{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 "" 261 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "System" 0 1 0 0 0 0 0 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 "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 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 "Normal" -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 14 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 "Text Output" -1 260 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text \+ Output" -1 261 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 3 1 3 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 "Derivation of an order 12 Runge-K utta scheme" }}{PARA 0 "" 0 "" {TEXT -1 46 "by Peter Stone, Gabriola I sland, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 30.11.20 11" }}{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 81 "load procedures for constru cting Runge-Kutta schemes and root-finding procedures " }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 262 9 "butcher.m" }{TEXT -1 5 " and " }{TEXT 262 7 "roots.m" }{TEXT -1 33 " are required by thi s worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 134 "They can be read into a Maple session by commands similar to those that follow, where each fi le path gives the location of the m-file." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "read \"C:\\\\Maple/procdrs/butcher.m\";\nread \"C:\\ \\Maple/procdrs/roots.m\";\nread \"C:\\\\Maple/procdrs/intg.m\";" }}} {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 91 "#===================================================== =====================================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "a modification of Hiroshi Ono's scheme with " }{XPPEDIT 18 0 "c[2]=39/88" "6#/&%\"cG6#\"\"#*&\"#R\"\"\"\"#))!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[4]=101/152" "6#/&%\"cG6#\"\"%*&\"$,\"\" \"\"\"$_\"!\"\"" }{TEXT -1 36 " and with a large stability region " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 30 "#-----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "checking the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Digits := 810:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157353 "ee := \{c[2]=.443181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 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7652111230162584946725882108573439834545166804164520563941383513647508 2312833345932360525985358984184274965931276186995271281824938700691180 9025686236794301283167041313847009553868409837276475830771065899056665 7312303777181295046911415017775848475003482868350105962466070847402554 7251950536652242753498527079567521364600499086860859065458203191282150 7960581353910394336202135221723559633930676642779610425844423310468406 5269441835116062847067925720934460786159015565249940341026000975605070 3783737404837153314340293071415852151098520714446828280320603401816655 5628734464367559769568517340527182568827,\nb[18]=.29,\nb[19]=.27,\nb[2 0]=.232142857142857142857142857142857142857142857142857142857142857142 8571428571428571428571428571428571428571428571428571428571428571428571 4285714285714285714285714285714285714285714285714285714285714285714285 7142857142857142857142857142857142857142857142857142857142857142857142 8571428571428571428571428571428571428571428571428571428571428571428571 4285714285714285714285714285714285714285714285714285714285714285714285 7142857142857142857142857142857142857142857142857142857142857142857142 8571428571428571428571428571428571428571428571428571428571428571428571 4285714285714285714285714285714285714285714285714285714285714285714285 7142857142857142857142857142857142857142857142857142857142857142857142 8571428571428571428571428571428571428571428571428571428571428571428571 42857142857142857142857142857142857142857143,\nb[21]=.21,\nb[22]=.19, \nb[23]=.17,\nb[24]=.11,\nb[25]=.2380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095238095238095238095238095238 0952380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095238095238095238095238095238 0952380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095238095238095238095238095238 0952380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095238095238095238095238095238 095e-1\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 74 ": The principal error norm for thi s scheme is approximately 0.4222219839 " }{TEXT 268 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-7);" "6#)\"#5,$\"\"(!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "RK12_25 := OrderConditions(12,25):\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%8y" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "nops(OrderConditions(11));\n7813-%; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%ZI" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%mZ" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "ee_ := `union`(\{c[1]=0.\},ee): ind := [ ]:\nfor ct from 7201 to 7813 do\n dd := subs(Sum=add,RK12_25[ct]):\n zr := subs(ee_,dd);\n ind := [op(ind),`if`(abs(lhs(zr)-rhs(zr))<1 0^(5-Digits),0,1)];\nend do:\nind;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# 7aam\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "ee_ := `union`(\{ c[1]=0.\},ee): ind := []:\nfor ct from 6601 to 7200 do\n dd := subs( Sum=add,RK12_25[ct]):\n zr := subs(ee_,dd);\n ind := [op(ind),`if` (abs(lhs(zr)-rhs(zr))<10^(5-Digits),0,1)];\nend do:\nind;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7d`m\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "ee_ := `union`(\{c[1 ]=0.\},ee): ind := []:\nfor ct from 6001 to 6600 do\n dd := subs(Sum =add,RK12_25[ct]):\n zr := subs(ee_,dd);\n ind := [op(ind),`if`(ab s(lhs(zr)-rhs(zr))<10^(5-Digits),0,1)];\nend do:\nind;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7d`m\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "ee_ := `union`(\{c[1]=0 .\},ee): ind := []:\nfor ct from 5401 to 6000 do\n dd := subs(Sum=ad d,RK12_25[ct]):\n zr := subs(ee_,dd);\n ind := [op(ind),`if`(abs(l hs(zr)-rhs(zr))<10^(5-Digits),0,1)];\nend do:\nind;" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#7d`m\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------- ------------" }}{PARA 0 "" 0 "" {TEXT -1 88 "We continue using the mat rix method for evaluating order conditions using the procedure " } {TEXT 0 19 "convert/Matrix_form" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 19 "convert/Matrix_form" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5000 "`convert/Matrix_form` := proc(ord,A::Matrix,B::Matrix,C::Matrix) \n local fact,flag,LS,RS,L,is_simple,G,H,nA,mA,nB,mB,nC,mC,dim,id,i, j;\n\n if type(ord,`=`(algebraic,rational)) then\n flag := 0;\n LS := op(1,ord);\n RS := op(2,ord);\n elif type(ord,`&+`( algebraic,rational)) then\n flag := 1;\n LS := op(1,ord);\n \+ RS := -op(2,ord);\n elif type(ord,`&*`(rational,specfunc(`&+`(a lgebraic,rational),``))) then\n flag := 2;\n fact := op(1,or d);\n LS := op([2,1,1],ord);\n RS := -op([2,1,2],ord);\n e lse\n error \"the 1st argument is invalid, it must be an order co ndition, or an error term, in abreviated form\"\n end if;\n\n ## a llow for order conditions that involve `b*`\n if has(LS,`b*`) then L S := subs(`b*`=b,LS) end if;\n\n ## check if we have a genuine order condition (probably not foolproof!)\n L := eval(subs(``=(u_->u_),LS ));\n if not (type(L,`*`) and indets(L) minus \{a,b,c,e\}=\{\} and c oeffs(L)=1 and type(1/RS,posint)) then\n error \"the 1st argument is invalid, it must be an order condition, or an error term, in abrev iated form\"\n end if;\n\n nA,mA := LinearAlgebra[Dimension](A);\n if nA<>mA then\n error \"the 2nd argument must be a square mat rix\"\n end if;\n dim := nA;\n nB,mB := LinearAlgebra[Dimension] (B);\n if nB<>1 or mB<>dim then\n error \"the 3rd argument must be a row vector with row dimension %1\",dim;\n end if;\n nC,mC := LinearAlgebra[Dimension](C);\n if nC<>mC or nC<>dim then\n err or \"the 4th argument must be a diagonal matrix with dimension %1\",di m;\n end if;\n for i to dim do\n for j to dim do\n if i<>j and C[i,j]<>0 then\n error \"the 4th argument must be a diagonal matrix with dimension %1\",dim;\n end if;\n e nd do;\n end do;\n id := Matrix([seq([1],i=1..dim)]);\n\n is_sim ple := proc(tt) local v;\n v := ListTools[Flatten](eval(subs(``=( u_->[u_]),subs(\{b=1,a=1,c=1,e=1\},[tt]))));\n if v=[1] then retu rn true else return false end if;\n end proc:\n\n ## We need to re place ordinary multiplication with matrix multiplication\n ## The \" hidden\" functions are replaced by the following procedure G \n if is_simple(LS) then\n G := proc(ff) local la,E,n,nn,i;\n \+ if patmatch(ff,b*E::algebraic,'la') then b.subs(la,E)\n elif f f=a*c then a.c\n elif patmatch(ff,a*c^n::posint,'la') then sub s(la,a.c^n)\n elif patmatch(ff,a*c^n::posint*E::algebraic,'la' ) then subs(la,a.c^n.E)\n elif patmatch(ff,c^n::posint*E::alge braic,'la') then subs(la,c^n.E) \n elif patmatch(ff,a*E:: algebraic,'la') then subs(la,a.E) \n else ff;\n end \+ if;\n end proc:\n else ## non-simple order condition\n\n # # procedure to form a diagonal matrix from row sums of a square matrix \n H := proc(U)\n LinearAlgebra[DiagonalMatrix](Lis tTools[Flatten](convert((U.id),listlist)))\n end proc;\n\n G := proc(ff) local la,E,n,gg,hh,nn,i;\n if patmatch(ff,b*E::al gebraic,'la') then\n gg := b.'H'(subs(la,E))\n elif ff=a*c then\n gg := 'H'(a.c)\n elif patmatch(ff,a* c^n::posint,'la') then\n gg := subs(la,'H'(a.c^n))\n \+ elif patmatch(ff,a*c^n::posint*E::algebraic,'la') then\n \+ hh,nn := op(subs(la,[E,n]));\n if type(hh,`*`) then\n \+ gg := op(1,hh);\n for i from 2 to nops(hh) do\n gg := gg.op(i,hh);\n end do;\n \+ gg := 'H'(a.c^nn.gg);\n else \n gg := \+ 