] = 2841955996551750120506796848975848061 34113/2492223886897871805620970575783604285450000-49566262846808342603 20274687068217/19170952976137475427853619813720032965*7^(1/2), a[18,10 ] = 78813744026468418092863662132343791904150881627/607405921692771033 827002477362806199218750000000-172459571507530930187559760312057215333 64483773/287816959817497659105718096965760475937500000000*7^(1/2), a[1 8,14] = 1720418743754626973815631730013574951008671541792083/286684375 57480405419751067042442370133000000000000000+6631256422044594437026593 9039199371363080688129/11467375022992162167900426816976948053200000000 00*7^(1/2), a[18,9] = 124038470496245645770949475501124572057558878067 8711331088767810247863088806999332515507/30553476572716320332431456121 614523605789653657517438727841957249705077232000000000000000-447108394 1352936412245516883652091474586377661473197192273896535773778256852603 02229/3802210417938030974702581206245362937609379121824392375020332457 7410762777600000000000*7^(1/2), a[18,7] = 4289307550825884559427247222 83447907984915278705435/2617987931180425311918711560704709622602855738 609664-76498733805890742729378834290483857132585978507041/117473817424 7626742527626982367497907578204498094080*7^(1/2), a[18,6] = 6945489182 7986654905811794573547009524702368493567653/51201966943475204145724977 2287235770160896000000000000-19278420727505794290006702097710586503291 7681968026949/3584137686043264290200748406010650391126272000000000000* 7^(1/2), a[18,1] = 773362477832243113237973980537115487292834281431555 5219/181614260456783667685936780491648991798344000000000000000-7549168 2558580568648425894347272875262003943330267987/18872197105495738747391 541292654483628168000000000000000*7^(1/2), a[18,16] = 1501373575980631 550465262783336687863485/29773768035473241837818528478694792530176+170 58357454582951976373885014055586875/3523522844434703176073198636531928 11008*7^(1/2), a[18,8] = 542622483675412027654022872591495582521733878 1/20007277028221159493848018729002772832256000000-96314521713121628743 334127199055813316326877/855011838812870063839658920042853539840000000 *7^(1/2), a[18,15] = -254618824596430425581156792109862619835/12209146 240504642810904193703275616662016+181972080991961001018895701030477170 625/7296590356751295407700139432135131851264*7^(1/2), a[18,13] = 84303 053727623559901892597740221920795377/355297271973568178618913445390000 00000000000-668871667581022998424770230088072148293/663554446548268972 642571338468750000000000*7^(1/2), a[18,11] = -611946677966621853095211 488458472723282243226327/141300530328851177674113909746125343740625000 0000+188814564128208460309798549660171819794928530747/1041161802423113 940756628808655660427562500000000*7^(1/2), a[18,12] = 9047237292228637 467633523053986950065115624317/862387983897830576895325196565664000000 000000000-5175814730144265227514848919003486791055011817/1207343177456 962807653455275191929600000000000000*7^(1/2), a[13,12] = -29746300739/ 247142463456, a[11,6] = 6566119246514996884067001154977284529/97030548 7021846325473990863582315520, a[11,3] = 0, a[8,3] = 0, a[9,8] = 136604 2683489166351293315549358278750/14463141822426771816505532646418083664 1, a[19,16] = -59220160570843125612523224803799201068911/4984447773795 74361124194115156720857090000-639182399323564197784537438328437546/191 70952976137475427853619813720032965*7^(1/2), a[19,8] = 545852935816848 763549582609751203523712604801307/488458911821805651705273894751044258 600000000000+7894550931548752199808312441931046629102173173/1017622732 96209510771932061406467553875000000000*7^(1/2), a[19,12] = 81659556018 3909054739803824796340399773958361526093/22254494604004075189901336249 265303906250000000000000+335220555235108533777253292179308805992523339 53/11354333981634732239745579719012910156250000000000*7^(1/2), a[19,13 ] = 6329822636137785234855809795487253116923767139/5256595381249568267 65286982193212890625000000000+5482497709033361096865692572130737451215 7/78975291184638946328919318238162994384765625*7^(1/2), a[19,15] = 201 545429091434053176048871670250775086069/344063551978506954213538940856 8184382990000-2272851721861698000685159054540758474/132332135376348828 543668823406468630115*7^(1/2), a[19,7] = 18039033114928576144292936355 618923845713121206510289/447410046832592216392357932737621273394042728 76630000+3281173673971439820995171767975524126665743950319/73163600018 411861654951257969914520112840582281250*7^(1/2), a[19,1] = 18778954190 432498641397367225501777310616443565686827961261/341413526737606015913 504201607836239464660351562500000000000+323797936373097675108367768125 0337532341535825485533/11753749990966634164859408191076463117431640625 00000000*7^(1/2), a[19,6] = 281798852147101242535948664028055771382218 40971048257987/8750336147566563208497920913111939431460625000000000000 0+8268874976067429137680756848115769916895601168366091/223222860907310 285931069411048773965088281250000000000*7^(1/2), a[19,14] = -361580679 9134503291043136722909584352046302608498253/87489128288209245055392660 651984772134399414062500000-139369221270545100600958791779240838739971 407839/3499565131528369802215706426079390885375976562500*7^(1/2), a[19 ,10] = 2498001978259689024640690501707586024679437753648551/2283703904 801922343978476111178519401359558105468750+384649894717462796365323255 35691058627774079964/932124042776294834276929024970824245452880859375* 7^(1/2), a[19,11] = -9277533642308681203831872862218187155851598190079 91/482960797022440548690819027452576858488464355468750-226761067735484 045528048017173212371351053066444/181564209406932537101811664455855961 8377685546875*7^(1/2), a[19,9] = -241406406266118697582192617523508379 615212463489714854766484816260099015472852370732495500147/788452752376 9812156098644804820935412919839823241194515461169143783756502496093750 00000000000+8687323558515392384516543121331604824498730528284220439682 172692980305015008846423783/107272483316596083756444147004366468202991 0180032815580334852944732483877890625000000000*7^(1/2), a[20,17] = 0, \+ a[20,18] = 0, a[20,3] = 0, a[20,4] = 0, a[20,5] = 0, a[20,2] = 0, a[15 ,4] = 0, a[15,5] = 0, a[15,2] = 0, a[15,3] = 0, a[16,2] = 0, a[16,4] = 0, a[16,3] = 0, c[17] = 993/2500, a[16,5] = 0, c[16] = 277/2500, a[14 ,8] = 12730301828542984375/4462520239258570944-23139795371828125/21803 844817224288*7^(1/2), a[14,11] = -315051553138064800/67942629698767761 +392422111155400/221311497390123*7^(1/2), a[14,9] = -24313006458133568 20990854267156732219942356459331979/2189818416332779722313164167138490 920989148784630720+49339902868206221533385089432866304648514986421/125 875747250399294250613000601944295132543520864*7^(1/2), a[14,10] = 2131 0748482922848/7398947051356315-5227019004392/4820160945509*7^(1/2), a[ 14,6] = 9785567316447209725/49721188064701989248-34388281046170925/566 853935591065024*7^(1/2), a[14,7] = 1016182669829177731250000000/421102 4590054414662359895513-23208493423698876953125/13716692475747279030488 259*7^(1/2), a[16,14] = 3780707924708291093575043739753526536125966223 87047615352035143989/3865246572631166547421340688737118752781069267141 3726806640625000000-73669448511920375490054459319905895570739042654813 8628337098463663/24737578064839465903496580407917560017798843309704785 156250000000000*7^(1/2), a[16,6] = -1903989014986649543258492920344376 660766478959434824857368133995942073946993/261054988044709844845132999 08841941049181178531837828644206625000000000000000+2892903583699286115 8293213150542073159989024626464512663622554495682276029/89227779116844 1852498013180477996031954434813099925783737531127929687500000*7^(1/2), a[16,9] = 20790434203496917090062867244491983719450919633584853119413 284106748179318736252691355190285977916572557692108955233/158776675835 9465966895356458877254530577780731744189320187958035866716805452345584 93166228541026757812500000000000000+1675029191712848711934999883357656 0400111643601891912316132035529723233388789270175212188770469349713697 18057209/7318101931533341081986828123220267795255614895281860241195526 74967375646957459693466445168584156250000000000000000*7^(1/2), a[16,13 ] = -46330543544553540874179448395431083082899647522368296822645347/50 87451352083124397391250759096940921758056970520019531250000000+1890919 683186766311626712563992017565381198275789277564189007889/113958910286 6619865015640170037714766473804761396484375000000000000*7^(1/2), a[16, 1] = 10138564041181259527635272051614987881801739392546613594359743095 85825318269273/1985646677392396603743891712109423383074590632683244641 3024679687500000000000000-19317542437109851317312675851814864892975915 3027347863824763410934840962491/27789327218966914032503107881773306705 6972748113814414737187500000000000000000*7^(1/2), a[16,10] = -17520323 06988852430152559893574903103295734904546828540573516708228831279/2433 262830102477113639615580994907630153630681641719830799102783203125000+ 8188399966658617269173597403932490693996444669750145084255060641364860 3/73372233030782386811286869826923368540017171323350321051788330078125 0000*7^(1/2), a[16,12] = -12481711262665060291317031264448114334877760 9285630839757851632016681433/39836047837350696581594272078075183566478 62336983160384375000000000000000+1948856827344727124966973306768815941 0742449986073234566291555857980323/29047118214734882924079156723596488 01722399620716887780273437500000000000*7^(1/2), a[16,11] = 33075872377 86214580372394397330733034806289881443017455732775116590947887/2161276 829253336041664932456066917395909700497105040226143341064453125000-159 249481348355430623321425412532136521666854903188329515308104047100201/ 5308399229745035891808606032445060270655404729731677748422241210937500 00*7^(1/2), c[20] = 9029/10000, c[15] = 1187/2500, c[19] = 1779/2500, \+ a[16,8] = -28591353782492183701826251126306766396461229874660236919089 505718702483/307389544220068686894086053026978697505927632660329659393 43750000000000+3262171864570727799722782444400697640060225376834843457 00377402630409/1743722897253444403020583118152408439159931363492594965 600000000000000*7^(1/2), a[20,9] = 35095786953682572377365865757719157 0597885286238759739828077612374286734702886125053531002333688587/11626 2089054500142128968176833967585224750790097585357847184215726577759883 206400000000000000000000+188404354024933101879058462970728598480065401 8527960079175320009299480163532738452671310798657/18601934248720022740 6349082934348136359601264156136572555494745162524415813130240000000000 000000*7^(1/2), a[20,13] = -248007790684578704085136683582127788955808 8410267941/116267479280630450771253235869423600000000000000000000+2426 53922955367016465634203061511070435162932747/2794891328861308912770510 47763037500000000000000000*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 9053 "dd := \{d[ 12,8] = 36311402269287109375000000/105435484914134126893463, d[20,8] = 324546670138750000000000000000000000000/87700765646973215870365578557 3133737, d[13,7] = -148790056173314860015625000000000/3717635847594421 27162561862673, d[15,4] = 0, d[10,1] = 0, d[12,3] = -32842678468662393 5099769171/1686967758626146030295408, d[4,6] = 0, d[6,1] = 0, d[5,7] = 0, d[12,2] = 360113655316927278109563/22343943822862861328416, d[15,8 ] = 0, d[16,4] = 0, d[9,1] = 0, d[2,1] = 0, d[3,1] = 0, d[2,6] = 0, d[ 14,1] = 0, d[16,5] = 0, d[12,1] = 0, d[18,1] = 0, d[15,2] = 0, d[13,1] = 0, d[15,3] = 0, d[3,6] = 0, d[5,2] = 0, d[3,4] = 0, d[4,4] = 0, d[5 ,3] = 0, d[2,5] = 0, d[3,7] = 0, d[2,4] = 0, d[21,1] = 0, d[5,4] = 0, \+ d[16,6] = 0, d[21,8] = 12618694117599487304687500000000000/21627933494 940466816676873343101, d[2,7] = 0, d[14,8] = 0, d[10,2] = 513028326435 6121609520487168/4028803827997056367944503, d[21,6] = 2609827467402595 93489990234375000000/64883800484821400450030620029303, d[2,2] = 0, d[4 ,2] = 0, d[3,2] = 0, d[14,2] = 0, d[5,1] = 0, d[14,6] = 0, d[15,7] = 0 , d[2,3] = 0, d[5,6] = 0, d[4,3] = 0, d[3,3] = 0, d[14,7] = 0, d[14,5] = 0, d[14,3] = 0, d[7,1] = 0, d[5,5] = 0, d[16,3] = 0, d[1,1] = 1, d[ 4,5] = 0, d[4,1] = 0, d[15,5] = 0, d[11,1] = 0, d[4,8] = 0, d[16,2] = \+ 0, d[16,8] = 0, d[15,1] = 0, d[2,8] = 0, d[14,4] = 0, d[10,5] = -67596 05648962890998964864000000/44316842107967620047389533, d[20,1] = 0, d[ 19,1] = 0, d[16,1] = 0, d[4,7] = 0, d[11,3] = 190624748347582489361049 53664000/1171523229268973284610999593, d[3,8] = 0, d[16,7] = 0, d[1,3] = 347534193914087512379783004010097614066187/436022514505975558470156 4664346063053148, d[17,1] = 0, d[15,6] = 0, d[5,8] = 0, d[6,2] = -7857 1809756168216208295874341625/1658264456321306922311964338528, d[6,3] = 474557441368491493117494799926375/829132228160653461155982169264, d[9 ,3] = 7862923848493474757857848265456377738162649411387158211338000296 1083/9436965640053329933174083027017072783412875204813675574669745448, d[3,5] = 0, d[21,3] = -1230653835045296296664166992187500/30897047849 91495259525267620443, d[18,3] = -2804152816102015499727532000000000/22 533895771657324671486752673813, d[1,2] = -1229397799019134731969346832 8224507565801/968938921124390129933681036521347345144, d[8,1] = 0, d[7 ,6] = -2130021387193375899623870849609375000000000/4471246266173435476 32672005465987891367, d[21,2] = 860242915667386608280053222656250/2162 7933494940466816676873343101, d[7,4] = -892172690009639751748055416870 11718750000/49680514068593727514741333940665321263, d[10,3] = -6816886 40666963003271839903232/44316842107967620047389533, d[18,2] = 26798405 976250515382706000000000/2048535979241574970135159333983, d[6,5] = 294 106151443761066098669361328125/51820764260040841322248885579, d[8,8] = 8519851174354553222656250000000/569505189101437830803808753, d[11,8] \+ = -33721254136250000000000000000000/1171523229268973284610999593, d[9, 8] = -5216026296049769324716090545283915468267624870345468994140625000 0000/3538862115019998724940281135131402293779828201805128340501154543, d[7,5] = 595746511455163889253309936523437500000000/14904154220578118 2544224001821995963789, d[7,7] = 1105309694129066467285156250000000000 000/380346626117560436678661324372574461, d[10,7] = -26462220664660560 00000000000000/23862914981213333871671287, d[17,5] = -4720745728420553 1729045312500000000/8801620389428653604230810638813, d[1,8] = -8451618 5931732458401186928710937500000000/10900562862649388961753911660865157 63287, d[19,8] = 216435982877960205078125000000000000/2727410433678107 81000752899939849, d[17,2] = 2784325485192364296497558593750/889052564 58875288931624349887, d[13,2] = 46061553822559411296853774044/11265563 174528549307956420081, d[19,3] = -15646848301434568771156943359375000/ 38963006195401540142964699991407, d[1,6] = -18599642380275229084638491 32281928515625000/3270168858794816688526173498259547289861, d[9,2] = - 9579494258976377964149333038911010720368333776857565355677179211/13888 102487201368555075913211209820137472958358813356254112944, d[17,8] = 1 0295562851974487304687500000000000/8801620389428653604230810638813, d[ 7,8] = -106279778281641006469726562500000000000000/1490415422057811825 44224001821995963789, d[13,3] = -882668511649710364946121842837/177030 27845687720341074374413, d[19,5] = -1614016396242443510347929687500000 00/38963006195401540142964699991407, d[18,7] = 12075626274750237500000 0000000000000/202805061944915922043380774064317, d[11,7] = 96197296099 5631000000000000000000/8200662604882812992276997151, d[20,3] = -204529 838958329654721984960000000000000/877007656469732158703655785573133737 , d[18,5] = -21525911193488337801772000000000000/202805061944915922043 380774064317, d[6,6] = -350513952178631200295251464843750/518207642600 40841322248885579, d[9,7] = 148798625352670979480040858203423313346364 1882013192630896484375000000/24772034805139991074581967945919816056458 797412635898383508081801, d[20,6] = 2295390965255985094800000000000000 000000/877007656469732158703655785573133737, d[13,6] = 235240943326464 407232357812500000/371763584759442127162561862673, d[8,4] = 5721636731 7309194344917724609375/1518680504270500882143490008, d[1,4] = -1177791 740067350517980088002864999653352375/436022514505975558470156466434606 3053148, d[12,6] = 727740165467810382910156250/31630645474240238068038 9, d[13,5] = -27365323924871811347707753250000/53109083537063161023223 123239, d[11,4] = -84922595341263987542080296000000/117152322926897328 4610999593, d[9,6] = -104537737529964942564283282030186432891267422301 5001606895917773437500/10616586345059996174820843405394206881339484605 415385021503463629, d[19,6] = 1379318133517687339294433593750000000/27 2741043367810781000752899939849, d[11,6] = -67582934095754635510000000 0000000/3514569687806919853832998779, d[11,5] = 1890229716167391066141 28000000000/1171523229268973284610999593, d[10,6] = 805605758178114198 9600000000000/44316842107967620047389533, d[6,7] = 1496760753105893242 309570312500000/362745349820285889255742199053, d[8,7] = -267085163100 0518798828125000000/43808091469341371600292981, d[8,6] = 1707518166719 26852035522460937500/1708515567304313492411426259, d[21,4] = 353152853 73953236288663256347656250/21627933494940466816676873343101, d[7,2] = \+ -184208817072179529433273204345703125000/55200571187326363905268148822 96146807, d[20,2] = 1549581054327356288971680000000000000/797279687699 75650791241435052103067, d[17,4] = 18882660366917419511969273925781250 /8801620389428653604230810638813, d[21,7] = -1562752062538456176757812 50000000000/64883800484821400450030620029303, d[11,2] = -1578073606726 309755316838496000/1171523229268973284610999593, d[20,7] = -1363585033 062990000000000000000000000000/877007656469732158703655785573133737, d [8,2] = 236271732114439879127251953125/337484556504555751587442224, d[ 18,4] = 61497889686293713601431894000000000/20280506194491592204338077 4064317, d[20,5] = -1968652567641128005964608000000000000000/877007656 469732158703655785573133737, d[9,4] = -3502902461370010077437335571677 84850454404933346063990393209364646375/9436965640053329933174083027017 072783412875204813675574669745448, d[8,3] = -1284329050165669489593741 9921875/1518680504270500882143490008, d[18,8] = -421931759182812500000 