'H'(a.c^nn.hh)\n end if;\n elif patmatch(ff,c^n::po sint*E::algebraic,'la') then\n hh,nn := op(subs(la,[E,n])); \n if type(hh,`*`) then\n gg := op(1,hh);\n \+ for i from 2 to nops(hh) do\n gg := gg.o p(i,hh);\n end do;\n gg := 'H'(c^nn.gg);\n else \n gg := 'H'(c^nn.hh)\n end \+ if; \n elif patmatch(ff,a*E::algebraic,'la') then\n \+ hh := subs(la,E);\n if type(hh,`*`) then\n \+ gg := op(1,hh);\n for i from 2 to nops(hh) do\n \+ gg := gg.op(i,hh);\n end do;\n \+ gg := 'H'(a.gg);\n else \n gg := 'H'(a.hh) \n end if;\n elif type(ff,`*`) then\n gg := op(1,ff);\n for i from 2 to nops(ff) do\n \+ gg := gg.op(i,ff);\n end do;\n gg := 'H'(gg);\n elif patmatch(ff,(E::algebraic)^n::posint,'la') then \+ \n gg := subs(la,'H'(E)^n); \n else\n \+ gg := 'H'(ff)\n end if;\n gg;\n end proc:\n end if;\n\n if flag=0 then\n subs(\{a=A,b=B,c=C\},(op(1,LS).eval(s ubs(\{b=1,``=G\},``(LS))).id))[1,1]=RS;\n elif flag=1 then\n su bs(\{a=A,b=B,c=C\},(op(1,LS).eval(subs(\{b=1,``=G\},``(LS))).id))[1,1] -RS;\n else ## flag=2\n fact*(subs(\{a=A,b=B,c=C\},(op(1,LS).ev al(subs(\{b=1,``=G\},``(LS))).id))[1,1]-RS);\n end if; \nend proc:" }}}{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 28 "RK12 := OrderConditions(12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "We need to set up the app ropriate matrices of coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "A := Matrix([seq([seq(a[i,j ],j=1..i-1),seq(0,j=i..25)],i=1..25)]):\nB := Matrix([[seq(b[i],i=1..2 5)]]):\nC := LinearAlgebra[DiagonalMatrix]([0,seq(c[i],i=2..25)]):" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A_ := subs(ee,A): B_ := subs(ee,B): C_ := subs(ee,C):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "ind := []:\nfor ct from 4801 to 5400 do\n zr := convert(RK12[ct ],'Matrix_form',A_,B_,C_);\n ind := [op(ind),`if`(abs(lhs(zr)-rhs(zr ))<10^(5-Digits),0,1)];\nend do:\nind;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7d`m\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "ind := []:\nfor ct from 4201 to 48 00 do\n zr := convert(RK12[ct],'Matrix_form',A_,B_,C_);\n ind := [ op(ind),`if`(abs(lhs(zr)-rhs(zr))<10^(5-Digits),0,1)];\nend do:\nind; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7d`m\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "ind := []:\nfor ct from 3601 to 4200 do\n zr := convert(RK12[ct],'Matrix_f orm',A_,B_,C_);\n ind := [op(ind),`if`(abs(lhs(zr)-rhs(zr))<10^(5-Di gits),0,1)];\nend do:\nind;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7d`m\" \"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "ind := []:\nfor ct from 3048 to 3600 do\n zr := convert(RK12[ct],'Matrix_form',A_,B_,C_);\n ind := [op(ind),`if`(ab s(lhs(zr)-rhs(zr))<10^(5-Digits),0,1)];\nend do:\nind;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7e]m\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 30 "#-----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "relations between the nodes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 98 "#------------------ ---------------------------------------------------------------------- ---------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 69 "relations between th e nodes resulting from the stage-order conditions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The stage-orders of stages 3 to 25 of the scheme \+ are given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[stage, `|`, 3, 4, 5, 6, 7 , 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 ], [`stage-order`, `|`, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 5 , 5, 5, 1, 1, 1, 1, 6]]);" "6#-%'matrixG6#7$7;%&stageG%\"|grG\"\"$\"\" %\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#? \"#@\"#A\"#B\"#C\"#D7;%,stage-orderGF)\"\"#F*F*F+F+F+F,F,F,F,F-F-F-F-F -F,F,F,\"\"\"FDFDFDF-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 92 "The zero linking coefficients in the \+ first 13 stages are indicated by the following tableau." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " matrix([[c[2], a[2,1], ``, ``, ``, ``, ``, ``, ``, ``, ``, ``, ``], [c [3], a[3,1], a[3,2], ``, ``, ``, ``, ``, ``, ``, ``, ``, ``], [c[4], a [4,1], 0, a[4,3], ``, ``, ``, ``, ``, ``, ``, ``, ``], [c[5], a[5,1], \+ 0, a[5,3], a[5,4], ``, ``, ``, ``, ``, ``, ``, ``], [c[6], a[6,1], 0, \+ 0, a[6,4], a[6,5], ``, ``, ``, ``, ``, ``, ``], [c[7], a[7,1], 0, 0, a [7,4], a[7,5], a[7,6], ``, ``, ``, ``, ``, ``], [c[8], a[8,1], 0, 0, 0 , a[8,5], a[8,6], a[8,7], ``, ``, ``, ``, ``], [c[9], a[9,1], 0, 0, 0, 0, a[9,6], a[9,7], a[9,8], ``, ``, ``, ``], [c[10], a[10,1], 0, 0, 0, 0, a[10,6], a[10,7], a[10,8], a[10,9], ``, ``, ``], [c[11], a[11,1], \+ 0, 0, 0, 0, 0, a[11,7], a[11,8], a[11,9], a[11,10], ``, ``], [c[12], a [12,1], 0, 0, 0, 0, 0, 0, a[12,8], a[12,9], a[12,10], a[12,11], ``], [ c[13], a[13,1], 0, 0, 0, 0, 0, 0, 0, a[13,9], a[13,10], a[13,11], a[13 ,12]]])" "6#-%'matrixG6#7.7/&%\"cG6#\"\"#&%\"aG6$F+\"\"\"%!GF0F0F0F0F0 F0F0F0F0F07/&F)6#\"\"$&F-6$F4F/&F-6$F4F+F0F0F0F0F0F0F0F0F0F07/&F)6#\" \"%&F-6$F " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 5 "" 0 "" {TEXT -1 20 "tableau construction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 459 "matrix([[c[2],a[2,1],``$11],[c[3], a[3,1],a[3,2],``$10],[c[4],a[4,1],0,a[4,3],``$9],[c[5],a[5,1],0,a[5,3] ,a[5,4],``$8],[c[6],a[6,1],0$2,a[6,4],a[6,5],``$7],[c[7],a[7,1],0$2,se q(a[7,i],i=4..6),``$6],[c[8],a[8,1],0$3,seq(a[8,i],i=5..7),``$5],[c[9] ,a[9,1],0$4,seq(a[9,i],i=6..8),``$4],[c[10],a[10,1],0$4,seq(a[10,i],i= 6..9),``$3],[c[11],a[11,1],0$5,seq(a[11,i],i=7..10),``$2],[c[12],a[12, 1],0$6,seq(a[12,i],i=8..11),``],[c[13],a[13,1],0$7,seq(a[13,i],i=9..12 )]]);" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "Taking account of the zero linking coefficients, the stage-order \+ conditions give rise to the following relations between the nodes." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3] = 2/3;" "6#/&%\" cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4];" "6#&%\"cG6#\"\"%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[5] = (4*c[4]-3*c[6] )*c[6]/(2*(3*c[4]-2*c[6]));" "6#/&%\"cG6#\"\"&*(,&*&\"\"%\"\"\"&F%6#F+ F,F,*&\"\"$F,&F%6#\"\"'F,!\"\"F,&F%6#F3F,*&\"\"#F,,&*&F0F,&F%6#F+F,F,* &F8F,&F%6#F3F,F4F,F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[8] = (20*c[6]*c[7 ]-15*c[7]*c[9]+12*c[9]^2-15*c[6]*c[9])*c[9]/(30*c[7]*c[6]-20*c[7]*c[9] +15*c[9]^2-20*c[6]*c[9]);" "6#/&%\"cG6#\"\")*(,**(\"#?\"\"\"&F%6#\"\"' F,&F%6#\"\"(F,F,*(\"#:F,&F%6#F2F,&F%6#\"\"*F,!\"\"*&\"#7F,*$&F%6#F9\" \"#F,F,*(F4F,&F%6#F/F,&F%6#F9F,F:F,&F%6#F9F,,**(\"#IF,&F%6#F2F,&F%6#F/ F,F,*(F+F,&F%6#F2F,&F%6#F9F,F:*&F4F,*$&F%6#F9F@F,F,*(F+F,&F%6#F/F,&F%6 #F9F,F:F:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[12] = (15*c[9]*c[13]*c[11 ]-20*c[9]*c[11]*c[10]-12*c[9]*c[13]^2+15*c[9]*c[13]*c[10]-12*c[11]*c[1 3]^2+15*c[11]*c[13]*c[10]+10*c[13]^3-12*c[10]*c[13]^2)*c[13]/(20*c[9]* c[13]*c[11]-30*c[9]*c[11]*c[10]-15*c[9]*c[13]^2+20*c[9]*c[13]*c[10]-15 *c[11]*c[13]^2+20*c[11]*c[13]*c[10]+12*c[13]^3-15*c[10]*c[13]^2)" "6#/ &%\"cG6#\"#7*(,2**\"#:\"\"\"&F%6#\"\"*F,&F%6#\"#8F,&F%6#\"#6F,F,**\"#? F,&F%6#F/F,&F%6#F5F,&F%6#\"#5F,!\"\"*(F'F,&F%6#F/F,&F%6#F2\"\"#F?**F+F ,&F%6#F/F,&F%6#F2F,&F%6#F>F,F,*(F'F,&F%6#F5F,&F%6#F2FEF?**F+F,&F%6#F5F ,&F%6#F2F,&F%6#F>F,F,*&F>F,*$&F%6#F2\"\"$F,F,*(F'F,&F%6#F>F,&F%6#F2FEF ?F,&F%6#F2F,,2**F7F,&F%6#F/F,&F%6#F2F,&F%6#F5F,F,**\"#IF,&F%6#F/F,&F%6 #F5F,&F%6#F>F,F?*(F+F,&F%6#F/F,&F%6#F2FEF?**F7F,&F%6#F/F,&F%6#F2F,&F%6 #F>F,F,*(F+F,&F%6#F5F,&F%6#F2FEF?**F7F,&F%6#F5F,&F%6#F2F,&F%6#F>F,F,*& F'F,*$&F%6#F2FgnF,F,*(F+F,&F%6#F>F,&F%6#F2FEF?F?" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "determination of relat ions between the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We specify the zero coefficients in stages 3 to 25. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 373 "e0 := \{seq(a[i,2]=0,i= [$4..20]),seq(a[i,3]=0,i=[$6..20]),seq(a[i,4]=0,i=[$8..20]),\n s eq(a[i,5]=0,i=[$9..20]),seq(a[i,6]=0,i=[$11..18]),seq(a[i,7]=0,i=[$12. .17]),\n seq(a[i,8]=0,i=[$13..17]),\n seq(a[21,j]=0,j=[2,3 ,7,8,$12..17]),seq(a[22,j]=0,j=[2,3,6,8,$11..18]),\n seq(a[23,j] =0,j=[3,4,5,$8..20]),seq(a[24,j]=0,j=[2,$4..22]),seq(a[25,j]=0,j=[4,5, 8])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Since stage 4 has stage-order 3, we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4, 3]*c[3] = 1/2;" "6#/*&&%\"aG6$\"\" %\"\"$\"\"\"&%\"cG6#F)F**&F*F*\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^2;" "6#*$&%\"cG6#\"\"%\"\"#" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "a[4,3]*c[3]^2 = 1/3;" "6#/*&&%\"aG6$\"\"%\"\"$\"\"\"*$& %\"cG6#F)\"\"#F**&F*F*F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^3; " "6#*$&%\"cG6#\"\"%\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "If we suppose that " }{XPPEDIT 18 0 "c[3] <> c[4];" "6#0&%\"cG 6#\"\"$&F%6#\"\"%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3] <> 0;" " 6#0&%\"cG6#\"\"$\"\"!" }{TEXT -1 106 " then, in order for these two eq uations to be consistent with a unique value for the linking coefficie nt " }{XPPEDIT 18 0 "a[4,3];" "6#&%\"aG6$\"\"%\"\"$" }{TEXT -1 32 " \+ the determinant of the matrix " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[c[3], c[4]^2/2], [c[3]^2, c[4]^3/3]]);" "6#-%' matrixG6#7$7$&%\"cG6#\"\"$*&&F)6#\"\"%\"\"#F0!