00000000000000/202805061944915922043380774064317, d[13,4] = 2801401786 8690622217347948135375/123921194919814042387520620891, d[7,3] = 200264 95166832942406987130681152343750000/4968051406859372751474133394066532 1263, d[10,8] = 1205897769990000000000000000000/4431684210796762004738 9533, d[12,4] = 1463128747811842734792753375/1686967758626146030295408 , d[1,5] = 63013073818110058843644348837967685562500/12111736514054876 6241710129565168418143, d[19,4] = 496654275845501365651289536132812500 /272741043367810781000752899939849, d[6,4] = -211413522687192615147418 0451671875/829132228160653461155982169264, d[19,7] = -8620469225620679 93164062500000000000/272741043367810781000752899939849, d[17,7] = -399 57881329811364746093750000000000/8801620389428653604230810638813, d[12 ,5] = -203541930343956522262390625/105435484914134126893463, d[13,8] = 12693444727793574218750000000000/123921194919814042387520620891, d[21 ,5] = -32178068167446154884164843750000000/926911435497448577857580286 1329, d[8,5] = -47757642115025832702087402343750/569505189101437830803 808753, d[17,3] = -1225245774180874854780557617187500/2933873463142884 534743603546271, d[1,7] = 64209798218394956729271375871093750000000/19 5651128303963391621224055451425906231, d[18,6] = -31983577389585720775 000000000000000/67601687314971974014460258021439, d[19,2] = 9975495334 07946864651528320312500/30304560374201197888972544437761, d[20,4] = 89 9784412670111646707904480000000000000/87700765646973215870365578557313 3737, d[12,7] = -4909301586807617187500000/3497859689094497100731, d[6 ,8] = -52467846585220184326171875000000/51820764260040841322248885579, d[9,5] = 292382005285650271934444689761491362357752371977293918976649 939593750/353886211501999872494028113513140229377982820180512834050115 4543, d[17,6] = 20462401498211190052490234375000000/293387346314288453 4743603546271, d[10,4] = 276081526101167849809781568000/40288038279970 56367944503\}: " }{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([s eq([seq(d[j,i],j=1..11)],i=1..8)])):\nevalf[8](%);\nsubs(dd,matrix([se q([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- $!)%3)o7!\"'F+F+F+F+$!)a>QZF/$!)<3PLF/$\")V'4+(!\"&$!)ki(*oF6$\")6St7! \"%$!)s-Z8F;7-$\")ibqzF/F+F+F+F+$\")NaBdF6$\")j0JSF6$!)X(oX)F;$\")n/K$ )F;$!)d@Q:!\"$$\")I:F;FK7-$!)x@,FF6F+F+F+F+$!)n\")\\DF;$!)-#ez\"F;$\") _]nPFK$!)\\*=r$FK$\")Ap_oFK$!)T!*[sFK7-$\")dk-_F6F+F+F+F+$\")*\\an&F;$ \")V=(*RF;$!)L\"eQ)FK$\")W.i#)FK$!)0HD:!\"#$\")/[8;Fdo7-$!).n(o&F6F+F+ F+F+$!)s'Rw'F;$!).#Qw%F;$\")A;%***FK$!)KkY)*FK$\")?$y\"=Fdo$!)i$H#>Fdo 7-$\")<&=G$F6F+F+F+F+$\")D?ETF;$\")*eg!HF;$!)&3n4'FK$\")zr1gFK$!)m#*36 Fdo$\")I/t6Fdo7-$!)#yLv(F/F+F+F+F+$!)p[75F;$!)H)38(F6$\")%4g\\\"FK$!)v #RZ\"FK$\")C3@FFK$!)6TyGFKQ(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7,$\"\"!F)F(F(F(F(F(F(F(F(F(7,$\")No6;!\"'$\")Tq)3 %!\"(F(F(F($\")&*yJJF-$\")O<38F-$\")tu\"H$F-$\")`eV>F-$\")CYxRF-7,$!)o %o%>!\"&$!)[(f)\\F-F(F(F($!)_?wTF>$!)bTW7F>$!);#e,%F>$!)H8KBF>$!)!zI)R F>7,$\")\"GJn)F>$\")F(F(F($\")=OX@!\"%$\")YOKIF>$\")P(4#=FR$\")5 (f-\"FR$\")`&Gj\"FR7,$!)z[I>FR$!)Om_^F>F(F(F($!)i\\j`FR$!)!491\"F>$!)H VUTFR$!)#QZC#FR$!)k`rMFR7,$\")Su+BFR$\")FqFjF>F(F(F($\")W`upFR$!).=JZF >$\")XCd]FR$\")()HF(F(F($!)C$)RXFR$ \")AIafF>$!)\"z1;$FR$!)a\"[b\"FR$!)(R&3CFR7,$\")^%RW$F>$\")fJC5F>F(F(F ($\")\\tp6FR$!)%z/3#F>$\")aeNzF>$\")Rh+PF>$\")GWMeF>Q(pprint36\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can ch eck which of the groups of order conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "nm := NULL:\nfor ct to nops(SO8_21 ) do\n eqn_group := convert(SO8_21[ct],'polynom_order_conditions',8) :\n tt := expand(subs(\{op(e26),op(dd)\},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 "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,e26):" }}}{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..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, 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_21 := PrincipalErrorTerms(8,21,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "sm := 0:\nfo r 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 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. " }}{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$$\"3`*****\\n5;\"o!#@$\"3))*R0 TU`@V*!#G7$$\"3#******\\8ABO\"!#?$\"3ItZp+NePP!#F7$$\"33+++-K[V?F1$\"3 m#**eXG.3L)F47$$\"3#)******pUkCFF1$\"3!*Rj\"GwQrY\"!#E7$$\"3s*****\\Sm p3%F1$\"3'pmg(>AFRKF?7$$\"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$$\"33+++XTFwSFS$\"3'=sB z#G)3*=F_o7$$\"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$$ 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[!)F_o7$$\"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$$\"37+++l*RRL )Faq$\"37q@Jg$>`'zF_o7$$\"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\\*)Faq$\"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);\n findmax(ssm,u=0.48..0.58);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+c& zi6(!#5$\"+/EaS%*!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+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 14 to 2 1" }}{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 4856 "e5 := \{a[14,1] = 380/8289, c[2] = 1/16, c[3] = 112/1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, a[7,1] = 21400899/350000000, a[7, 2] = 0, a[7,3] = 0, a[7,4] = 3064329829899/27126050000000, a[6,2] = 0, a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a[5,3 ] = -352806597/250880000, a[5,4] = 178077159/125440000, a[6,1] = 12089 /252720, a[4,2] = 0, a[4,3] = 42/355, a[5,1] = 94495479/250880000, a[5 ,2] = 0, c[13] = 1, a[2,1] = 1/16, a[3,1] = 18928/1134225, a[3,2] = 10 0352/1134225, a[4,1] = 14/355, c[7] = 39/250, c[8] = 24/25, c[9] = 144 35868/16178861, c[10] = 11/12, c[11] = 19/20, c[12] = 1, a[9,6] = 7632 051964154290925661849798370645637589377834346780/173408725741881158304 9800347581865260479233950396659, a[9,5] = -875325048502130441118613421 785266742862694404520560000/170212030428894418395571677575961339495435 011888324169, a[9,7] = 75198347919711375170485321796523477298993035137 50000/1045677303502317596597890707812349832637339039997351, a[9,4] = - 14764960804048657303638372252908780219281424435/2981692102565021975611 711269209606363661854518, a[9,1] = -1840911252282376584438157336464708 426954728061551/2991923615171151921596253813483118262195533733898, a[9 ,2] = 0, a[9,3] = 0, a[8,7] = 1993321838240/380523459069, a[8,4] = -7/ 5, a[8,5] = -8339128164608/939060038475, a[8,6] = 341936800488/4795112 6225, a[7,5] = -21643947/592609375, a[7,6] = 124391943/6756250000, a[8 ,1] = -15365458811/13609565775, a[8,2] = 0, a[10,8] = 1908158550070998 850625/117087067039189929394176, a[10,9] = -52956818288156668227044990 077324877908565/2912779959477433986349822224412353951940608, a[11,1] = -10116106591826909534781157993685116703/95628199450368940304422314118 71744000, a[11,2] = 0, a[10,5] = -3378604805394255292453489375/5170426 70569824692230499952, a[10,6] = 1001587844183325981198091450220795/184 232684207722503701669953872896, a[10,7] = 1870230752313499007680148902 74453125/25224698849808178010752575653374848, a[10,1] = -6307773670525 4280154824845013881/78369357853786633855112190394368, a[10,2] = 0, a[1 0,3] = 0, a[10,4] = -31948346510820970247215/6956009216960026632192, a [12,10] = 7958341351371843889152/3284467988443203581305, a[12,6] = 182 7357820434213461438077550902273440/13938101391424531770956768083964169 7, a[12,7] = 643504802814241550941949227194107500000/24212460911883655 0860494007545333945331, a[12,8] = 162259938151380266113750/59091082835 244183497007, a[12,9] = -230282516328735238185454148568570156166785755 54130463402/20013169183191444503443905240405603349978424504151629055, \+ a[12,3] = 0, a[12,4] = 26290092604284231996745/5760876126062860430544, a[12,5] = -697069297560926452045586710000/411079677552454305940365023 19, a[12,2] = 0, a[12,1] = -3218022174758599831659045535578571/1453396 753634469525663775847094384, a[11,8] = 39747262782380466933662225/1756 032802431424164410720256, a[11,9] = 4817577141926095533524468380517154 8038966866545122229/19897864205138151465288801659520641189038528436121 60000, a[11,10] = -2378292068163246/47768728487211875, a[11,7] = 22264 55130519213549256016892506730559375/3648804431596752555774356483800473 55776, a[11,4] = -9623541317323077848129/3864449564977792573440, a[11, 5] = -4823348333146829406881375/576413233634141239944816, a[13,9] = 40 5932030463777247926705030596175437402459637909765779/78803919436321841 083201886041201537229769115088303952, a[13,10] = -10290327637248/10820 76946951, a[13,11] = 863264105888000/85814662253313, a[13,6] = 1432721 9974204125/40489566827933216, a[13,7] = 2720762324010009765625000/1091 7367480696813922225349, a[13,8] = -498533005859375/95352091037424, a[1 3,5] = 0, a[13,2] = 0, a[13,3] = 0, a[13,4] = 0, a[13,1] = 46316748798 41/103782082379976, a[12,11] = -507974327957860843878400/1215556548191 79042718967, a[14,4] = 0, c[14] = 1/2-1/14*7^(1/2), a[14,5] = 0, a[14, 2] = 0, a[14,3] = 0, a[14,13] = 3/392-3/392*7^(1/2), a[14,12] = 110061 3127343/48439922837376-29746300739/1424703612864*7^(1/2), a[13,12] = - 29746300739/247142463456, a[11,6] = 6566119246514996884067001154977284 529/970305487021846325473990863582315520, a[11,3] = 0, a[8,3] = 0, a[9 ,8] = 1366042683489166351293315549358278750/14463141822426771816505532 6464180836641, a[14,8] = 12730301828542984375/4462520239258570944-2313 9795371828125/21803844817224288*7^(1/2), a[14,11] = -31505155313806480 0/67942629698767761+392422111155400/221311497390123*7^(1/2), a[14,9] = -2431300645813356820990854267156732219942356459331979/218981841633277 9722313164167138490920989148784630720+49339902868206221533385089432866 304648514986421/125875747250399294250613000601944295132543520864*7^(1/ 2), a[14,10] = 21310748482922848/7398947051356315-5227019004392/482016 0945509*7^(1/2), a[14,6] = 9785567316447209725/49721188064701989248-34 388281046170925/566853935591065024*7^(1/2), a[14,7] = 1016182669829177 731250000000/4211024590054414662359895513-23208493423698876953125/1371 6692475747279030488259*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 := Si mpleOrderConditions(7,15,'expanded'):\nSO7_16 := SimpleOrderConditions (7,16,'expanded'):\nSO7_17 := SimpleOrderConditions(7,17,'expanded'): \nSO7_18 := SimpleOrderConditions(7,18,'expanded'):\nSO7_19 := SimpleO rderConditions(7,19,'expanded'):\nSO7_20 := SimpleOrderConditions(7,20 ,'expanded'):\nSO8_21 := SimpleOrderConditions(8,21,'expanded'):\nerrt erms8_21 := PrincipalErrorTerms(8,21,'expanded'):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1799 "whch := [1 ,2,4,8,16,21,27,31,32,64]:\ninterp_order_eqns15 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_14[ct],'interpolation_order_condition' ):\n interp_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],'interpolati on_order_condition'):\n interp_order_eqns16 := [op(interp_order_eqns 16),temp_eqn];\nend do:\nwhch := [1,2,4,8,16,17,25,27,32,61,63,64]:\ni nterp_order_eqns17 := []:\nfor ct in whch do\n temp_eqn := convert(S O7_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_order_eqns18 := [op(interp_order_eqns18),temp_eqn];\nend do:\nw hch := [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:\nwhch := [1,2,4,8,16,17,25,27,32,61,63,64]:\ninte rp_order_eqns20 := []:\nfor ct in whch do\n temp_eqn := convert(SO7_ 19[ct],'interpolation_order_condition'):\n interp_order_eqns20 := [o p(interp_order_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 te mp_eqn := convert(SO7_20[ct],'interpolation_order_condition'):\n int erp_order_eqns21 := [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],'polynom_order_conditions',8):\n ordeqns := [op(ordeqns),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 1960 "calc_coeffs := proc()\n \+ local eqns,pols,e_u,eu,ct,eqs_15,eqs_16,eqs_17,eqs_18,eqs_19,eqs_20,eq s_21;\n global 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`(e 5,\{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 := `uni on`(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,i nterp_order_eqns17)):\n e13 := solve(\{op(eqs_17)\}):\n e14 := `un ion`(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 e1 8 := `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 := `unio n`(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_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(subs(e24,interp_order_eqns21)):\n e25 := solve(\{op(eq s_21)\}):\n e26 := `union`(e24,e25):\n eqns := []:\n for ct to n ops(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 s m := 0:\n for ct to 286 do\n sm := sm+expand(subs(eu,errterms8_ 21[ct]))^2;\n end 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 0 195 "a14_1 := 380/8289:\nc_15 := 4748/10000:\nc_16 := 1108/10000:\nc_17 := 3972/10000:\nc_18 := 4531/10000:\nc_19 := 709 2/10000:\nc_20 := 9020/10000:\nc_21 := 1008/10000:\ncalc_coeffs();\nss mA := sqrt(sm)*u^9:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"#:# \"%(=\"\"%+D/&F%6#\"#;#\"$x#F*/&F%6#\"#<#\"$$**F*/&F%6#\"#=#\"%JX\"&++ \"/&F%6#\"#>#\"%t " 0 "" {MPLTEXT 1 0 136 "c_17 := 3972/10000:\nc_18 := 4530/10000:\nc_19 := 7116/10000:\nc_ 20 := 9029/10000:\nc_21 := 1012/10000:\ncalc_coeffs();\nssmB := sqrt(s m)*u^9:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"#:#\"%(=\"\"%+D/ &F%6#\"#;#\"$x#F*/&F%6#\"#<#\"$$**F*/&F%6#\"#=#\"$`%\"%+5/&F%6#\"#>#\" %z " 0 "" {MPLTEXT 1 0 136 "c_17 := 397 1/10000:\nc_18 := 4531/10000:\nc_19 := 7164/10000:\nc_20 := 9046/10000 :\nc_21 := 1032/10000:\ncalc_coeffs();\nssmC := sqrt(sm)*u^9:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"cG6#\"#:#\"%(=\"\"%+D/&F%6#\"#;# \"$x#F*/&F%6#\"#<#\"%rR\"&++\"/&F%6#\"#=#\"%JXF7/&F%6#\"#>#\"%\"z\"F*/ &F%6#\"#?#\"%BX\"%+]/&F%6#\"#@#\"$H\"\"%]7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot([ssmA,ssmB,ss mC],u=0..1,color=[blue,brown,magenta]);" }}{PARA 13 "" 1 "" {GLPLOT2D 553 374 374 {PLOTDATA 2 "6'-%'CURVESG6$7dq7$$\"3`*****\\n5;\"o!#@$\"3# QU,u,%3f#*!#G7$$\"3#******\\8ABO\"!#?$\"3S(QzG7**)oO!#F7$$\"33+++-K[V? 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6#)\\,CY?F\\\\l/&F&6$FgoF=#!D++ +++++]x?de*QxN)>$\"AR9-e-Y9S(>(\\J(o,w'/&F&6$F[pF=#\"F+++]PfLWHRto/&F&6$F,FAF-/&F&6$F1FAF-/&F&6$F5FAF-/&F&6$ F9FAF-/&F&6$F=FA#\"C++]7.d4BC$*e5`2w'\\\"\"?`!*>UdD*)eG?)\\`ui$/&F&6$F AFA#\"I++++++]i:&Gnk1HTp4`5\"\"EhWdsVKh'ymVgv6EmM!Q/&F&6$FEFA#!@+++D\" G))z=0+J;&3n#\";\")HH+;PT$p943Q%/&F&6$FIFA#\"ao+++vV['*3j#>8?)=kjM8LU. #e3/![z4n_`i)z[\"\"\\o,=33NQ)*ej7uzek0;)>f%z'>eu5**R^![.sZ#/&F&6$FMFA# !@+++++++g0mk1Aik#\";(Gr;(QL87)\\\"H'Q#/&F&6$FQFA#\"B+++++++++Jc*4'H(> '*\"=^r*pF#*H\"G)[gi1?)/&F&6$FUFA#!:++](=&)Fgel/&F&6$F IFE#!_o+++]iST**oaMq[in#oa\"RGX04;ZKp(\\gHEg@&F]fl/&F&6$FMFE#\"@++++++ +++!**px*e?\"F\\z/&F&6$FQFE#!A+++++++++]i8a7sLFis/&F&6$FUFE#\";+++v$4r GpAS6j$F]gl/&F&6$FYFE#\"A+++++v=Ud$zFZW$p7F^al/&F&6$FgnFEF-/&F&6$F[oFE F-/&F&6$F_oFEF-/&F&6$FcoFE#\"D+++++](o/t[u>&Gc&H5F]bl/&F&6$FgoFE#!D+++ ++++++]7G=f<$>UFcbl/&F&6$F[pFE#\"E++++++D\"y]?gz(G)fV;#Fibl/&F&6$F_pFE #\"H++++++++++++](Q,nYXKF^]l/&F&6$FcpFE#\"D+++++](o/t[*f " 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 .. 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 l inking coefficients: ee" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24598 "ee := \{c[2] = 1/16, c[3] = 112/ 1065, c[4] = 56/355, c[5] = 39/100, c[6] = 7/15, c[7] = 39/250, c[8] = 24/25, c[9] = 14435868/16178861, c[10] = 11/12, c[11] = 19/20, c[12] \+ = 1, c[13] = 1, c[14] = 1/2-1/14*7^(1/2), c[15] = 1187/2500, c[16] = 2 77/2500, c[17] = 993/2500, c[18] = 453/1000, c[19] = 1779/2500, c[20] \+ = 9029/10000, c[21] = 253/2500,\na[2,1] = 1/16, a[3,1] = 18928/1134225 , a[3,2] = 100352/1134225, a[4,1] = 14/355, a[4,2] = 0, a[4,3] = 42/35 5, a[5,1] = 94495479/250880000, a[5,2] = 0, a[5,3] = -352806597/250880 000, a[5,4] = 178077159/125440000, a[6,1] = 12089/252720, a[6,2] = 0, \+ a[6,3] = 0, a[6,4] = 2505377/10685520, a[6,5] = 960400/5209191, a[7,1] = 21400899/350000000, a[7,2] = 0, a[7,3] = 0, a[7,4] = 3064329829899/ 27126050000000, a[7,5] = -21643947/592609375, a[7,6] = 124391943/67562 50000, a[8,1] = -15365458811/13609565775, a[8,2] = 0, a[8,3] = 0, a[8, 4] = -7/5, a[8,5] = -8339128164608/939060038475, a[8,6] = 341936800488 /47951126225, a[8,7] = 1993321838240/380523459069, a[9,1] = -184091125 2282376584438157336464708426954728061551/29919236151711519215962538134 83118262195533733898, a[9,2] = 0, a[9,3] = 0, a[9,4] = -14764960804048 657303638372252908780219281424435/298169210256502197561171126920960636 3661854518, a[9,5] = -875325048502130441118613421785266742862694404520 560000/170212030428894418395571677575961339495435011888324169, a[9,6] \+ = 7632051964154290925661849798370645637589377834346780/173408725741881 1583049800347581865260479233950396659, a[9,7] = 7519834791971137517048 532179652347729899303513750000/104567730350231759659789070781234983263 7339039997351, a[9,8] = 1366042683489166351293315549358278750/14463141 8224267718165055326464180836641, a[10,1] = -63077736705254280154824845 013881/78369357853786633855112190394368, a[10,2] = 0, a[10,3] = 0, a[1 0,4] = -31948346510820970247215/6956009216960026632192, a[10,5] = -337 8604805394255292453489375/517042670569824692230499952, a[10,6] = 10015 87844183325981198091450220795/184232684207722503701669953872896, a[10, 7] = 187023075231349900768014890274453125/2522469884980817801075257565 3374848, a[10,8] = 1908158550070998850625/117087067039189929394176, a[ 10,9] = -52956818288156668227044990077324877908565/2912779959477433986 349822224412353951940608, a[11,1] = -101161065918269095347811579936851 16703/9562819945036894030442231411871744000, a[11,2] = 0, a[11,3] = 0, a[11,4] = -9623541317323077848129/3864449564977792573440, a[11,5] = - 4823348333146829406881375/576413233634141239944816, a[11,6] = 65661192 46514996884067001154977284529/970305487021846325473990863582315520, a[ 11,7] = 2226455130519213549256016892506730559375/364880443159675255577 435648380047355776, a[11,8] = 39747262782380466933662225/1756032802431 424164410720256, a[11,9] = 4817577141926095533524468380517154803896686 