\"\"7$*$&F)6#F+F0*&&F)6# F/F+F+F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "must be zero. This gives the relation " }{XPPEDIT 18 0 "c[3] = 2/3;" "6#/&%\"cG6# \"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4];" "6#& %\"cG6#\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "subs(e0,[add(a[4,j]*c[j],j= 2..3)=1/2*c[4]^2,add(a[4,j]*c[j]^2,j=2..3)=1/3*c[4]^3]);\nlinalg[genma trix](%,[a[4,3]],flag);\nlinalg[det](%)=0:\nop(solve(\{%,c[3]<>c[4],c[ 3]<>0\},c[3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/*&&%\"aG6$\"\"% \"\"$\"\"\"&%\"cG6#F*F+,$*&#F+\"\"#F+*$)&F-6#F)F2F+F+F+/*&F&F+)F,F2F+, $*&#F+F*F+*$)F5F*F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7$7$&%\"cG6#\"\"$,$*&#\"\"\"\"\"#F/*$)&F)6#\"\"%F0F/F/F/7$*$)F(F0F/,$ *&#F/F+F/*$)F3F+F/F/F/Q(pprint56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"cG6#\"\"$,$*&#\"\"#F'\"\"\"&F%6#\"\"%F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 55 ": Thi s relation can be expressed in the integral form: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*(x-c[3]),x=0..c[4])=0" "6#/-%$I ntG6$*&%\"xG\"\"\",&F(F)&%\"cG6#\"\"$!\"\"F)/F(;\"\"!&F,6#\"\"%F2" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Int(x*(x-c [3]),x=0..c[4])=0;\nc[3]=solve(value(%),c[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\",&F(F)&%\"cG6#\"\"$!\"\"F)/F(;\" \"!&F,6#\"\"%F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$,$*&# \"\"#F'\"\"\"&F%6#\"\"%F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "#---------------------------------------------- ---------------------" }}{PARA 0 "" 0 "" {TEXT -1 42 "We obtain a rela tion involving the nodes " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 47 " from \+ the fact that stage 6 has stage-order 4." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "subs(e0,[seq(add(a[6,j]*c[j]^(k-1),j=2..5)=1/k*c[6]^ k,k=2..4)]);\nlinalg[genmatrix](%,[a[6,4],a[6,5]],flag);\nop(solve(\{l inalg[det](%)=0,c[5]<>c[4],c[5]<>0\},c[5]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/,&*&&%\"aG6$\"\"'\"\"%\"\"\"&%\"cG6#F+F,F,*&&F(6$F* \"\"&F,&F.6#F3F,F,,$*&#F,\"\"#F,*$)&F.6#F*F9F,F,F,/,&*&F'F,)F-F9F,F,*& F1F,)F4F9F,F,,$*&#F,\"\"$F,*$)F " 0 "" {MPLTEXT 1 0 64 "Int(x*(x-c [4])*(x-c[5]),x=0..c[6])=0;\nc[5]=solve(value(%),c[5]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$IntG6$*(%\"xG\"\"\",&F(F)&%\"cG6#\"\"%!\"\"F ),&F(F)&F,6#\"\"&F/F)/F(;\"\"!&F,6#\"\"'F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&,$*&#\"\"\"\"\"#F+*(&F%6#\"\"'F+,&*&\"\"$ F+F.F+!\"\"*&\"\"%F+&F%6#F6F+F+F+,&*&F3F+F7F+F+*&F,F+F.F+F4F4F+F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "#-------- ---------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 40 "We obtain relation involving the nodes " }{XPPEDIT 18 0 "c[6]" "6 #&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7];" "6#&%\"cG6#\" \"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[9];" "6#&%\"cG6#\"\"*" }{TEXT -1 47 " from the fact that stage 9 has stage-order 5." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 180 "subs(e0,[seq(add(a[9,j]*c[j]^(k-1),j=2..8)=1/ k*c[9]^k,k=2..5)]);\nlinalg[genmatrix](%,[a[9,6],a[9,7],a[9,8]],flag); \nop(solve(\{linalg[det](%)=0,c[8]<>c[6],c[8]<>c[7],c[8]<>0\},c[8])); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/,(*&&%\"aG6$\"\"*\"\"'\"\"\"&% \"cG6#F+F,F,*&&F(6$F*\"\"(F,&F.6#F3F,F,*&&F(6$F*\"\")F,&F.6#F9F,F,,$*& #F,\"\"#F,*$)&F.6#F*F?F,F,F,/,(*&F'F,)F-F?F,F,*&F1F,)F4F?F,F,*&F7F,)F: F?F,F,,$*&#F,\"\"$F,*$)FBFOF,F,F,/,(*&F'F,)F-FOF,F,*&F1F,)F4FOF,F,*&F7 F,)F:FOF,F,,$*&#F,\"\"%F,*$)FBFgnF,F,F,/,(*&F'F,)F-FgnF,F,*&F1F,)F4Fgn F,F,*&F7F,)F:FgnF,F,,$*&#F,\"\"&F,*$)FBFeoF,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7&&%\"cG6#\"\"'&F)6#\"\"(&F)6#\"\"),$*&# \"\"\"\"\"#F5*$)&F)6#\"\"*F6F5F5F57&*$)F(F6F5*$)F,F6F5*$)F/F6F5,$*&#F5 \"\"$F5*$)F9FFF5F5F57&*$)F(FFF5*$)F,FFF5*$)F/FFF5,$*&#F5\"\"%F5*$)F9FS F5F5F57&*$)F(FSF5*$)F,FSF5*$)F/FSF5,$*&#F5\"\"&F5*$)F9FjnF5F5F5Q)pprin t346\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"),$*&#\"\"\"\" \"&F+*(&F%6#\"\"*F+,**(\"#:F+&F%6#\"\"(F+F.F+F+*(\"#?F+F4F+&F%6#\"\"'F +!\"\"*&\"#7F+)F.\"\"#F+F<*(F3F+F9F+F.F+F+F+,**(F;F+F4F+F9F+F<*(\"\"%F +F4F+F.F+F+*(FEF+F9F+F.F+F+*&\"\"$F+F?F+F " 0 "" {MPLTEXT 1 0 73 "I nt(x*(x-c[6])*(x-c[7])*(x-c[8]),x=0..c[9])=0;\nc[8]=solve(value(%),c[8 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$**%\"xG\"\"\",&F(F)&% \"cG6#\"\"'!\"\"F),&F(F)&F,6#\"\"(F/F),&F(F)&F,6#\"\")F/F)/F(;\"\"!&F, 6#\"\"*F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"),$*&#\"\"\" \"\"&F+*(&F%6#\"\"*F+,**(\"#:F+&F%6#\"\"(F+F.F+F+*(\"#?F+F4F+&F%6#\"\" 'F+!\"\"*&\"#7F+)F.\"\"#F+F<*(F3F+F9F+F.F+F+F+,**(F;F+F4F+F9F+F<*(\"\" %F+F4F+F.F+F+*(FEF+F9F+F.F+F+*&\"\"$F+F?F+F " 0 "" {MPLTEXT 1 0 220 "subs(e0,[seq(add(a[13,j]*c[j]^(k-1),j=2..12)=1/k*c[13]^k,k=2. .6)]):\nlinalg[genmatrix](%,[a[13,9],a[13,10],a[13,11],a[13,12]],flag) ;\n``;\nop(solve(\{linalg[det](%%)=0,c[12]<>0,c[12]<>c[9],c[12]<>c[10] ,c[12]<>c[11]\},c[12]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7'7'&%\"cG6#\"\"*&F)6#\"#5&F)6#\"#6&F)6#\"#7,$*&#\"\"\"\"\"#F8*$)&F) 6#\"#8F9F8F8F87'*$)F(F9F8*$)F,F9F8*$)F/F9F8*$)F2F9F8,$*&#F8\"\"$F8*$)F " 0 "" {MPLTEXT 1 0 88 "Int(x*(x-c[9])*(x-c[10])*(x- c[11])*(x-c[12]),x=0..c[13])=0;\nc[12]=solve(value(%),c[12]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*,%\"xG\"\"\",&F(F)&%\"cG6# \"\"*!\"\"F),&F(F)&F,6#\"#5F/F),&F(F)&F,6#\"#6F/F),&F(F)&F,6#\"#7F/F)/ F(;\"\"!&F,6#\"#8F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7*( ,2**\"#:\"\"\"&F%6#\"\"*F,&F%6#\"#8F,&F%6#\"#5F,F,**\"#?F,F-F,F3F,&F%6 #\"#6F,!\"\"*(F'F,F-F,)F0\"\"#F,F;**F+F,F-F,F0F,F8F,F,*(F'F,F3F,F=F,F; **F+F,F3F,F0F,F8F,F,*&F5F,)F0\"\"$F,F,*(F'F,F8F,F=F,F;F,F0F,,2**F7F,F- F,F0F,F3F,F,**\"#IF,F-F,F3F,F8F,F;*(F+F,F-F,F=F,F;**F7F,F-F,F0F,F8F,F, *(F+F,F3F,F=F,F;**F7F,F3F,F0F,F8F,F,*&F'F,FCF,F,*(F+F,F8F,F=F,F;F;" }} }{PARA 0 "" 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 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 98 "#--------------------------------------------------- ----------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 76 "relations between the nodes resulting from the colum n simplifying conditions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The \+ scheme satisfies the following column simplifying conditions: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,j],i=j+1..25)=b[j]*(1-c[j])" "6#/-%$SumG6$ *&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#D*&&F)6#F0F,,&F ,F,&%\"cG6#F0!\"\"F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 14;" "6# /%\"jG\"#9" }{TEXT -1 19 " . . 24, " }{XPPEDIT 18 0 "Sum(b[i ]*c[i]*a[i,j],i=j+1..25)=1/2" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG 6#F+F,&%\"aG6$F+%\"jGF,/F+;,&F3F,F,F,\"#D*&F,F,\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "b[j]*(1-c[j]^2)" "6#*&&%\"bG6#%\"jG\"\"\",&F(F(* $&%\"cG6#F'\"\"#!\"\"F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 14;" "6#/%\"jG\"#9" }{TEXT -1 9 " . . 23, " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a [i,j],i=j+1..25)=1/3" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\" \"#F,&%\"aG6$F+%\"jGF,/F+;,&F5F,F,F,\"#D*&F,F,\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "b[j]*(1-c[j]^3)" "6#*&&%\"bG6#%\"jG\"\"\",&F(F(*$& %\"cG6#F'\"\"$!\"\"F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 14;" "6 #/%\"jG\"#9" }{TEXT -1 24 " . . 22, " }{XPPEDIT 18 0 "S um(b[i]*c[i]^3*a[i,j],i=j+1..25)=1/4" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\" \"*$&%\"cG6#F+\"\"$F,&%\"aG6$F+%\"jGF,/F+;,&F5F,F,F,\"#D*&F,F,\"\"%!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "b[j]*(1-c[j]^4)" "6#*&&%\"bG6#%\"j G\"\"\",&F(F(*$&%\"cG6#F'\"\"%!\"\"F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 14;" "6#/%\"jG\"#9" }{TEXT -1 9 " . . 20, " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum (b[i]*c[i]^4*a[i,j],i=j+1..25)=1/5" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\" *$&%\"cG6#F+\"\"%F,&%\"aG6$F+%\"jGF,/F+;,&F5F,F,F,\"#D*&F,F,\"\"&!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "b[j]*(1-c[j]^5)" "6#*&&%\"bG6#%\"jG \"\"\",&F(F(*$&%\"cG6#F'\"\"&!\"\"F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 14;" "6#/%\"jG\"#9" }{TEXT -1 9 " . . 17. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The zero linking coeff icients in columns 8 to 25 of the scheme are indicated by the followin g tableau." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[a[9,8], ``, ``, ``, ``, ``, ``, ``, \+ ``, ``, ``, ``, ``, ``, ``, ``, ``], [a[10,8], a[10,9], ``, ``, ``, `` , ``, ``, ``, ``, ``, ``, ``, ``, ``, ``, ``], [a[11,8], a[11,9], a[11 ,10], ``, ``, ``, ``, ``, ``, ``, ``, ``, ``, ``, ``, ``, ``], [a[12,8 ], a[12,9], a[12,10], a[12,11], ``, ``, ``, ``, ``, ``, ``, ``, ``, `` , ``, ``, ``], [0, a[13,9], a[13,10], a[13,11], a[13,12], ``, ``, ``, \+ ``, ``, ``, ``, ``, ``, ``, ``, ``], [0, a[14,9], a[14,10], a[14,11], \+ a[14,12], a[14,13], ``, ``, ``, ``, ``, ``, ``, ``, ``, ``, ``], [0, a [15,9], a[15,10], a[15,11], a[15,12], a[15,13], a[15,14], ``, ``, ``, \+ ``, ``, ``, ``, ``, ``, ``], [0, a[16,9], a[16,10], a[16,11], a[16,12] , a[16,13], a[16,14], a[16,15], ``, ``, ``, ``, ``, ``, ``, ``, ``], [ 0, a[17,9], a[17,10], a[17,11], a[17,12], a[17,13], a[17,14], a[17,15] , a[17,16], ``, ``, ``, ``, ``, ``, ``, ``], [a[18,8], a[18,9], a[18,1 0], a[18,11], a[18,12], a[18,13], a[18,14], a[18,15], a[18,16], a[18,1 7], ``, ``, ``, ``, ``, ``, ``], [a[19,8], a[19,9], a[19,10], a[19,11] , a[19,12], a[19,13], a[19,14], a[19,15], a[19,16], a[19,17], a[19,18] , ``, ``, ``, ``, ``, ``], [a[20,8], a[20,9], a[20,10], a[20,11], a[20 ,12], a[20,13], a[20,14], a[20,15], a[20,16], a[20,17], a[20,18], a[20 ,19], ``, ``, ``, ``, ``], [0, a[21,9], a[21,10], a[21,11], 0, 0, 0, 0 , 0, 0, a[21,18], a[21,19], a[21,20], ``, ``, ``, ``], [0, a[22,9], a[ 22,10], 0, 0, 0, 0, 0, 0, 0, 0, a[22,19], a[22,20], a[22,21], ``, ``, \+ ``], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a[23,21], a[23,22], ``, ` `], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a[24,23], ``], [0, a [25,9], a[25,10], a[25,11], a[25,12], a[25,13], a[25,14], a[25,15], a[ 25,16], a[25,17], a[25,18], a[25,19], a[25,20], a[25,21], a[25,22], a[ 25,23], a[25,24]]])" "6#-%'matrixG6#7373&%\"aG6$\"\"*\"\")%!GF-F-F-F-F -F-F-F-F-F-F-F-F-F-F-73&F)6$\"#5F,&F)6$F1F+F-F-F-F-F-F-F-F-F-F-F-F-F-F -F-73&F)6$\"#6F,&F)6$F7F+&F)6$F7F1F-F-F-F-F-F-F-F-F-F-F-F-F-F-73&F)6$ \"#7F,&F)6$F?F+&F)6$F?F1&F)6$F?F7F-F-F-F-F-F-F-F-F-F-F-F-F-73\"\"!