6545122229/1989786420513815146528880165952064118903852843612160000, a[ 11,10] = -2378292068163246/47768728487211875, a[12,1] = -3218022174758 599831659045535578571/1453396753634469525663775847094384, a[12,2] = 0, a[12,3] = 0, a[12,4] = 26290092604284231996745/5760876126062860430544 , a[12,5] = -697069297560926452045586710000/41107967755245430594036502 319, a[12,6] = 1827357820434213461438077550902273440/13938101391424531 7709567680839641697, a[12,7] = 643504802814241550941949227194107500000 /242124609118836550860494007545333945331, a[12,8] = 162259938151380266 113750/59091082835244183497007, a[12,9] = -230282516328735238185454148 56857015616678575554130463402/2001316918319144450344390524040560334997 8424504151629055, a[12,10] = 7958341351371843889152/328446798844320358 1305, a[12,11] = -507974327957860843878400/121555654819179042718967, a [13,1] = 4631674879841/103782082379976, a[13,2] = 0, a[13,3] = 0, a[13 ,4] = 0, a[13,5] = 0, a[13,6] = 14327219974204125/40489566827933216, a [13,7] = 2720762324010009765625000/10917367480696813922225349, a[13,8] = -498533005859375/95352091037424, a[13,9] = 405932030463777247926705 030596175437402459637909765779/788039194363218410832018860412015372297 69115088303952, a[13,10] = -10290327637248/1082076946951, a[13,11] = 8 63264105888000/85814662253313, a[13,12] = -29746300739/247142463456, a [14,1] = 380/8289, a[14,2] = 0, a[14,3] = 0, a[14,4] = 0, a[14,5] = 0, a[14,6] = 9785567316447209725/49721188064701989248-34388281046170925/ 566853935591065024*7^(1/2), a[14,7] = 1016182669829177731250000000/421 1024590054414662359895513-23208493423698876953125/13716692475747279030 488259*7^(1/2), a[14,8] = 12730301828542984375/4462520239258570944-231 39795371828125/21803844817224288*7^(1/2), a[14,9] = -24313006458133568 20990854267156732219942356459331979/2189818416332779722313164167138490 920989148784630720+49339902868206221533385089432866304648514986421/125 875747250399294250613000601944295132543520864*7^(1/2), a[14,10] = 2131 0748482922848/7398947051356315-5227019004392/4820160945509*7^(1/2), a[ 14,11] = -315051553138064800/67942629698767761+392422111155400/2213114 97390123*7^(1/2), a[14,12] = 1100613127343/48439922837376-29746300739/ 1424703612864*7^(1/2), a[14,13] = 3/392-3/392*7^(1/2), a[15,1] = 13094 6139152859396534950567713097/2815269161783203125000000000000000+906709 44595916412828478989403/66148981703437500000000000000000*7^(1/2), a[15 ,2] = 0, a[15,3] = 0, a[15,4] = 0, a[15,5] = 0, a[15,6] = 335394319040 2140976803568136084793/19770296302701765625000000000000000-25574898891 7794999289135717669191/8649504632432022460937500000000000*7^(1/2), a[1 5,7] = 10596614540600968744792362758669/454890311695700580092722875000 00-13058425932946007192467094947/1451071902055187778276093750000*7^(1/ 2), a[15,8] = 16161913072172934315785836819/99325094830650000000000000 00-75980668436324671626317237501/49917227145660000000000000000*7^(1/2) , a[15,9] = -372785014307838622905807656152260936782776633069641710843 5354320431359/80163492265138591596681606080323829375985834847314404296 87500000000000+8142513497965121274481241741350161529949699991737637172 29208919303/1589842118264019833619870377392538080518560957031250000000 000000000*7^(1/2), a[15,10] = 14462059099710033235500874587/1031948992 6824569702148437500-348155288848452187540396337457/2525749282789230346 67968750000*7^(1/2), a[15,11] = -207941800836756311792092915907/818392 39362061500549316406250+10891006522099224595926000559657/4582997404275 444030761718750000*7^(1/2), a[15,12] = 12163051029345238407733418657/3 016875774609375000000000000000, a[15,13] = -579945478542537523/5493164 06250000000000+3727856565327672037/2197265625000000000000*7^(1/2), a[1 5,14] = 26094995957293704259/549316406250000000000*7^(1/2), a[16,1] = \+ 1013856404118125952763527205161498788180173939254661359435974309585825 318269273/198564667739239660374389171210942338307459063268324464130246 79687500000000000000-1931754243710985131731267585181486489297591530273 47863824763410934840962491/2778932721896691403250310788177330670569727 48113814414737187500000000000000000*7^(1/2), a[16,2] = 0, a[16,3] = 0, a[16,4] = 0, a[16,5] = 0, a[16,6] = -19039890149866495432584929203443 76660766478959434824857368133995942073946993/2610549880447098448451329 9908841941049181178531837828644206625000000000000000+28929035836992861 158293213150542073159989024626464512663622554495682276029/892277791168 441852498013180477996031954434813099925783737531127929687500000*7^(1/2 ), a[16,7] = 147164997510791325111769333459204570445091856853535796848 57180786483042792373/1608898833767467146570215659505253270165679923696 70063678825639487979375000000+3509716113154202879194528857399000228802 41691060058741934303404427219789/7486803689896095933707556250309345701 6256710599371634570393697387568750000*7^(1/2), a[16,8] = -285913537824 92183701826251126306766396461229874660236919089505718702483/3073895442 2006868689408605302697869750592763266032965939343750000000000+32621718 6457072779972278244440069764006022537683484345700377402630409/17437228 97253444403020583118152408439159931363492594965600000000000000*7^(1/2) , a[16,9] = 2079043420349691709006286724449198371945091963358485311941 3284106748179318736252691355190285977916572557692108955233/15877667583 5946596689535645887725453057778073174418932018795803586671680545234558 493166228541026757812500000000000000+167502919171284871193499988335765 6040011164360189191231613203552972323338878927017521218877046934971369 718057209/731810193153334108198682812322026779525561489528186024119552 674967375646957459693466445168584156250000000000000000*7^(1/2), a[16,1 0] = -1752032306988852430152559893574903103295734904546828540573516708 228831279/243326283010247711363961558099490763015363068164171983079910 2783203125000+81883999666586172691735974039324906939964446697501450842 550606413648603/733722330307823868112868698269233685400171713233503210 517883300781250000*7^(1/2), a[16,11] = 3307587237786214580372394397330 733034806289881443017455732775116590947887/216127682925333604166493245 6066917395909700497105040226143341064453125000-15924948134835543062332 1425412532136521666854903188329515308104047100201/53083992297450358918 0860603244506027065540472973167774842224121093750000*7^(1/2), a[16,12] = -124817112626650602913170312644481143348777609285630839757851632016 681433/398360478373506965815942720780751835664786233698316038437500000 0000000000+19488568273447271249669733067688159410742449986073234566291 555857980323/290471182147348829240791567235964880172239962071688778027 3437500000000000*7^(1/2), a[16,13] = -46330543544553540874179448395431 083082899647522368296822645347/508745135208312439739125075909694092175 8056970520019531250000000+18909196831867663116267125639920175653811982 75789277564189007889/1139589102866619865015640170037714766473804761396 484375000000000000*7^(1/2), a[16,14] = 3780707924708291093575043739753 52653612596622387047615352035143989/3865246572631166547421340688737118 7527810692671413726806640625000000-73669448511920375490054459319905895 5707390426548138628337098463663/24737578064839465903496580407917560017 798843309704785156250000000000*7^(1/2), a[16,15] = 4608882876307429368 775251443876403372924872509408621400283/759222789493608716485758601053 18720457058558153486785997500-8491470298642874191233142902597241747531 2941024930181294/53145595264552610154003102073723104319940990707440750 19825*7^(1/2), a[17,1] = 142969271097328003658032802388115489924269556 60898056576097/3414135267376060159135042016078362394646603515625000000 00000-35498109931158869858570877257888472099215653607797737/1047152271 9224819528692927297504485322802734375000000000*7^(1/2), a[17,2] = 0, a [17,3] = 0, a[17,4] = 0, a[17,5] = 0, a[17,6] = 1168094170732515767795 5382237279696768268247094386992999/87503361475665632084979209131119394 314606250000000000000-997172433570276305925254457325512017866796579074 407189/21875840368916408021244802282779848578651562500000000000*7^(1/2 ), a[17,7] = 7292471180619236115859359700346738458665766464861503/4474 1004683259221639235793273762127339404272876630000-39568816155893758057 7353827374940476509802536449201/71700328018043624421852232810516229710 58377063562500*7^(1/2), a[17,8] = 158994316538680760818928006714630116 934243743639/488458911821805651705273894751044258600000000000-64764040 35945812133836473370020724114834745161/6784151553080634051462137427097 8369250000000000*7^(1/2), a[17,9] = -777365420445936318089732417738641 0045148764290716888867400388432148819282908940948159254319/78845275237 6981215609864480482093541291983982324119451546116914378375650249609375 000000000000-349211535189884795491496750852447695276558730568496460079 234407077243879427767491465619/350423445500880540271050880214263796129 77065881071975624271862861261140011093750000000000*7^(1/2), a[17,10] = 513652394689855627502026559415560286064929495444927/22837039048019223 43978476111178519401359558105468750-2108466842104252676352514760416833 25933811084598/4152188917821676989051774747597308002471923828125*7^(1/ 2), a[17,11] = -259187471300326360195328568552656084453334237831807/48 2960797022440548690819027452576858488464355468750+19532790685141980036 42387066372792263574848990934/1270949465848527759712681651190991732864 3798828125*7^(1/2), a[17,12] = 262214308744442251330599379201914827178 216725701161/22254494604004075189901336249265303906250000000000000-404 2541431743211698640695349601479330547680120287/11127247302002037594950 66812463265195312500000000000*7^(1/2), a[17,13] = 55038056541544477790 82385289578389472169301/1734849960808438372162663307568359375000000000 -44976428168893445098944962904416522821649/526501941230926308859462121 58775329589843750*7^(1/2), a[17,14] = 88988313395506756310627650962047 52299374508560508163/1749782565764184901107853213039695442687988281250 00000+343000404310309138700676556673479752657859745169/699913026305673 9604431412852158781770751953125000*7^(1/2), a[17,15] = -77127127250912 159388465163390223095790721/119229943754928152450236266633550943965000 0+19577936100246207024301582455131137389/92632494763444179980568176384 5280410805*7^(1/2), a[17,16] = 312951003846638093776403168468573978963 277/5815189069428367546448931343495076666050000+1835267423444683727710 449934904130527/44732223610987442664991779565346743585*7^(1/2), a[18,1 ] = 7733624778322431132379739805371154872928342814315555219/1816142604 56783667685936780491648991798344000000000000000-7549168255858056864842 5894347272875262003943330267987/18872197105495738747391541292654483628 168000000000000000*7^(1/2), a[18,2] = 0, a[18,3] = 0, a[18,4] = 0, a[1 8,5] = 0, a[18,6] = 69454891827986654905811794573547009524702368493567 653/512019669434752041457249772287235770160896000000000000-19278420727 5057942900067020977105865032917681968026949/35841376860432642902007484 06010650391126272000000000000*7^(1/2), a[18,7] = 428930755082588455942 724722283447907984915278705435/261798793118042531191871156070470962260 2855738609664-76498733805890742729378834290483857132585978507041/11747 38174247626742527626982367497907578204498094080*7^(1/2), a[18,8] = 542 6224836754120276540228725914955825217338781/20007277028221159493848018 729002772832256000000-96314521713121628743334127199055813316326877/855 011838812870063839658920042853539840000000*7^(1/2), a[18,9] = 12403847 0496245645770949475501124572057558878067871133108876781024786308880699 9332515507/30553476572716320332431456121614523605789653657517438727841 957249705077232000000000000000-447108394135293641224551688365209147458 637766147319719227389653577377825685260302229/380221041793803097470258 12062453629376093791218243923750203324577410762777600000000000*7^(1/2) , a[18,10] = 78813744026468418092863662132343791904150881627/607405921 692771033827002477362806199218750000000-172459571507530930187559760312 05721533364483773/287816959817497659105718096965760475937500000000*7^( 1/2), a[18,11] = -611946677966621853095211488458472723282243226327/141 3005303288511776741139097461253437406250000000+18881456412820846030979 8549660171819794928530747/10411618024231139407566288086556604275625000 00000*7^(1/2), a[18,12] = 90472372922286374676335230539869500651156243 17/862387983897830576895325196565664000000000000000-517581473014426522 7514848919003486791055011817/12073431774569628076534552751919296000000 00000000*7^(1/2), a[18,13] = 84303053727623559901892597740221920795377 /35529727197356817861891344539000000000000000-668871667581022998424770 230088072148293/663554446548268972642571338468750000000000*7^(1/2), a[ 18,14] = 1720418743754626973815631730013574951008671541792083/28668437 557480405419751067042442370133000000000000000+663125642204459443702659 39039199371363080688129/1146737502299216216790042681697694805320000000 000*7^(1/2), a[18,15] = -254618824596430425581156792109862619835/12209 146240504642810904193703275616662016+181972080991961001018895701030477 170625/7296590356751295407700139432135131851264*7^(1/2), a[18,16] = 15 01373575980631550465262783336687863485/2977376803547324183781852847869 4792530176+17058357454582951976373885014055586875/35235228444347031760 7319863653192811008*7^(1/2), a[18,17] = 0, a[19,1] = 18778954190432498 641397367225501777310616443565686827961261/341413526737606015913504201 607836239464660351562500000000000+323797936373097675108367768125033753 2341535825485533/11753749990966634164859408191076463117431640625000000 00*7^(1/2), a[19,2] = 0, a[19,3] = 0, a[19,4] = 0, a[19,5] = 0, a[19,6 ] = 28179885214710124253594866402805577138221840971048257987/875033614 75665632084979209131119394314606250000000000000+8268874976067429137680 756848115769916895601168366091/223222860907310285931069411048773965088 281250000000000*7^(1/2), a[19,7] = 18039033114928576144292936355618923 845713121206510289/447410046832592216392357932737621273394042728766300 00+3281173673971439820995171767975524126665743950319/73163600018411861 654951257969914520112840582281250*7^(1/2), a[19,8] = 54585293581684876 3549582609751203523712604801307/48845891182180565170527389475104425860 0000000000+7894550931548752199808312441931046629102173173/101762273296 209510771932061406467553875000000000*7^(1/2), a[19,9] = -2414064062661 1869758219261752350837961521246348971485476648481626009901547285237073 2495500147/78845275237698121560986448048209354129198398232411945154611 6914378375650249609375000000000000+86873235585153923845165431213316048 24498730528284220439682172692980305015008846423783/1072724833165960837 564441470043664682029910180032815580334852944732483877890625000000000* 7^(1/2), a[19,10] = 24980019782596890246406905017075860246794377536485 51/2283703904801922343978476111178519401359558105468750+38464989471746 279636532325535691058627774079964/932124042776294834276929024970824245 452880859375*7^(1/2), a[19,11] = -927753364230868120383187286221818715 585159819007991/482960797022440548690819027452576858488464355468750-22 6761067735484045528048017173212371351053066444/18156420940693253710181 16644558559618377685546875*7^(1/2), a[19,12] = 81659556018390905473980 3824796340399773958361526093/22254494604004075189901336249265303906250 000000000000+33522055523510853377725329217930880599252333953/113543339 81634732239745579719012910156250000000000*7^(1/2), a[19,13] = 63298226 36137785234855809795487253116923767139/5256595381249568267652869821932 12890625000000000+54824977090333610968656925721307374512157/7897529118 4638946328919318238162994384765625*7^(1/2), a[19,14] = -36158067991345 03291043136722909584352046302608498253/8748912828820924505539266065198 4772134399414062500000-13936922127054510060095879177924083873997140783 9/3499565131528369802215706426079390885375976562500*7^(1/2), a[19,15] \+ = 201545429091434053176048871670250775086069/3440635519785069542135389 408568184382990000-2272851721861698000685159054540758474/1323321353763 48828543668823406468630115*7^(1/2), a[19,16] = -5922016057084312561252 3224803799201068911/498444777379574361124194115156720857090000-6391823 99323564197784537438328437546/19170952976137475427853619813720032965*7 ^(1/2), a[19,17] = 0, a[19,18] = 0, a[20,1] = 897330932052009060320705 19160411508217914220092982292430255743/1525559787836982808561868956129 851531106089600000000000000000000+234076470257947541762894493668275072 391709451556335985408969/679399095797846594906095486535561410614048000 00000000000000000*7^(1/2), a[20,2] = 0, a[20,3] = 0, a[20,4] = 0, a[20 ,5] = 0, a[20,6] = 647054740433591047501139839926183281833250508652625 223013947/143365507441730571608029936240426015645050880000000000000000 0+66418276131729569040513905514010579101397635988537328052807/14336550 74417305716080299362404260156450508800000000000000000*7^(1/2), a[20,7] = 5849126156722383963541665513374372113516941982763467945837/13194659 173149343572070306265951736497918392922592706560000+711597078575774626 289414881120414051622031127652898916601/126871722818743688192983714095 68977401844608579416064000000*7^(1/2), a[20,8] = -17936059277760954417 1797254107830289034341017967853947/72026197301596174177852867424409982 196121600000000000+349410771363369319140735005695090042067326461163496 83/360130986507980870889264337122049910980608000000000000*7^(1/2), a[2 0,9] = 350957869536825723773658657577191570597885286238759739828077612 374286734702886125053531002333688587/116262089054500142128968176833967 585224750790097585357847184215726577759883206400000000000000000000+188 4043540249331018790584629707285984800654018527960079175320009299480163 532738452671310798657/186019342487200227406349082934348136359601264156 136572555494745162524415813130240000000000000000*7^(1/2), a[20,10] = - 8838763885586903327477514842388396732758843333309547477/17007365807397 58894715606936615857357812500000000000000+7724080662523013714735975132 2705182070781822501923107/14966481910509878273497341042219544748750000 00000000000*7^(1/2), a[20,11] = 58803614160031168862776673639561218650 451486878573351031/118692445476234989246255684186745288742125000000000 