&F)6 $\"#8F+&F)6$FJF1&F)6$FJF7&F)6$FJF?F-F-F-F-F-F-F-F-F-F-F-F-73FG&F)6$\"# 9F+&F)6$FTF1&F)6$FTF7&F)6$FTF?&F)6$FTFJF-F-F-F-F-F-F-F-F-F-F-73FG&F)6$ \"#:F+&F)6$FjnF1&F)6$FjnF7&F)6$FjnF?&F)6$FjnFJ&F)6$FjnFTF-F-F-F-F-F-F- F-F-F-73FG&F)6$\"#;F+&F)6$FhoF1&F)6$FhoF7&F)6$FhoF?&F)6$FhoFJ&F)6$FhoF T&F)6$FhoFjnF-F-F-F-F-F-F-F-F-73FG&F)6$\"#F,&F)6$F`sF+&F)6 $F`sF1&F)6$F`sF7&F)6$F`sF?&F)6$F`sFJ&F)6$F`sFT&F)6$F`sFjn&F)6$F`sFho&F )6$F`sFhp&F)6$F`sFjqF-F-F-F-F-F-73&F)6$\"#?F,&F)6$FhtF+&F)6$FhtF1&F)6$ FhtF7&F)6$FhtF?&F)6$FhtFJ&F)6$FhtFT&F)6$FhtFjn&F)6$FhtFho&F)6$FhtFhp&F )6$FhtFjq&F)6$FhtF`sF-F-F-F-F-73FG&F)6$\"#@F+&F)6$FbvF1&F)6$FbvF7FGFGF GFGFGFG&F)6$FbvFjq&F)6$FbvF`s&F)6$FbvFhtF-F-F-F-73FG&F)6$\"#AF+&F)6$F` wF1FGFGFGFGFGFGFGFG&F)6$F`wF`s&F)6$F`wFht&F)6$F`wFbvF-F-F-73FGFGFGFGFG FGFGFGFGFGFGFGFG&F)6$\"#BFbv&F)6$F\\xF`wF-F-73FGFGFGFGFGFGFGFGFGFGFGFG FGFGFG&F)6$\"#CF\\xF-73FG&F)6$\"#DF+&F)6$FfxF1&F)6$FfxF7&F)6$FfxF?&F)6 $FfxFJ&F)6$FfxFT&F)6$FfxFjn&F)6$FfxFho&F)6$FfxFhp&F)6$FfxFjq&F)6$FfxF` s&F)6$FfxFht&F)6$FfxFbv&F)6$FfxF`w&F)6$FfxF\\x&F)6$FfxFbx" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 20 "tableau con struction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "e0 := \{seq(a[i,2]=0,i=[$4..20]),seq(a[i,3]=0,i=[$6. .20]),seq(a[i,4]=0,i=[$8..20]),\n seq(a[i,5]=0,i=[$9..20]),seq(a [i,6]=0,i=[$11..18]),seq(a[i,7]=0,i=[$12..17]),\n seq(a[i,8]=0,i =[$13..17]),\n seq(a[21,j]=0,j=[2,3,7,8,$12..17]),seq(a[22,j]=0, j=[2,3,6,8,$11..18]),\n seq(a[23,j]=0,j=[3,4,5,$8..20]),seq(a[24 ,j]=0,j=[2,$4..22]),seq(a[25,j]=0,j=[4,5,8])\}:\nsubs(%,matrix([seq([s eq(a[i,j],j=8..i-1),``$(25-i)],i=9..25)]));" }}}{PARA 0 "" 0 "" {TEXT -1 2 "OR" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "matrix([[a[9,8] ,``$16],[a[10,8],a[10,9],``$15],[seq(a[11,i],i=8..10),``$14],\n[seq(a[ 12,i],i=8..11),``$13],[0,seq(a[13,i],i=9..12),``$12],[0,seq(a[14,i],i= 9..13),``$11],\n[0,seq(a[15,i],i=9..14),``$10],[0,seq(a[16,i],i=9..15) ,``$9],[0,seq(a[17,i],i=9..16),``$8],\n[seq(a[18,i],i=8..17),``$7],[se q(a[19,i],i=8..18),``$6],[seq(a[20,i],i=8..19),``$5],\n[0,seq(a[21,i], i=9..11),0$6,seq(a[21,i],i=18..20),``$4],\n[0,a[22,9],a[22,10],0$8,seq (a[22,i],i=19..21),``$3],[0$13,a[23,21],a[23,22],``$2],\n[0$15,a[24,23 ],``],[0,seq(a[25,i],i=9..24)]]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 138 "Taking account of the zero linking c oefficients, the column simplifying conditions give rise to the follow ing relations between the nodes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[6] = (3*c[3]-1)/2; " "6#/&%\"cG6#\"\"'*&,&*&\"\"$\"\"\"&F%6#F+F,F,F,!\"\"F,\"\"#F/" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[7]=(3*c[9]^2-4*c[6]*c[9]+2*c[9]-2*c[6 ]+1)/( 4*c[9]+2-6*c[6])" "6#/&%\"cG6#\"\"(*&,,*&\"\"$\"\"\"*$&F%6#\"\" *\"\"#F,F,*(\"\"%F,&F%6#\"\"'F,&F%6#F0F,!\"\"*&F1F,&F%6#F0F,F,*&F1F,&F %6#F6F,F9F,F,F,,(*&F3F,&F%6#F0F,F,F1F,*&F6F,&F%6#F6F,F9F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[11]=(12*c[17]^3-15*c[10]*c[17]^2+9*c[17]^2-15*c[8 ]*c[17]^2-10*c[17]*c[10]+6*c[17]+20*c[10]*c[8]*c[17]-10*c[8]*c[17]+3-5 *c[8]+10*c[10]*c[8]-5*c[10])/(5*(3*c[17]^2+2*c[17]-4*c[17]*c[10]-4*c[8 ]*c[17]-2*c[10]+1-2*c[8]+6*c[10]*c[8]))" "6#/&%\"cG6#\"#6*&,:*&\"#7\" \"\"*$&F%6#\"#<\"\"$F,F,*(\"#:F,&F%6#\"#5F,&F%6#F0\"\"#!\"\"*&\"\"*F,* $&F%6#F0F9F,F,*(F3F,&F%6#\"\")F,&F%6#F0F9F:*(F6F,&F%6#F0F,&F%6#F6F,F:* &\"\"'F,&F%6#F0F,F,**\"#?F,&F%6#F6F,&F%6#FCF,&F%6#F0F,F,*(F6F,&F%6#FCF ,&F%6#F0F,F:F1F,*&\"\"&F,&F%6#FCF,F:*(F6F,&F%6#F6F,&F%6#FCF,F,*&FgnF,& F%6#F6F,F:F,*&FgnF,,2*&F1F,*$&F%6#F0F9F,F,*&F9F,&F%6#F0F,F,*(\"\"%F,&F %6#F0F,&F%6#F6F,F:*(F\\pF,&F%6#FCF,&F%6#F0F,F:*&F9F,&F%6#F6F,F:F,F,*&F 9F,&F%6#FCF,F:*(FLF,&F%6#F6F,&F%6#FCF,F,F,F:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "determination of relations b etween the nodes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 373 "e0 := \{seq(a[i,2]=0,i=[$4..20]),seq(a[i,3]=0 ,i=[$6..20]),seq(a[i,4]=0,i=[$8..20]),\n seq(a[i,5]=0,i=[$9..20] ),seq(a[i,6]=0,i=[$11..18]),seq(a[i,7]=0,i=[$12..17]),\n seq(a[i ,8]=0,i=[$13..17]),\n seq(a[21,j]=0,j=[2,3,7,8,$12..17]),seq(a[2 2,j]=0,j=[2,3,6,8,$11..18]),\n seq(a[23,j]=0,j=[3,4,5,$8..20]),s eq(a[24,j]=0,j=[2,$4..22]),seq(a[25,j]=0,j=[4,5,8])\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "#------------------- -----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 33 "Th e column simplifying equations " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^k*a[i,j],i=j+1..25)=1/(k+1)" "6#/-%$SumG6 $*(&%\"bG6#%\"iG\"\"\")&%\"cG6#F+%\"kGF,&%\"aG6$F+%\"jGF,/F+;,&F5F,F,F ,\"#D*&F,F,,&F1F,F,F,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "b[j]*(1-c[ j]^(k+1))))" "6#*&&%\"bG6#%\"jG\"\"\",&F(F()&%\"cG6#F',&%\"kGF(F(F(!\" \"F(" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"!" }{TEXT -1 8 " . . 2, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j = 22;" "6#/%\"jG\"#A" }{TEXT -1 45 " give rise to a relation between the nodes " }{XPPEDIT 18 0 "c[2 3];" "6#&%\"cG6#\"#B" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[22];" "6 #&%\"cG6#\"#A" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "Then, be cause " }{XPPEDIT 18 0 "c[23] = c[3];" "6#/&%\"cG6#\"#B&F%6#\"\"$" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[22] = c[6];" "6#/&%\"cG6#\"#A&F%6# \"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[25]=1" "6#/&%\"cG6#\"#D \"\"\"" }{TEXT -1 32 ", we obtain a relation between " }{XPPEDIT 18 0 "c[3];" "6#&%\"cG6#\"\"$" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6] ;" "6#&%\"cG6#\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "[seq(eval(subs(j=22,'add'( b[i]*c[i]^k*a[i,j],i=j+1..25)=b[j]/(k+1)*(1-c[j]^(k+1)))),k=0..2)]:\ne xpand(subs(e0,%)):\nlinalg[genmatrix](%,[a[23,22],a[25,22]],flag);\n`` ;\nop(solve(\{linalg[det](%%)=0,c[23]<>c[22],c[23]<>c[25]\},c[23]));\n ``;\nfactor(subs(\{c[23]=c[3],c[22]=c[6],c[25]=1\},%%)):\nc[6]=solve(% ,c[6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%&%\"bG6#\"# B&F)6#\"#D,&&F)6#\"#A\"\"\"*&F0F3&%\"cGF1F3!\"\"7%*&F(F3&F6F*F3*&F,F3& F6F-F3,&*&#F3\"\"#F3F0F3F3*&#F3F@F3*&F0F3)F5F@F3F3F77%*&F(F3)F:F@F3*&F ,F3)F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "#------------------------ ------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 33 "The col umn simplifying equations " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^k*a[i,j],i=j+1..25)=1/(k+1)" "6#/-%$SumG6 $*(&%\"bG6#%\"iG\"\"\")&%\"cG6#F+%\"kGF,&%\"aG6$F+%\"jGF,/F+;,&F5F,F,F ,\"#D*&F,F,,&F1F,F,F,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "b[j]*(1-c[ j]^(k+1))))" "6#*&&%\"bG6#%\"jG\"\"\",&F(F()&%\"cG6#F',&%\"kGF(F(F(!\" \"F(" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"!" }{TEXT -1 8 " . . 3, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j = 20;" "6#/%\"jG\"#?" }{TEXT -1 48 " give rise to a relation connecting the nodes " }{XPPEDIT 18 0 " c[20]" "6#&%\"cG6#\"#?" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[21]" "6#&% \"cG6#\"#@" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[22]" "6#&%\"cG6#\"#A" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[25];" "6#&%\"cG6#\"#D" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Then, because " }{XPPEDIT 18 0 "c[20]=c[9]" "6#/&%\"cG6#\"#?&F%6# \"\"*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[21] = c[7];" "6#/&%\"cG6#\" #@&F%6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[22]=c[6]" "6#/&%\"cG 6#\"#A&F%6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[25]=1" "6#/& %\"cG6#\"#D\"\"\"" }{TEXT -1 35 ", we obtain a relation connecting " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7];" "6#&%\"cG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c [9];" "6#&%\"cG6#\"\"*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "[seq(eval(subs(j=20,'ad d'(b[i]*c[i]^k*a[i,j],i=j+1..25)=b[j]/(k+1)*(1-c[j]^(k+1)))),k=0..3)]: \nsubs(e0,%):\nlinalg[genmatrix](%,[a[21,20],a[22,20],a[25,20]],flag); \n``;\nop(solve(\{linalg[det](%%)=0,c[21]<>c[22],c[21]<>c[25]\},c[21]) );\n``;\nfactor(subs(\{c[20]=c[9],c[21]=c[7],c[22]=c[6],c[25]=1\},%%)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7&&%\"bG6#\"#@&F)6# \"#A&F)6#\"#D*&&F)6#\"#?\"\"\",&F6F6&%\"cGF4!\"\"F67&*&F(F6&F9F*F6*&F, F6&F9F-F6*&F/F6&F9F0F6,$*&#F6\"\"#F6*&F3F6,&F6F6*$)F8FEF6F:F6F6F67&*&F (F6)F=FEF6*&F,F6)F?FEF6*&F/F6)FAFEF6,$*&#F6\"\"$F6*&F3F6,&F6F6*$)F8FTF 6F:F6F6F67&*&F(F6)F=FTF6*&F,F6)F?FTF6*&F/F6)FAFTF6,$*&#F6\"\"%F6*&F3F6 ,&F6F6*$)F8F]oF6F:F6F6F6Q)pprint396\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#@,$*&#\"\"\"\"\" #F+*&,:*(\"\"%F+&F%6#\"#AF+)&F%6#\"#?F,F+!\"\"*(F0F+F1F+F5F+F8**\"\"'F +F1F+&F%6#\"#DF+F5F+F+*&F0F+F1F+F8*(F;F+F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "#------------------------------ ------------------------" }}{PARA 0 "" 0 "" {TEXT -1 32 "The column si mplifying equations" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^k*a[i,j],i=j+1..25)=1/(k+1)" "6#/-%$SumG6$*(&%\"bG6#% \"iG\"\"\")&%\"cG6#F+%\"kGF,&%\"aG6$F+%\"jGF,/F+;,&F5F,F,F,\"#D*&F,F,, &F1F,F,F,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "b[j]*(1-c[j]^(k+1)))) " "6#*&&%\"bG6#%\"jG\"\"\",&F(F()&%\"cG6#F',&%\"kGF(F(F(!\"\"F(" } {TEXT -1 4 ", " }{XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"!" }{TEXT -1 7 " . . 