00000-195151947898929484657927742905600534538199744564712363/124939416 2907736728907954570386792513075000000000000000*7^(1/2), a[20,12] = -54 175810754967243498377218440527251354798928847932480401/109385291877600 8303734030479323888217600000000000000000000+80777995073373267105869591 4872912518198545978092589343/21877058375520166074680609586477764352000 0000000000000000*7^(1/2), a[20,13] = -24800779068457870408513668358212 77889558088410267941/1162674792806304507712532358694236000000000000000 00000+242653922955367016465634203061511070435162932747/279489132886130 891277051047763037500000000000000000*7^(1/2), a[20,14] = -160034668689 97045499992466728944319268578868436842768854363/3096191256207883785333 11524058377597436400000000000000000000-6168445492427758485270907087123 35253938511951736215769/1238476502483153514133246096233510389745600000 0000000000*7^(1/2), a[20,15] = -60108815292626420255491361488348250991 962662149/986499016232775139121058851224669826290892800000-13040199690 83779956900885316764115910875407/6070763176817077779206516007536429700 2516480*7^(1/2), a[20,16] = -58959967558221710656211400321861759969650 5263937/3858680337397332142181281290838845111910809600000-990151637428 2546563618749633979325112870981/23745725153214351644192500251315969919 4511360*7^(1/2), a[20,17] = 0, a[20,18] = 0, a[20,19] = 0, a[21,1] = 3 0997081338430842970238050305554363685394154010924527196929/83801502017 4123857242237585764688951413257226562500000000000+45653360920018349732 18894721195940768038738096387/2137045452903024393610801489286629657714 84375000000000*7^(1/2), a[21,2] = 0, a[21,3] = 0, a[21,4] = 0, a[21,5] = 0, a[21,6] = -33098372644113630504863314574774471907135574827406634 01/87503361475665632084979209131119394314606250000000000000+1282442166 57060373956178084993533317618563230091239/4464457218146205718621388220 97547930176562500000000000*7^(1/2), a[21,7] = -60428090962983762028102 222659035092334443807259518019/120800712644799898425936641839157743816 3915367669010000+50888609242748421551480430147730676944005130651/14632 7200036823723309902515939829040225681164562500*7^(1/2), a[21,8] = -290 2819118865630209141335036817092955423296512547/13188390619188752596042 395158278194982200000000000+122438723890010359901658026695190975148385 817/203524546592419021543864122812935107750000000000*7^(1/2), a[21,9] \+ = 23732329796428448946843245175452232812803764034385082866772338070762 88410234603005442258460187/2128822431417849282146634097301652561488356 7522751225191745156688216142556739453125000000000000+13473404880745746 6338865854369659975898277761095422843616676876929084259549554219507/21 4544966633192167512888294008732936405982036006563116066970588946496775 5781250000000000*7^(1/2), a[21,10] = -57276921633460666879267544795444 795665922000283843/207609445891083849452588737379865400123596191406250 +27116541674232832942490198536154073869071698/847385493432995303888117 29542802204132080078125*7^(1/2), a[21,11] = 56957259697860176155424674 7325447992832658077510179/14488823910673216460724570823577305754653930 66406250-1758449340703794994640531807795275987505836638/18156420940693 25371018116644558559618377685546875*7^(1/2), a[21,12] = -4353666069118 87300278335427807133982215783539255317/6676348381201222556970400874779 5911718750000000000000+519902618408155831782878999596105234295116437/2 2708667963269464479491159438025820312500000000000*7^(1/2), a[21,13] = \+ -1482763015196564094793258170458172355589102597/5256595381249568267652 86982193212890625000000000+850295386076203794036039140204173286753/157 950582369277892657838636476325988769531250*7^(1/2), a[21,14] = -560785 24815090382874736120910671348469312408461137/1749782565764184901107853 21303969544268798828125000000-2161514917044480886135528855393203658546 310731/6999130263056739604431412852158781770751953125000*7^(1/2), a[21 ,15] = 672440045006965423277162352231808676922717/17203177598925347710 676947042840921914950000-17625135794859273212919783295280073/132332135 376348828543668823406468630115*7^(1/2), a[21,16] = 2841955996551750120 50679684897584806134113/2492223886897871805620970575783604285450000-49 56626284680834260320274687068217/1917095297613747542785361981372003296 5*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 "int erpolation coefficients: dd" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9053 "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[21,1] = 0, d[1,2] = -1229397799019134731969346832822 4507565801/968938921124390129933681036521347345144, d[2,2] = 0, d[3,2] = 0, d[4,2] = 0, d[5,2] = 0, d[6,2] = -785718097561682162082958743416 25/1658264456321306922311964338528, d[7,2] = -184208817072179529433273 204345703125000/5520057118732636390526814882296146807, d[8,2] = 236271 732114439879127251953125/337484556504555751587442224, d[9,2] = -957949 4258976377964149333038911010720368333776857565355677179211/13888102487 201368555075913211209820137472958358813356254112944, d[10,2] = 5130283 264356121609520487168/4028803827997056367944503, d[11,2] = -1578073606 726309755316838496000/1171523229268973284610999593, d[12,2] = 36011365 5316927278109563/22343943822862861328416, d[13,2] = 460615538225594112 96853774044/11265563174528549307956420081, d[14,2] = 0, d[15,2] = 0, d [16,2] = 0, d[17,2] = 2784325485192364296497558593750/8890525645887528 8931624349887, d[18,2] = 26798405976250515382706000000000/204853597924 1574970135159333983, d[19,2] = 997549533407946864651528320312500/30304 560374201197888972544437761, d[20,2] = 1549581054327356288971680000000 000000/79727968769975650791241435052103067, d[21,2] = 8602429156673866 08280053222656250/21627933494940466816676873343101, d[1,3] = 347534193 914087512379783004010097614066187/436022514505975558470156466434606305 3148, d[2,3] = 0, d[3,3] = 0, d[4,3] = 0, d[5,3] = 0, d[6,3] = 4745574 41368491493117494799926375/829132228160653461155982169264, d[7,3] = 20 026495166832942406987130681152343750000/496805140685937275147413339406 65321263, d[8,3] = -12843290501656694895937419921875/15186805042705008 82143490008, d[9,3] = 786292384849347475785784826545637773816264941138 71582113380002961083/9436965640053329933174083027017072783412875204813 675574669745448, d[10,3] = -681688640666963003271839903232/44316842107 967620047389533, d[11,3] = 19062474834758248936104953664000/1171523229 268973284610999593, d[12,3] = -328426784686623935099769171/16869677586 26146030295408, d[13,3] = -882668511649710364946121842837/177030278456 87720341074374413, d[14,3] = 0, d[15,3] = 0, d[16,3] = 0, d[17,3] = -1 225245774180874854780557617187500/2933873463142884534743603546271, d[1 8,3] = -2804152816102015499727532000000000/225338957716573246714867526 73813, d[19,3] = -15646848301434568771156943359375000/3896300619540154 0142964699991407, d[20,3] = -204529838958329654721984960000000000000/8 77007656469732158703655785573133737, d[21,3] = -1230653835045296296664 166992187500/3089704784991495259525267620443, d[1,4] = -11777917400673 50517980088002864999653352375/4360225145059755584701564664346063053148 , d[2,4] = 0, d[3,4] = 0, d[4,4] = 0, d[5,4] = 0, d[6,4] = -2114135226 871926151474180451671875/829132228160653461155982169264, d[7,4] = -892 17269000963975174805541687011718750000/4968051406859372751474133394066 5321263, d[8,4] = 57216367317309194344917724609375/1518680504270500882 143490008, d[9,4] = -3502902461370010077437335571677848504544049333460 63990393209364646375/9436965640053329933174083027017072783412875204813 675574669745448, d[10,4] = 276081526101167849809781568000/402880382799 7056367944503, d[11,4] = -84922595341263987542080296000000/11715232292 68973284610999593, d[12,4] = 1463128747811842734792753375/168696775862 6146030295408, d[13,4] = 28014017868690622217347948135375/123921194919 814042387520620891, d[14,4] = 0, d[15,4] = 0, d[16,4] = 0, d[17,4] = 1 8882660366917419511969273925781250/8801620389428653604230810638813, d[ 18,4] = 61497889686293713601431894000000000/20280506194491592204338077 4064317, d[19,4] = 496654275845501365651289536132812500/27274104336781 0781000752899939849, d[20,4] = 899784412670111646707904480000000000000 /877007656469732158703655785573133737, d[21,4] = 353152853739532362886 63256347656250/21627933494940466816676873343101, d[1,5] = 630130738181 10058843644348837967685562500/121117365140548766241710129565168418143, d[2,5] = 0, d[3,5] = 0, d[4,5] = 0, d[5,5] = 0, d[6,5] = 294106151443 761066098669361328125/51820764260040841322248885579, d[7,5] = 59574651 1455163889253309936523437500000000/14904154220578118254422400182199596 3789, d[8,5] = -47757642115025832702087402343750/569505189101437830803 808753, d[9,5] = 29238200528565027193444468976149136235775237197729391 8976649939593750/35388621150199987249402811351314022937798282018051283 40501154543, d[10,5] = -6759605648962890998964864000000/44316842107967 620047389533, d[11,5] = 189022971616739106614128000000000/117152322926 8973284610999593, d[12,5] = -203541930343956522262390625/1054354849141 34126893463, d[13,5] = -27365323924871811347707753250000/5310908353706 3161023223123239, d[14,5] = 0, d[15,5] = 0, d[16,5] = 0, d[17,5] = -47 207457284205531729045312500000000/8801620389428653604230810638813, d[1 8,5] = -21525911193488337801772000000000000/20280506194491592204338077 4064317, d[19,5] = -161401639624244351034792968750000000/3896300619540 1540142964699991407, d[20,5] = -19686525676411280059646080000000000000 00/877007656469732158703655785573133737, d[21,5] = -321780681674461548 84164843750000000/9269114354974485778575802861329, d[1,6] = -185996423 8027522908463849132281928515625000/32701688587948166885261734982595472 89861, d[2,6] = 0, d[3,6] = 0, d[4,6] = 0, d[5,6] = 0, d[6,6] = -35051 3952178631200295251464843750/51820764260040841322248885579, d[7,6] = - 2130021387193375899623870849609375000000000/44712462661734354763267200 5465987891367, d[8,6] = 170751816671926852035522460937500/170851556730 4313492411426259, d[9,6] = -104537737529964942564283282030186432891267 4223015001606895917773437500/10616586345059996174820843405394206881339 484605415385021503463629, d[10,6] = 8056057581781141989600000000000/44 316842107967620047389533, d[11,6] = -675829340957546355100000000000000 /3514569687806919853832998779, d[12,6] = 727740165467810382910156250/3 16306454742402380680389, d[13,6] = 235240943326464407232357812500000/3 71763584759442127162561862673, d[14,6] = 0, d[15,6] = 0, d[16,6] = 0, \+ d[17,6] = 20462401498211190052490234375000000/293387346314288453474360 3546271, d[18,6] = -31983577389585720775000000000000000/67601687314971 974014460258021439, d[19,6] = 1379318133517687339294433593750000000/27 2741043367810781000752899939849, d[20,6] = 229539096525598509480000000 0000000000000/877007656469732158703655785573133737, d[21,6] = 26098274 6740259593489990234375000000/64883800484821400450030620029303, d[1,7] \+ = 64209798218394956729271375871093750000000/19565112830396339162122405 5451425906231, d[2,7] = 0, d[3,7] = 0, d[4,7] = 0, d[5,7] = 0, d[6,7] \+ = 1496760753105893242309570312500000/362745349820285889255742199053, d [7,7] = 1105309694129066467285156250000000000000/380346626117560436678 661324372574461, d[8,7] = -2670851631000518798828125000000/43808091469 341371600292981, d[9,7] = 14879862535267097948004085820342331334636418 82013192630896484375000000/2477203480513999107458196794591981605645879 7412635898383508081801, d[10,7] = -2646222066466056000000000000000/238 62914981213333871671287, d[11,7] = 961972960995631000000000000000000/8 200662604882812992276997151, d[12,7] = -4909301586807617187500000/3497 859689094497100731, d[13,7] = -148790056173314860015625000000000/37176 3584759442127162561862673, d[14,7] = 0, d[15,7] = 0, d[16,7] = 0, d[17 ,7] = -39957881329811364746093750000000000/880162038942865360423081063 8813, d[18,7] = 120756262747502375000000000000000000/20280506194491592 2043380774064317, d[19,7] = -862046922562067993164062500000000000/2727 41043367810781000752899939849, d[20,7] = -1363585033062990000000000000 000000000000/877007656469732158703655785573133737, d[21,7] = -15627520 6253845617675781250000000000/64883800484821400450030620029303, d[1,8] \+ = -84516185931732458401186928710937500000000/1090056286264938896175391 166086515763287, d[2,8] = 0, d[3,8] = 0, d[4,8] = 0, d[5,8] = 0, d[6,8 ] = -52467846585220184326171875000000/51820764260040841322248885579, d [7,8] = -106279778281641006469726562500000000000000/149041542205781182 544224001821995963789, d[8,8] = 8519851174354553222656250000000/569505 189101437830803808753, d[9,8] = -5216026296049769324716090545283915468 2676248703454689941406250000000/35388621150199987249402811351314022937 79828201805128340501154543, d[10,8] = 1205897769990000000000000000000/ 44316842107967620047389533, d[11,8] = -3372125413625000000000000000000 0/1171523229268973284610999593, d[12,8] = 36311402269287109375000000/1 05435484914134126893463, d[13,8] = 12693444727793574218750000000000/12 3921194919814042387520620891, d[14,8] = 0, d[15,8] = 0, d[16,8] = 0, d [17,8] = 10295562851974487304687500000000000/8801620389428653604230810 638813, d[18,8] = -42193175918281250000000000000000000/202805061944915 922043380774064317, d[19,8] = 216435982877960205078125000000000000/272 741043367810781000752899939849, d[20,8] = 3245466701387500000000000000 00000000000/877007656469732158703655785573133737, d[21,8] = 1261869411 7599487304687500000000000/21627933494940466816676873343101\}: " } {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 "Supp ose 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$%\"xG%\"yG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(x[k])=y[k] " "6#/-%\"yG6#&%\"xG6#%\"kG&F%6#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 33 "When a Runge-Kutta step of width " }{TEXT 271 1 "h" } {TEXT -1 69 " has been made using the basic scheme, we wish to obtain \+ the result " }{XPPEDIT 18 0 "y[k](u)" "6#-&%\"yG6#%\"kG6#%\"uG" } {TEXT -1 21 " of a step of width " }{XPPEDIT 18 0 "h*u" "6#*&%\"hG\" \"\"%\"uGF%" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 107 "ee2 := map(_U->lhs(_U)=rhs(_U)/u,ee):\nsubs(ee2,matr ix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..8)]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*,$*&\"\"\"F**&\"#;F*%\"uGF*!\"\" F*F(%!GF/F/F/F/F/7*,$*(\"$7\"F*\"%l5F.F-F.F*,$*(\"&G*=F*\"(DU8\"F.F-F. F*,$*(\"'_.5F*F8F.F-F.F*F/F/F/F/F/7*,$*(\"#cF*\"$b$F.F-F.F*,$*(\"#9F*F @F.F-F.F*\"\"!,$*(\"#UF*F@F.F-F.F*F/F/F/F/7*,$*(\"#RF*\"$+\"F.F-F.F*,$ *(\")za\\%*F*\"*++)3DF.F-F.F*FD,$*(\"*(f1GNF*FPF.F-F.F.,$*(\"*fr2y\"F* \"*++WD\"F.F-F.F*F/F/F/7*,$*(\"\"(F*\"#:F.F-F.F*,$*(\"&*37F*\"'?FDF.F- F.F*FDFD,$*(\"(x`]#F*\")?bo5F.F-F.F*,$*(\"'+/'*F*\"(\">4_F.F-F.F*F/F/7 *,$*(FKF*\"$]#F.F-F.F*,$*(\")**3S@F*\"*+++]$F.F-F.F*FDFD,$*(\".**)H)HV 1$F*\"/+++]g7FF.F-F.F*,$*(\")ZRk@F*\"*v$4EfF.F-F.F.,$*(\"*V>RC\"F*\"++ +DcnF.F-F.F*F/7*,$*(\"#CF*\"#DF.F-F.F*,$*(\",6)eaO:F*\",vdc4O\"F.F-F.F .FDFD,$*(FenF*\"\"&F.F-F.F.,$*(\".3Y;G\"R$)F*\"-v%Q+1R*F.F-F.F.,$*(\"- )[+o$>MF*\",Di7^z%F.F-F.F*,$*(\".S#Q=K$*>F*\"-p!fM_!QF.F-F.F*Q(pprint2 6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The new polynomials (of degree " }{XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 55 " ) are obtained as follows (re-using the weight symbol " }{XPPEDIT 18 0 "b[j ]" "6#&%\"bG6#%\"jG" }{TEXT -1 3 "). " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "[seq(b[j]=add(d[j,i]*u^(i-1),i=1..8),j=1..21)]:\npols := eval(subs(dd,%)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The first few non-zero polynomials with rough approx imations for the coefficients are . . . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct to 12 do\n if rhs(pols[ct])<>0 then print(ev alf[6](pols[ct])) end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"\",2$F'\"\"!F'*&$\"'\")o7!\"%F'%\"uGF'!\"\"*&$\"'cqzF.F') F/\"\"#F'F'*&$\"'A,F!\"$F')F/\"\"$F'F0*&$\"'l-_F9F')F/\"\"%F'F'*&$\"'n (o&F9F')F/\"\"&F'F0*&$\"'&=G$F9F')F/\"\"'F'F'*&$\"'Q`xF.F')F/\"\"(F'F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"',0*&$\"'?QZ!\"%\"\" \"%\"uGF-!\"\"*&$\"'aBd!\"$F-)F.\"\"#F-F-*&$\"'#)\\D!\"#F-)F.\"\"$F-F/ *&$\"'XvcF9F-)F.\"\"%F-F-*&$\"'(Rw'F9F-)F.\"\"&F-F/*&$\"'?ETF9F-)F.F'F -F-*&$\"'\\75F9F-)F.\"\"(F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" bG6#\"\"(,0*&$\"'3PL!\"%\"\"\"%\"uGF-!\"\"*&$\"'1JS!\"$F-)F.\"\"#F-F-* &$\"'#ez\"!\"#F-)F.\"\"$F-F/*&$\"'=(*RF9F-)F.\"\"%F-F-*&$\"'#Qw%F9F-)F .\"\"&F-F/*&$\"'11HF9F-)F.\"\"'F-F-*&$\"')38(F3F-)F.F'F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"),0*&$\"''4+(!\"$\"\"\"%\"uGF-F -*&$\"'(oX)!\"#F-)F.\"\"#F-!\"\"*&$\"'^nPF5F-)F.\"\"$F-F-*&$\"'\"eQ)F5 F-)F.\"\"%F-F5*&$\"';%***F5F-)F.\"\"&F-F-*&$\"'r'4'F5F-)F.\"\"'F-F5*&$ \"','\\\"F5F-)F.\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6# \"\"*,0*&$\"'j(*o!\"$\"\"\"%\"uGF-!\"\"*&$\"'0K$)!\"#F-)F.\"\"#F-F-*&$ \"'*=r$F/F-)F.\"\"$F-F/*&$\"'.i#)F/F-)F.\"\"%F-F-*&$\"'kY)*F/F-)F.\"\" &F-F/*&$\"'s1gF/F-)F.\"\"'F-F-*&$\"'$RZ\"F/F-)F.\"\"(F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5,0*&$\"'St7!\"#\"\"\"%\"uGF-F-*&$ \"'AQ:!\"\"F-)F.\"\"#F-F2*&$\"'p_oF2F-)F.\"\"$F-F-*&$\"'HD:\"\"!F-)F. \"\"%F-F2*&$\"'$y\"=F=F-)F.\"\"&F-F-*&$\"'$*36F=F-)F.\"\"'F-F2*&$\"'3@ FF2F-)F.\"\"(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6,0*& $\"'.Z8!\"#\"\"\"%\"uGF-!\"\"*&$\"':F;F/F-)F.\"\"#F-F-*&$\"'!*[sF/F-)F .\"\"$F-F/*&$\"'[8;\"\"!F-)F.\"\"%F-F-*&$\"'%H#>F=F-)F.\"\"&F-F/*&$\"' /t6F=F-)F.\"\"'F-F-*&$\"'TyGF/F-)F.\"\"(F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7,0*&$\"'o6;!\"%\"\"\"%\"uGF-F-*&$\"'&o%>! \"$F-)F.\"\"#F-!\"\"*&$\"'8t')F2F-)F.\"\"$F-F-*&$\"'\\I>!\"#F-)F.\"\"% F-F5*&$\"'u+BF>F-)F.\"\"&F-F-*&$\"'_.9F>F-)F.\"\"'F-F5*&$\"'&RW$F2F-)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 ca n now check that this scheme satisfies the order conditions (and row s um conditions) for a 21 stage, order 8 Runge-Kutta scheme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RK8_21eqs := [op(RowSumConditions(2 1,'expanded')),op(OrderConditions(8,21,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "simplif y(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 }