4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "f or " }{XPPEDIT 18 0 "j = 17;" "6#/%\"jG\"#<" }{TEXT -1 48 " give ris e to a relation connecting the nodes " }{XPPEDIT 18 0 "c[17]" "6#&%\" cG6#\"#<" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[18];" "6#&%\"cG6#\"#=" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[19]" "6#&%\"cG6#\"#>" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[20]" "6#&%\"cG6#\"#?" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[25];" "6#&%\"cG6#\"#D" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "Then, by using the symmetry relations " }{XPPEDIT 18 0 "c[20] = c[9];" "6#/&%\"cG6#\"#?&F%6#\"\"*" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[19]=c[10]" "6#/&%\"cG6#\"#>&F%6#\"#5" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[18]=c[11]" "6#/&%\"cG6#\"#=&F%6#\"#6" }{TEXT -1 21 ", and the fact that " }{XPPEDIT 18 0 "c[25]=1" "6#/&%\"cG6#\"#D\" \"\"" }{TEXT -1 35 " we obtain a relation connecting " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10]" "6 #&%\"cG6#\"#5" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11]" "6#&%\"cG6#\"# 6" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[17];" "6#&%\"cG6#\"#<" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "[seq(eval(s ubs(j=17,'add'(b[i]*c[i]^k*a[i,j],i=j+1..25)=b[j]/(k+1)*(1-c[j]^(k+1)) )),k=0..4)]:\nexpand(subs(e0,%)):\nlinalg[genmatrix](%,[a[18,17],a[19, 17],a[20,17],a[25,17]],flag);\n``;\nop(solve(\{linalg[det](%%)=0,c[18] <>c[19],c[18]<>c[20],c[18]<>c[25]\},c[18]));\n``;\nsimplify(subs(\{c[2 0]=c[9],c[19]=c[10],c[18]=c[11],c[25]=1\},%%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7'&%\"bG6#\"#=&F)6#\"#>&F)6#\"#?&F)6#\"#D ,&&F)6#\"#<\"\"\"*&F6F9&%\"cGF7F9!\"\"7'*&F(F9&FF+&F%6#\"#DF+)&F%6#\"#<\" \"#F+F0*(\"#:F+&F%6#F2F+F:F+F+*(F?F+F@F+)F:\"\"$F+F+*(F?F+F6F+FCF+F+*& F/F+)F:\"\"%F+F0*&F/F+F9F+F0*&F/F+FCF+F0*&F?F+F3F+F+**F2F+F6F+F@F+F9F+ F0**F2F+F6F+F@F+F:F+F0*(F?F+F6F+F9F+F+*(F2F+F6F+F@F+F0*&F?F+F6F+F+*(F? F+F@F+F9F+F+*&F?F+F@F+F+*(F?F+F3F+F:F+F+*(F?F+F3F+F9F+F+*(F?F+F3F+FCF+ F+*(F2F+F3F+F6F+F0*(F2F+F3F+F@F+F0*&F/F+F:F+F0*,\"#IF+F3F+F6F+F@F+F:F+ F+**F2F+F3F+F:F+F6F+F0**FZF+F3F+F6F+F@F+F+**F2F+F3F+F@F+F:F+F0**F2F+F3 F+F@F+F9F+F0*(F?F+F:F+F6F+F+F+,J**\"\"'F+F3F+F:F+F6F+F0**F/F+F3F+F6F+F @F+F+*(F\\oF+F3F+F6F+F0*(FHF+F3F+F9F+F+*(FHF+F3F+F:F+F+**F\\oF+F3F+F@F +F:F+F0*&FHF+F3F+F+*(F\\oF+F3F+F@F+F0*(FHF+F6F+F9F+F+*(FHF+F:F+F6F+F+* *F\\oF+F6F+F@F+F:F+F0*&FHF+F6F+F+*(F\\oF+F6F+F@F+F0*&FDF+FCF+F0*&FDF+F 9F+F0*(FHF+F@F+F9F+F+*&FDF+F:F+F0*(FHF+F@F+F:F+F+FDF0*&FHF+F@F+F+F0F+F +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6,$*&#\"\"\"\"\"&F+*&,:*&\"#7F+)&F%6#\"#<\"\"$F+F+*& \"\"*F+)F2\"\"#F+F+*(\"#:F+&F%6#\"#5F+F8F+!\"\"*(F;F+&F%6#F7F+F8F+F?*( F>F+FAF+F2F+F?*&\"\"'F+F2F+F+*(F>F+F2F+FF+F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 98 "#---------------------------------------- ---------------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "some consequencial relations" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The relations " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6 #\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#& %\"cG6#\"\"%" }{TEXT -1 14 " ------- (i) " }}{PARA 256 "" 0 "" {TEXT 267 8 "________" }{TEXT -1 16 " " }}{PARA 257 "" 0 "" {TEXT -1 5 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " c[5] = (4*c[4]-3*c[6])*c[6]/(6*c[4]-4*c[6]);" "6#/&%\"cG6#\"\"&*(,&*& \"\"%\"\"\"&F%6#F+F,F,*&\"\"$F,&F%6#\"\"'F,!\"\"F,&F%6#F3F,,&*&F3F,&F% 6#F+F,F,*&F+F,&F%6#F3F,F4F4" }{TEXT -1 15 " ------- (ii) " }}{PARA 0 "" 0 "" {TEXT -1 68 "arising from the stage-order conditions, together with the relation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[6] = (3*c[3]-1)/2;" "6#/&%\"cG6#\"\"'*&,&*&\"\"$\"\"\"&F%6#F+F,F, F,!\"\"F,\"\"#F/" }{TEXT -1 16 " ------- (iii) " }}{PARA 0 "" 0 "" {TEXT -1 87 "arising from the column simplifying conditions give rise \+ to other relations as follows." }}{PARA 0 "" 0 "" {TEXT -1 18 "Substit uting for " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 26 " f rom (i) in (iii) gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[6] = c[4]-1/2;" "6#/&%\"cG6#\"\"',&&F%6#\"\"%\"\"\"*&F,F,\"\"# !\"\"F/" }{TEXT -1 14 " ----- (iv). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 8 "________" }{TEXT -1 16 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Substituting for \+ " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 26 " from (iv) i n (ii) gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[5] = (c[4]+3/2)*(c[4]-1/2)/(2*c[4]+2)" "6#/&%\"cG6#\"\"&*(,&&F%6#\"\"%\" \"\"*&\"\"$F-\"\"#!\"\"F-F-,&&F%6#F,F-*&F-F-F0F1F1F-,&*&F0F-&F%6#F,F-F -F0F-F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[5] = (2*c[4]+3)*(2* c[4]-1)/(8*(c[4]+1))" "6#/&%\"cG6#\"\"&*(,&*&\"\"#\"\"\"&F%6#\"\"%F,F, \"\"$F,F,,&*&F+F,&F%6#F/F,F,F,!\"\"F,*&\"\")F,,&&F%6#F/F,F,F,F,F5" } {TEXT -1 13 " ------- (v)." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 265 14 "______________" }{TEXT -1 17 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "c[5]=(4 *c[4]-3*c[6])*c[6]/(6*c[4]-4*c[6]);\nsubs(c[6]=c[4]-1/2,%);\nsimplify( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&*(,&*&\"\"%\"\"\" &F%6#F+F,F,*&\"\"$F,&F%6#\"\"'F,!\"\"F,F1F,,&*&F3F,F-F,F,*&F+F,F1F,F4F 4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&*(,&&F%6#\"\"%\"\" \"#\"\"$\"\"#F-F-,&F*F-#F-F0!\"\"F-,&*&F0F-F*F-F-F0F-F3" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&,$*&#\"\"\"\"\")F+*(,&*&\"\"#F+&F% 6#\"\"%F+F+\"\"$F+F+,&*&F0F+F1F+F+F+!\"\"F+,&F1F+F+F+F7F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 41 "#------------------------------------ ----" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 51 "#--------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "construction of the scheme .. par t 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Th e scheme constructed here is a modification of one due to Hiroshi Ono. " }}{PARA 0 "" 0 "" {TEXT -1 76 "See: On the 25 stage 12th order expli cit Runge-Kutta method, by Hiroshi Ono." }}{PARA 0 "" 0 "" {TEXT -1 120 " Transactions of the Japan Society for Industrial and appli ed Mathematics, Vol. 6, No. 3, (2006) pages 177 to 186." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The scheme has the f ollowing rational values for specific nodes and weights." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 39/88;" "6#/&%\"cG6#\"\" #*&\"#R\"\"\"\"#))!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 101/ 228;" "6#/&%\"cG6#\"\"$*&\"$,\"\"\"\"\"$G#!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4] = 101/152;" "6#/&%\"cG6#\"\"%*&\"$,\"\"\"\"\"$_\"! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 8255/76912;" "6#/&%\"cG 6#\"\"&*&\"%b#)\"\"\"\"&7p(!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 6] = 25/152;" "6#/&%\"cG6#\"\"'*&\"#D\"\"\"\"$_\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 1/5;" "6#/&%\"cG6#\"\"**&\"\"\"F)\"\"&!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10]=1/3" "6#/&%\"cG6#\"#5*&\"\" \"F)\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[19]=1/3" "6#/&%\"c G6#\"#>*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[20] = 1/5;" "6#/&%\"cG6#\"#?*&\"\"\"F)\"\"&!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[22] = 25/152;" "6#/&%\"cG6#\"#A*&\"#D\"\"\"\"$_\"!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[23] = 101/228;" "6#/&%\"cG6#\" #B*&\"$,\"\"\"\"\"$G#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[24] = \+ 39/88;" "6#/&%\"cG6#\"#C*&\"#R\"\"\"\"#))!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[25]=1" "6#/&%\"cG6#\"#D\"\"\"" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2] = -11/100;" "6#/ &%\"bG6#\"\"#,$*&\"#6\"\"\"\"$+\"!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3] = -17/100;" "6#/&%\"bG6#\"\"$,$*&\"#<\"\"\"\"$+\"!\"\"F-" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[5] = 0;" "6#/&%\"bG6#\"\"&\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "b[6] = -19/100;" "6#/&%\"bG6#\"\"',$* &\"#>\"\"\"\"$+\"!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7] = -21 /100;" "6#/&%\"bG6#\"\"(,$*&\"#@\"\"\"\"$+\"!\"\"F-" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "b[8] = 0;" "6#/&%\"bG6#\"\")\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "b[9] = -13/56;" "6#/&%\"bG6#\"\"*,$*&\"#8\"\"\"\"#c! \"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[10] = -27/100;" "6#/&%\"bG 6#\"#5,$*&\"#F\"\"\"\"$+\"!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b [11] = -29/100;" "6#/&%\"bG6#\"#6,$*&\"#H\"\"\"\"$+\"!\"\"F-" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[12]=0" "6#/&%\"bG6#\"#7\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[18] = 29/10 0;" "6#/&%\"bG6#\"#=*&\"#H\"\"\"\"$+\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[19] = 27/100;" "6#/&%\"bG6#\"#>*&\"#F\"\"\"\"$+\"!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[20] = 13/56;" "6#/&%\"bG6#\"#? *&\"#8\"\"\"\"#c!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[21] = 21/10 0;" "6#/&%\"bG6#\"#@*&F'\"\"\"\"$+\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[22] = 19/100;" "6#/&%\"bG6#\"#A*&\"#>\"\"\"\"$+\"!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[23] = 17/100;" "6#/&%\"bG6#\"# B*&\"#<\"\"\"\"$+\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[24] = 11 /100;" "6#/&%\"bG6#\"#C*&\"#6\"\"\"\"$+\"!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The nodes :" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[13] = 1/2-sqrt (495+66*sqrt(15))/66" "6#/&%\"cG6#\"#8,&*&\"\"\"F*\"\"#!\"\"F**&-%%sqr tG6#,&\"$&\\F**&\"#mF*-F/6#\"#:F*F*F*F4F,F," }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[14] = 1/2-sqrt(495-66*sqrt(15))/66" "6#/&%\"cG6#\"#9, &*&\"\"\"F*\"\"#!\"\"F**&-%%sqrtG6#,&\"$&\\F**&\"#mF*-F/6#\"#:F*F,F*F4 F,F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[15]=1/2" "6#/&%\"cG6#\"#:*& \"\"\"F)\"\"#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[16] = 1/2+sqrt (495-66*sqrt(15))/66" "6#/&%\"cG6#\"#;,&*&\"\"\"F*\"\"#!\"\"F**&-%%sqr tG6#,&\"$&\\F**&\"#mF*-F/6#\"#:F*F,F*F4F,F*" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[17] = 1/2+sqrt(495+66*sqrt(15))/66" "6#/&%\"cG6#\"#<, &*&\"\"\"F*\"\"#!\"\"F**&-%%sqrtG6#,&\"$&\\F**&\"#mF*-F/6#\"#:F*F*F*F4 F,F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 33 "are the zeros of the derivative " }{XPPEDIT 18 0 "`P'` [6](x) = Diff(P[6](x),x);" "6#/-&%#P'G6#\"\"'6#%\"xG-%%DiffG6$-&%\"PG6 #F(6#F*F*" }{TEXT -1 9 " of the " }{TEXT 260 19 "Legendre polynomial " }{TEXT -1 2 " " }{XPPEDIT 18 0 "P[6](x);" "6#-&%\"PG6#\"\"'6#%\"xG " }{TEXT -1 36 " mapped linearly from the interval " }{XPPEDIT 18 0 " [-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 19 " to the interval " } {XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "They provide nodes for " }{TEXT 260 25 "Gauss-Lobatt o integration" }{TEXT -1 18 " on the interval " }{XPPEDIT 18 0 "[0,1] " "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(orthopoly[P](6,x),x);\n simplify(subs(x=2*z-1,%));\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"$$p\"\")\"\"\"*$)%\"xG\"\"&F(F(F(*&#\"$:$\"\"%F(*$)F+\"\"$F( F(!\"\"*&#\"$0\"F'F(F+F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&\"%s F\"\"\")%\"zG\"\"&F&F&*&\"%IpF&)F(\"\"%F&!\"\"*&\"%+jF&)F(\"\"$F&F&*& \"%?DF&)F(\"\"#F&F.*&\"$?%F&F(F&F&\"#@F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6'#\"\"\"\"\"#,&F#F$*&\"#m!\"\",&\"$&\\F$*&F(F$\"#:F#F$F#F),&F#F$* &F(F)F*F#F$,&F#F$*&F(F),&F+F$*&F(F$F-F#F)F#F),&F#F$*&F(F)F2F#F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "These nodes and their corresponding weights can be obtained from the quadrature conditions. " }}{PARA 0 "" 0 "" {TEXT -1 44 "We assume the following symmetry conditions." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2]=c[24]" "6#/&%\"cG6#\" \"#&F%6#\"#C" }{TEXT -1 12 ", " }{XPPEDIT 18 0 "c[3]=c[23]" "6#/&%\"cG6#\"\"$&F%6#\"#B" }{TEXT -1 12 ", " }{XPPEDIT 18 0 "c[6] = c[22];" "6#/&%\"cG6#\"\"'&F%6#\"#A" }{TEXT -1 12 ", \+ " }{XPPEDIT 18 0 "c[7]=c[21]" "6#/&%\"cG6#\"\"(&F%6#\"#@" }{TEXT -1 12 ", " }{XPPEDIT 18 0 "c[9] = c[20];" "6#/&%\"cG6#\"\"*&F%6 #\"#?" }{TEXT -1 12 ", " }{XPPEDIT 18 0 "c[10]=c[19]" "6#/&% \"cG6#\"#5&F%6#\"#>" }{TEXT -1 12 ", " }{XPPEDIT 18 0 "c[11] =c[18]" "6#/&%\"cG6#\"#6&F%6#\"#=" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]+ b[24]=0" "6#/,&&%\"bG6#\"\"#\"\"\"&F&6#\"#CF)\"\"!" }{TEXT -1 6 ", \+ " }{XPPEDIT 18 0 "b[3]+b[23]=0" "6#/,&&%\"bG6#\"\"$\"\"\"&F&6#\"#BF) \"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "b[6]+b[22] = 0;" "6#/,&&% \"bG6#\"\"'\"\"\"&F&6#\"#AF)\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "b[7]+b[21]=0" "6#/,&&%\"bG6#\"\"(\"\"\"&F&6#\"#@F)\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "b[9]+b[20] = 0;" "6#/,&&%\"bG6#\"\"*\"\"\"& F&6#\"#?F)\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "b[10]+b[19]=0" " 6#/,&&%\"bG6#\"#5\"\"\"&F&6#\"#>F)\"\"!" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "b[11]+b[18]=0" "6#/,&&%\"bG6#\"#6\"\"\"&F&6#\"#=F)\"\"! " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "We also initially specify that " }{XPPEDIT 18 0 "b[4]=0 " "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[6]=0" "6 #/&%\"bG6#\"\"'\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[9]=0" "6#/&% \"bG6#\"\"*\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[12]=0" "6#/&%\"b G6#\"#7\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[17]=1" "6#/&%\"c G6#\"#<\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "sym_eqs := [c[2]=c[24],c[3] =c[23],c[6]=c[22],c[7]=c[21],c[9]=c[20],c[10]=c[19],c[11]=c[18],\nb[2] =-b[24],b[3]=-b[23],b[6]=-b[22],b[7]=-b[21],b[9]=-b[20],b[10]=-b[19],b [11]=-b[18]]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "e1 := \{b[4]=0,b[5]=0,b[8]=0,b[12]=0,c[25]=1\}: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "Qeqs := QuadratureConditions(12,25,'expanded'):\nqua deqns := subs(e1,subs(sym_eqs,Qeqs)):\nconvert(ListTools[Enumerate](qu adeqns),matrix);\n``;\nindets(quadeqns);\nnops(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7.7$\"\"\"/,0&%\"bG6#\"#DF(&F,6#\"# \+ " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 130 "solve(\{op(quadeqns),c[13]+c[17]=1,c[14]+c[16 ]=1,b[1]=b[25],b[13]=b[17],b[14]=b[16]\}):\nsol := allvalues(%):\ninfo level[solve] := 0: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 29 "We select the solution with " }{XPPEDIT 18 0 "c[13] < \+ c[14];" "6#2&%\"cG6#\"#8&F%6#\"#9" }{XPPEDIT 18 0 "`` < c[16];" "6#2%! G&%\"cG6#\"#;" }{XPPEDIT 18 0 "`` < c[17];" "6#2%!G&%\"cG6#\"#<" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "phi := u_->evalb(subs(evalf(u_),c[13]%$phiGf*6#%#u_G6\"6$%)operatorG%&arrowGF(-%&evalbG6#-% %subsG6$-%&evalfG6#9$332&%\"cG6#\"#8&F:6#\"#92F=&F:6#\"#;2FA&F:6#\"# " 0 "" {MPLTEXT 1 0 81 "e2 := expand(rationalize(simplify(op(select(phi,[sol] )))));\ne3 := `union`(e1,e2):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e2 G<./&%\"cG6#\"#:#\"\"\"\"\"#/&F(6#\"#8,&F+F,*&\"#m!\"\",&\"$&\\F,*&F4F ,F*F+F,F+F5/&F(6#\"#<,&F+F,*&F4F5F6F+F,/&F(6#\"#;,&F+F,*&F4F5,&F7F,*&F 4F,F*F+F5F+F,/&F(6#\"#9,&F+F,*&F4F5FEF+F5/&%\"bG6#F,#F,\"#U/&FOFA,&#\" #J\"$v\"F,*&\"$+\"F5F*F+F,/&FOF0,&FVF,*&FZF5F*F+F5/&FOF;Fgn/&FOFIFU/&F O6#\"#DFQ/&FOF)#\"$G\"\"$D&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 408 "e3 := \{c[15] = 1/2, c[13] = 1/2-1/66*(495+66 *15^(1/2))^(1/2), c[17] = 1/2+1/66*(495+66*15^(1/2))^(1/2), b[4] = 0, \+ b[5] = 0, b[8] = 0, b[12] = 0, c[25] = 1, c[16] = 1/2+1/66*(495-66*15^ (1/2))^(1/2), c[14] = 1/2-1/66*(495-66*15^(1/2))^(1/2), b[1] = 1/42, b [16] = 31/175+1/100*15^(1/2), b[13] = 31/175-1/100*15^(1/2), b[17] = 3 1/175-1/100*15^(1/2), b[14] = 31/175+1/100*15^(1/2), b[25] = 1/42, b[1 5] = 128/525\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We now specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 39/88;" "6#/&%\"cG6#\"\"#*&\"#R \"\"\"\"#))!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 101/152;" " 6#/&%\"cG6#\"\"%*&\"$,\"\"\"\"\"$_\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[9] = 1/5;" "6#/&%\"cG6#\"\"**&\"\"\"F)\"\"&!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 1/3;" "6#/&%\"cG6#\"#5*&\"\" \"F)\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "e4 := \{c[2]=39/88,c[4]=101/ 152,c[9]=1/5,c[10]=1/3\}:\ne5 := `union`(e3,e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 463 "e5 := \{b[8] = 0, b[5] = 0, b[4] = 0, b[16] = 31/175+1/100*15^(1/2), b[25] = 1/42, b[13] = 31/175 -1/100*15^(1/2), b[1] = 1/42, c[16] = 1/2+1/66*(495-66*15^(1/2))^(1/2) , c[14] = 1/2-1/66*(495-66*15^(1/2))^(1/2), c[25] = 1, b[12] = 0, b[17 ] = 31/175-1/100*15^(1/2), c[17] = 1/2+1/66*(495+66*15^(1/2))^(1/2), b [14] = 31/175+1/100*15^(1/2), c[15] = 1/2, c[13] = 1/2-1/66*(495+66*15 ^(1/2))^(1/2), b[15] = 128/525, c[9] = 1/5, c[10] = 1/3, c[2] = 39/88, c[4] = 101/152\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "The nodes " }{XPPEDIT 18 0 "c[3]" "6#&% \"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5];" "6#&%\"cG6#\"\"& " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" } {TEXT -1 35 " are determined by the relations " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\" \"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = (2*c[4]+3)*(2*c[4]-1)/(8*(c[4] +1))" "6#/&%\"cG6#\"\"&*(,&*&\"\"#\"\"\"&F%6#\"\"%F,F,\"\"$F,F,,&*&F+F ,&F%6#F/F,F,F,!\"\"F,*&\"\")F,,&&F%6#F/F,F,F,F,F5" }{TEXT -1 11 " a nd " }{XPPEDIT 18 0 "c[6] = c[4]-1/2;" "6#/&%\"cG6#\"\"',&&F%6#\"\" %\"\"\"*&F,F,\"\"#!\"\"F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "e6 := simplify(subs(e5,\{c[3]=2/3*c[4],c[5]= (2*c[4]+3)*(2*c[4]-1)/(8*(c[4]+1)),c[6]=c[4]-1/2\}));\ne7 := `union`(e 5,e6):\nc[3]=subs(e7,c[3]),c[5]=subs(e7,c[5]),c[6]=subs(e7,c[6]);\n" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e6G<%/&%\"cG6#\"\"'#\"#D\"$_\"/&F( 6#\"\"&#\"%D#)\"&7p(/&F(6#\"\"$#\"$,\"\"$G#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"cG6#\"\"$#\"$,\"\"$G#/&F%6#\"\"&#\"%D#)\"&7p(/&F%6 #\"\"'#\"#D\"$_\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "e7 := \{b[8] = 0, b[5] = 0, b[4] = 0, b[16] = 31/175+1/100*15 ^(1/2), b[25] = 1/42, b[13] = 31/175-1/100*15^(1/2), b[1] = 1/42, c[16 ] = 1/2+1/66*(495-66*15^(1/2))^(1/2), c[14] = 1/2-1/66*(495-66*15^(1/2 ))^(1/2), c[25] = 1, b[12] = 0, b[17] = 31/175-1/100*15^(1/2), c[17] = 1/2+1/66*(495+66*15^(1/2))^(1/2), b[14] = 31/175+1/100*15^(1/2), c[15 ] = 1/2, c[13] = 1/2-1/66*(495+66*15^(1/2))^(1/2), b[15] = 128/525, c[ 9] = 1/5, c[10] = 1/3, c[2] = 39/88, c[4] = 101/152, c[6] = 25/152, c[ 5] = 8225/76912, c[3] = 101/228\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Substituting fo r " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 5 " in " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[7]=(3*c[9]^2-4*c[6]*c[9]+ 2*c[9]-2*c[6]+1)/( 4*c[9]+2-6*c[6])" "6#/&%\"cG6#\"\"(*&,,*&\"\"$\"\" \"*$&F%6#\"\"*\"\"#F,F,*(\"\"%F,&F%6#\"\"'F,&F%6#F0F,!\"\"*&F1F,&F%6#F 0F,F,*&F1F,&F%6#F6F,F9F,F,F,,(*&F3F,&F%6#F0F,F,F1F,*&F6F,&F%6#F6F,F9F9 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "gives " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 28 ". and then subsituting for " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6# \"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 5 " in " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]=( 20*c[6]*c[7]-15*c[7]*c[9]+12*c[9]^2-15*c[6]*c[9])*c[9]/(30*c[6]*c[7]-2 0*c[7]*c[9]+15*c[9]^2-20*c[6]*c[9])" "6#/&%\"cG6#\"\")*(,**(\"#?\"\"\" &F%6#\"\"'F,&F%6#\"\"(F,F,*(\"#:F,&F%6#F2F,&F%6#\"\"*F,!\"\"*&\"#7F,*$ &F%6#F9\"\"#F,F,*(F4F,&F%6#F/F,&F%6#F9F,F:F,&F%6#F9F,,**(\"#IF,&F%6#F/ F,&F%6#F2F,F,*(F+F,&F%6#F2F,&F%6#F9F,F:*&F4F,*$&F%6#F9F@F,F,*(F+F,&F%6 #F/F,&F%6#F9F,F:F:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "give s " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 285 "e 8 := `union`(e7,subs(e7,\{c[7]=(3*c[9]^2-4*c[6]*c[9]+2*c[9]-2*c[6]+1)/ (4*c[9]+2-6*c[6])\})):\ne9 := `union`(e8,subs(e8,\{c[8]=(20*c[6]*c[7]- 15*c[7]*c[9]+12*c[9]^2-15*c[6]*c[9])*c[9]/\n (30*c[6]*c[7]- 20*c[7]*c[9]+15*c[9]^2-20*c[6]*c[9])\})):\nc[7]=subs(e9,c[7]),c[8]=sub s(e9,c[8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"cG6#\"\"(#\"%8?\"% XM/&F%6#\"\")#\"&Fx\"\"']xF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " The nodes " }{XPPEDIT 18 0 "c[11]" "6#&%\"cG6#\"#6" }{TEXT -1 7 " an d " }{XPPEDIT 18 0 "c[12]" "6#&%\"cG6#\"#7" }{TEXT -1 37 " can be ob tained from the equations " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[11] = (12*c[17]^3+9*c[17] ^2-15*c[10]*c[17]^2-15*c[9]*c[17]^2-10*c[9]*c[17]+6*c[17]-10*c[17]*c[1 0]+20*c[10]*c[9]*c[17]+3-5*c[9]-5*c[10]+10*c[10]*c[9])/(5*(3*c[17]^2-4 *c[17]*c[10]+2*c[17]-4*c[9]*c[17]-2*c[9]-2*c[10]+1+6*c[10]*c[9]))" "6# /&%\"cG6#\"#6*&,:*&\"#7\"\"\"*$&F%6#\"#<\"\"$F,F,*&\"\"*F,*$&F%6#F0\" \"#F,F,*(\"#:F,&F%6#\"#5F,&F%6#F0F7!\"\"*(F9F,&F%6#F3F,&F%6#F0F7F?*(F< F,&F%6#F3F,&F%6#F0F,F?*&\"\"'F,&F%6#F0F,F,*(FF,F,*(F'F,&F%6#F5F,&F%6#F2FEF?**F+F,&F%6#F5F,&F%6#F2F,&F %6#F>F,F,*&F>F,*$&F%6#F2\"\"$F,F,*(F'F,&F%6#F>F,&F%6#F2FEF?F,&F%6#F2F, ,2**F7F,&F%6#F/F,&F%6#F2F,&F%6#F5F,F,**\"#IF,&F%6#F/F,&F%6#F5F,&F%6#F> F,F?*(F+F,&F%6#F/F,&F%6#F2FEF?**F7F,&F%6#F/F,&F%6#F2F,&F%6#F>F,F,*(F+F ,&F%6#F5F,&F%6#F2FEF?**F7F,&F%6#F5F,&F%6#F2F,&F%6#F>F,F,*&F'F,*$&F%6#F 2FgnF,F,*(F+F,&F%6#F>F,&F%6#F2FEF?F?" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 550 "eqA := c[11] = (12*c[17]^3+9*c[17]^2-15*c[10 ]*c[17]^2-15*c[9]*c[17]^2-10*c[9]*c[17]+6*c[17]-10*c[17]*c[10]+20*c[10 ]*c[9]*c[17]+3-5*c[9]-5*c[10]+10*c[10]*c[9])/(5*(3*c[17]^2-4*c[17]*c[1 0]+2*c[17]-4*c[9]*c[17]-2*c[9]-2*c[10]+1+6*c[10]*c[9])):\neqB := c[12] = (15*c[9]*c[13]*c[11]-20*c[9]*c[11]*c[10]-12*c[9]*c[13]^2+15*c[9]*c[ 13]*c[10]-12*c[11]*c[13]^2+15*c[11]*c[13]*c[10]+10*c[13]^3-12*c[10]*c[ 13]^2)*c[13]/(20*c[9]*c[13]*c[11]-30*c[9]*c[11]*c[10]-15*c[9]*c[13]^2+ 20*c[9]*c[13]*c[10]-15*c[11]*c[13]^2+20*c[11]*c[13]*c[10]+12*c[13]^3-1 5*c[10]*c[13]^2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 225 "e10 := `union`(e9,\{simplify(expand(rationa lize((subs(e9,eqA)))))\}):\ne11 := `union`(e10,\{simplify(expand(ratio nalize((subs(e10,eqB)))))\}):\nc[11]=subs(e11,c[11]);\n``;\nevalf[60]( %%);\n``;\nc[12]=subs(e11,c[12]);\n``;\nevalf[60](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6,*#\")v!*y^\")ZF(\\'\"\"\"*(\"'SKLF,F +!\"\"\"#:#F,\"\"#F,*(\"(eu0$F,\"*<-q9(F/,&\"$&\\F,*&\"#mF,F0F1F,F1F,* *\"&3/(F,F5F/F6F1F0F1F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6$\"gnTP'[&z_aZ3Wk8fStB?dP Zo\\jb1ag6Y%*!#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7,***\",a^*y#)[\"\"\"\"0l+E<'z9@!\" \",&\"$&\\F+*&\"#mF+\"#:#F+\"\"#F+F3F2F3F-*(\".Z*GiF!3\"F+\"/bLv0K\\qF -F.F3F-#\".JNe9O_#\".0.Dt%3kF+*(\",$35B(Q$F+\".h+l%p\"G\"F-F2F3F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#7$\"fnxVm=bD%Go*pv\\u?&=gahNe.h%G/o%ez^!#g" }}}{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 0 883 "e11 := \{c[12] = -4882 7895154/211479617260065*(495+66*15^(1/2))^(1/2)*15^(1/2)-1080276228947 /70493205753355*(495+66*15^(1/2))^(1/2)+2523614583531/6408473250305+33 872310083/1281694650061*15^(1/2), b[8] = 0, b[5] = 0, b[4] = 0, b[16] \+ = 31/175+1/100*15^(1/2), b[25] = 1/42, b[13] = 31/175-1/100*15^(1/2), \+ b[1] = 1/42, c[16] = 1/2+1/66*(495-66*15^(1/2))^(1/2), c[14] = 1/2-1/6 6*(495-66*15^(1/2))^(1/2), c[25] = 1, b[12] = 0, b[17] = 31/175-1/100* 15^(1/2), c[17] = 1/2+1/66*(495+66*15^(1/2))^(1/2), b[14] = 31/175+1/1 00*15^(1/2), c[15] = 1/2, c[13] = 1/2-1/66*(495+66*15^(1/2))^(1/2), b[ 15] = 128/525, c[11] = 51789075/64972747+333240/64972747*15^(1/2)+3057 458/714700217*(495+66*15^(1/2))^(1/2)+70408/714700217*(495+66*15^(1/2) )^(1/2)*15^(1/2), c[9] = 1/5, c[10] = 1/3, c[2] = 39/88, c[4] = 101/15 2, c[8] = 17727/277750, c[6] = 25/152, c[5] = 8225/76912, c[3] = 101/2 28, c[7] = 2013/3445\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The nodes " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }{TEXT -1 6 " to " }{XPPEDIT 18 0 "c[1 7]" "6#&%\"cG6#\"#<" }{TEXT -1 23 " are given as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "[[se q(c[i]=subs(e11,c[i]),i=2..17)]]:\nlinalg[transpose](%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#727#/&%\"cG6#\"\"##\"#R\"#))7#/&F*6# \"\"$#\"$,\"\"$G#7#/&F*6#\"\"%#F6\"$_\"7#/&F*6#\"\"&#\"%D#)\"&7p(7#/&F *6#\"\"'#\"#DF>7#/&F*6#\"\"(#\"%8?\"%XM7#/&F*6#\"\")#\"&Fx\"\"']xF7#/& F*6#\"\"*#\"\"\"FC7#/&F*6#\"#5#F^oF47#/&F*6#\"#6,*#\")v!*y^\")ZF(\\'F^ o*(\"'SKLF^oF]p!\"\"\"#:#F^oF,F^o*(\"(eu0$F^o\"*<-q9(F`p,&\"$&\\F^o*& \"#mF^oFapFbpF^oFbpF^o**\"&3/(F^oFepF`pFfpFbpFapFbpF^o7#/&F*6#\"#7,*** \",a^*y#)[F^o\"0l+E<'z9@F`pFfpFbpFapFbpF`p*(\".Z*GiF!3\"F^o\"/bLv0K\\q F`pFfpFbpF`p#\".JNe9O_#\".0.Dt%3kF^o*(\",$35B(Q$F^o\".h+l%p\"G\"F`pFap FbpF^o7#/&F*6#\"#8,&FbpF^o*&FipF`pFfpFbpF`p7#/&F*6#\"#9,&FbpF^o*&FipF` p,&FgpF^o*&FipF^oFapFbpF`pFbpF`p7#/&F*6#FapFbp7#/&F*6#\"#;,&FbpF^o*&Fi pF`pF\\sFbpF^o7#/&F*6#\"#<,&FbpF^o*&FipF`pFfpFbpF^oQ)pprint216\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "#-------- ------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 11 "The nodes " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\" #" }{TEXT -1 6 " to " }{XPPEDIT 18 0 "c[17]" "6#&%\"cG6#\"#<" } {TEXT -1 37 " are given approximately as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "[[seq(c[i]=e valf[65](subs(e11,c[i])),i=2..17)]]:\nlinalg[transpose](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#727#/&%\"cG6#\"\"#$\"\\o======== =======================V%!#l7#/&F*6#\"\"$$\"\\o9cC)H>x3NShX#)H>x3NShX# )H>x3NShX#)HWF/7#/&F*6#\"\"%$\"\\o@%ot%*y:j_5Uot%*y:j_5Uot%*y:j_5UotWm F/7#/&F*6#\"\"&$\"\\oP'[ms,J()H_)H&z6>$Q%*4Y7Vh@B7;vT*RSp5F/7#/&F*6#\" \"'$\"\\o@%ot%*y:j_5Uot%*y:j_5Uot%*y:j_5UotW;F/7#/&F*6#\"\"($\"\\o\\JZ myA(=RqN'ek&*pi7.-u5M&)3^K%eF/7#/&F*6#\"\")$\"\\oCeBeBeBeBeBeBeBe BeBeBeBeBeBeBeBQ'!#m7#/&F*6#\"\"*$\"\\o+++++++++++++++++++++++++++++++ +#F/7#/&F*6#\"#5$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL$F/7#/&F*6#\"#6$ \"\\o-i2uj[&z_aZ3Wk8fStB?dPZo\\jb1ag6Y%*F/7#/&F*6#\"#7$\"\\o()e$oPk'=b D%Go*pv\\u?&=gahNe.h%G/o%ez^FZ7#/&F*6#\"#8$\"[o(RR$f/U&>+kv\"[T1-Vni,$ *Q)R1Nlrg=0))[)F/7#/&F*6#\"#9$\"\\oJq_m[9ixS;k7?(HNohX!fS6)4$*GkkKgvbE F/7#/&F*6#\"#:$\"\\o++++++++++++++++++++++++++++++++&F/7#/&F*6#\"#;$\" \\opHZL^&yB#f$et)z-Z;$Qa4%f)=!p5d`t'RCWtF/7#/&F*6#\"#<$\"\\o.1mS&zX!)* fV#=&e$zpDt$)p5;g$\\Y$GR\"[>6:*F/Q)pprint456\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 51 "#---------------------------------------------- ----" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "construction of the schem e .. part 2" }}{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 0 883 "e11 := \{c[12] = -48827895154/211479617260065*(495+66*15^(1/2))^( 1/2)*15^(1/2)-1080276228947/70493205753355*(495+66*15^(1/2))^(1/2)+252 3614583531/6408473250305+33872310083/1281694650061*15^(1/2), b[8] = 0, b[5] = 0, b[4] = 0, b[16] = 31/175+1/100*15^(1/2), b[25] = 1/42, b[13 ] = 31/175-1/100*15^(1/2), b[1] = 1/42, c[16] = 1/2+1/66*(495-66*15^(1 /2))^(1/2), c[14] = 1/2-1/66*(495-66*15^(1/2))^(1/2), c[25] = 1, b[12] = 0, b[17] = 31/175-1/100*15^(1/2), c[17] = 1/2+1/66*(495+66*15^(1/2) )^(1/2), b[14] = 31/175+1/100*15^(1/2), c[15] = 1/2, c[13] = 1/2-1/66* (495+66*15^(1/2))^(1/2), b[15] = 128/525, c[11] = 51789075/64972747+33 3240/64972747*15^(1/2)+3057458/714700217*(495+66*15^(1/2))^(1/2)+70408 /714700217*(495+66*15^(1/2))^(1/2)*15^(1/2), c[9] = 1/5, c[10] = 1/3, \+ c[2] = 39/88, c[4] = 101/152, c[8] = 17727/277750, c[6] = 25/152, c[5] = 8225/76912, c[3] = 101/228, c[7] = 2013/3445\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "The zero linking coefficients in stages 2 to 13 are indicated by the \+ following array." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[c[2], a[2,1], ``, ``, ``, ``, \+ ``, ``, ``, ``, ``, ``, ``], [c[3], a[3,1], a[3,2], ``, ``, ``, ``, `` , ``, ``, ``, ``, ``], [c[4], a[4,1], 0, a[4,3], ``, ``, ``, ``, ``, ` `, ``, ``, ``], [c[5], a[5,1], 0, a[5,3], a[5,4], ``, ``, ``, ``, ``, \+ ``, ``, ``], [c[6], a[6,1], 0, 0, a[6,4], a[6,5], ``, ``, ``, ``, ``, \+ ``, ``], [c[7], a[7,1], 0, 0, a[7,4], a[7,5], a[7,6], ``, ``, ``, ``, \+ ``, ``], [c[8], a[8,1], 0, 0, 0, a[8,5], a[8,6], a[8,7], ``, ``, ``, ` `, ``], [c[9], a[9,1], 0, 0, 0, 0, a[9,6], a[9,7], a[9,8], ``, ``, ``, ``], [c[10], a[10,1], 0, 0, 0, 0, a[10,6], a[10,7], a[10,8], a[10,9], ``, ``, ``], [c[11], a[11,1], 0, 0, 0, 0, 0, a[11,7], a[11,8], a[11,9 ], a[11,10], ``, ``], [c[12], a[12,1], 0, 0, 0, 0, 0, 0, a[12,8], a[12 ,9], a[12,10], a[12,11], ``], [c[13], a[13,1], 0, 0, 0, 0, 0, 0, 0, a[ 13,9], a[13,10], a[13,11], a[13,12]]])" "6#-%'matrixG6#7.7/&%\"cG6#\" \"#&%\"aG6$F+\"\"\"%!GF0F0F0F0F0F0F0F0F0F07/&F)6#\"\"$&F-6$F4F/&F-6$F4 F+F0F0F0F0F0F0F0F0F0F07/&F)6#\"\"%&F-6$F " 0 "" {MPLTEXT 1 0 196 "e12 := \{seq(a[i,2]=0,i=4..13),seq (a[i,3]=0,i=6..13),seq(a[i,4]=0,i=8..13),\n seq(a[i,5]=0,i=9..1 3),a[11,6]=0,seq(a[13,j]=0,j=6..8),a[12,10]=-11/342,a[12,11]=1/5499\}: \ne13 := `union`(e11,e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We set up a system of equations from the " } {TEXT 260 22 "stage order conditions" }{TEXT -1 144 " that apply to ro ws 3 to 13, but remove three of the stage-order conditions so that the system of equations we construct is not over-determined." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[stage, `|`, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1 6, 17, 18, 19, 20, 21, 22, 23, 24, 25], [`stage-order`, `|`, 2, 3, 3, \+ 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 5, 5, 5, 1, 1, 1, 1, 6]]);" "6#-%' matrixG6#7$7;%&stageG%\"|grG\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"# 6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D7;%,stage-ord erGF)\"\"#F*F*F+F+F+F,F,F,F,F-F-F-F-F-F,F,F,\"\"\"FDFDFDF-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "RSeqs := RowSumConditions (13,'expanded'):\nSOeqs := [op(StageOrderConditions(2,13,'expanded')), op(StageOrderConditions(3,5..13,'expanded')),\n op(StageOrderC onditions(4,7..13,'expanded')),op(StageOrderConditions(5,10..13,'expan ded'))]:\ncdns := [op(RSeqs),op(SOeqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "There are 43 equations and 43 unkn own coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "eqns := simplify(expand(subs(e13,cdns))):\nnops(%);\nindets(eqns);\nnops(%); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#V" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "e14 := solve(\{op(eqns)\}):\ne15 := `union`(e13,e14): \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 "e15" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11801 "e15 := \{c[12] = -48827895154/211479617260065*(495 +66*15^(1/2))^(1/2)*15^(1/2)-1080276228947/70493205753355*(495+66*15^( 1/2))^(1/2)+2523614583531/6408473250305+33872310083/1281694650061*15^( 1/2), b[8] = 0, b[5] = 0, b[4] = 0, b[16] = 31/175+1/100*15^(1/2), b[2 5] = 1/42, b[13] = 31/175-1/100*15^(1/2), b[1] = 1/42, c[16] = 1/2+1/6 6*(495-66*15^(1/2))^(1/2), c[14] = 1/2-1/66*(495-66*15^(1/2))^(1/2), c [25] = 1, b[12] = 0, b[17] = 31/175-1/100*15^(1/2), c[17] = 1/2+1/66*( 495+66*15^(1/2))^(1/2), b[14] = 31/175+1/100*15^(1/2), c[15] = 1/2, c[ 13] = 1/2-1/66*(495+66*15^(1/2))^(1/2), b[15] = 128/525, a[11,6] = 0, \+ c[11] = 51789075/64972747+333240/64972747*15^(1/2)+3057458/714700217*( 495+66*15^(1/2))^(1/2)+70408/714700217*(495+66*15^(1/2))^(1/2)*15^(1/2 ), a[13,5] = 0, a[12,5] = 0, a[11,5] = 0, a[10,5] = 0, a[9,5] = 0, a[1 2,3] = 0, a[11,3] = 0, a[10,3] = 0, a[9,3] = 0, a[8,3] = 0, a[7,3] = 0 , a[6,3] = 0, a[13,4] = 0, a[12,4] = 0, a[11,4] = 0, a[10,4] = 0, a[9, 4] = 0, a[8,4] = 0, a[10,2] = 0, a[9,2] = 0, a[8,2] = 0, a[7,2] = 0, a [6,2] = 0, a[5,2] = 0, a[4,2] = 0, a[13,3] = 0, a[13,2] = 0, a[12,2] = 0, a[11,2] = 0, a[13,8] = 0, a[13,7] = 0, a[13,6] = 0, c[9] = 1/5, c[ 10] = 1/3, a[10,7] = 5620721356384469075/3545698345096452809913, a[10, 1] = -17689789589/433566079650, c[2] = 39/88, a[10,8] = 10436097770007 465488281250/34480588269890588098919643, a[8,1] = 56734933487268443970 56253/158826937830690185546875000, a[8,5] = 83874430427399451044535741 888/1824862440224861393218994140625, a[10,6] = -46012134893475328/7661 6158064651325, a[8,6] = -986894364933740544726981696/54826948196505455 902099609375, a[9,1] = 2526695731/133816691250, a[9,6] = 2446788660275 2/291937806983125, a[7,4] = 46426400659946403657/610018749211776263125 , a[7,1] = 1534965908593628117013/2925190767628355078125, a[7,6] = 110 05843237812473721/5193844391648046875, a[8,7] = 5672317641427470160608 354913689/40381865475711712996627696578125000, a[9,8] = 10950503629357 18750/11254693571026117461, a[10,9] = 110139186925/164279737386, a[7,5 ] = -156479396217570600333298656/73273972406867506457421875, a[5,1] = \+ 8719384129925/100440274363616, a[5,4] = -2900926450625/200880548727232 , a[5,3] = 6944404306875/200880548727232, a[9,7] = 11944081475573/1322 478962029186830, a[6,1] = 1167025/30304848, a[6,4] = 390625/3949854672 , a[6,5] = 404856925/3216589572, a[3,1] = 28078/126711, a[3,2] = 11221 1/506844, a[4,1] = 101/608, a[4,3] = 303/608, a[12,9] = -2812112309092 4227718193133410699967950378769045003781451294718634251047915103875622 2724747330456172743380029/17772194148441545239165648144261871500391656 432256173921137707234507471866579929942450646636509884128799622+676374 0877307754299524494087133025918627479792756821750588456196513199369023 804229800976446366900597141/384929481231135915944675073516609735767633 88417275663680176970401792228430972341222548509067597756397660*(495+66 *15^(1/2))^(1/2)*15^(1/2)+36476119417835236391068396812936741450147529 01446257471407287711249070189350233573351451468961784464641/6415491353 8522652657445845586101622627938980695459439466961617336320380718287235 37091418177932959399610*(495+66*15^(1/2))^(1/2)-5945858035882701232660 1158015692968164149411139000290745613838388905948341143108254364095422 742138234247/128309827077045305314891691172203245255877961390918878933 92323467264076143657447074182836355865918799220*15^(1/2), a[11,7] = 18 993974973485227904952341910291889381118708193023824550/120600346650805 22994492944014191159522924996678137100092619*(495+66*15^(1/2))^(1/2)*1 5^(1/2)+37364999682821940340524213896266047309233288831638123551875/42 880123253619637313752689828235233859288877077820800329312+838708650164 81720496920100044907813868655001976949702225/1340003851675613666054771 557132351058102777408681900010291*15^(1/2)+101882217739425975585850562 279744393868968529126539562125/402001155502684099816431467139705317430 8332226045700030873*(495+66*15^(1/2))^(1/2), a[12,7] = 120395605263827 6744439598162243722245159009039634735838771465034509936951515837384129 84486608167592412726499223108027/2950634943857949765325588035995009291 4469387576187449738113057156690702051511248959452612515855642377422918 7413128270-45742907135783126373462537143661743332646830734536211959233 4365398748049518495290294028445214608755002615341437/12781611192800302 2106371584838423621028673976938217239497998948047176530437562265364750 324954973542895485894482620*(495+66*15^(1/2))^(1/2)*15^(1/2)-194091304 4462983950305441883315112017888766063727491806262689934670829763302390 46596838838518978497037496895389/1278161119280030221063715848384236210 2867397693821723949799894804717653043756226536475032495497354289548589 448262*(495+66*15^(1/2))^(1/2)+127804725683516766786140751534474181571 5639522782419860193188824482580443365812010191019950554809391125710454 6897/12781611192800302210637158483842362102867397693821723949799894804 7176530437562265364750324954973542895485894482620*15^(1/2), a[12,6] = \+ 2510855210832371087772481156946088716323360997203425638520449710419038 37740861882751285544579556435153690263824896/1090595815805663574044725 3172792547498045678798706251505169170513806180033868049458919144773539 359368893655280625-172242031650876272198839853212806695960945558133287 845511282815171786743955334377646964770699567651065020416/690469019186 8715251945079564920891103542689964359766701594916437990617305392877150 31284885947411166121788875*(495+66*15^(1/2))^(1/2)*15^(1/2)-2483265050 9666020262656713720023165781119954282152925003984702088217050133522422 78156416063385897318005911552/2992032416476443275842867811465719478201 832317889232237357797123129267499003580098468901172438781719861085125* (495+66*15^(1/2))^(1/2)+9894140133479791042465990020353916090461157452 5495141245531026292768496305967031119826379359226270888023523328/14960 1620823822163792143390573285973910091615894461611867889856156463374950 17900492344505862193908599305425625*15^(1/2), a[12,1] = 45360785864184 2934653550493130067801432656091871963472207879414392900586486760399119 2705032200412298285836937/11726066565369264241035137088606164503095345 11569707813821852440031214272905699213661387392662314110822638750-5364 4358955165736694206216046324955151216394767306749400786848404923755537 638953128683448772183267658/133671517907153930451935491132384488709863 3780843915294531482553071844639268719963591518065572670919625*(495+66* 15^(1/2))^(1/2)*15^(1/2)-352791365485048385267153663396939334514393712 8361020345607398270646119601220687075114545149449903002764/25397588402 3592467858677433151530528548740418360343905960981685083650481461056793 08238843245880747472875*(495+66*15^(1/2))^(1/2)+1361420982903372918496 3097602829580265823874321154447805758789754253663967458230331500911195 3003119707396/12698794201179623392933871657576526427437020918017195298 0490842541825240730528396541194216229403737364375*15^(1/2), a[13,1] = \+ 49032552280925998159/1175735428193602866744+1095947848209481903/176360 3142290404300116*15^(1/2)-7475873553226772611/6466544855064815767092*( 495+66*15^(1/2))^(1/2)+5352672249559690835/116397807391166683807656*(4 95+66*15^(1/2))^(1/2)*15^(1/2), a[13,10] = 1036874421023669248333689/1 9655651658225505790722624+587392382348212140203319/3931130331645101158 1445248*15^(1/2)-259903711964372646898395/432424336480961127395897728* (495+66*15^(1/2))^(1/2)*15^(1/2)-185211952023363244825707/108106084120 240281848974432*(495+66*15^(1/2))^(1/2), a[13,9] = -162384252323883015 198625/10650929594044147603352784*15^(1/2)+122383789982155868115425/46 8640902137942494547522496*(495+66*15^(1/2))^(1/2)*15^(1/2)+22650501572 36155663680125/14201239458725530137803712-120963267116719524969875/260 35605674330138585973472*(495+66*15^(1/2))^(1/2), a[11,9] = 28743974673 994042985226536878679230542480801150/175398340822784745177474854434859 7377943356553311*(495+66*15^(1/2))^(1/2)*15^(1/2)+29199171561549944970 6434755611025178229310468246875/56127469063291118456791953419155116094 187409705952+1071091749312998743815087165568469189531652126075/1753983 408227847451774748544348597377943356553311*15^(1/2)+353259784704351651 896312581655773178339865859125/175398340822784745177474854434859737794 3356553311*(495+66*15^(1/2))^(1/2), a[13,11] = 14514838889000707433801 4877677070795676982463/1292134149111656271966122106166449116832367968+ 289384776308176063330366809130393474082987849/155056097893398752635934 65273997389401988415616*15^(1/2)-2741150225892371789772317271486124535 01574585/511685123048215883698584354041913850265617715328*(495+66*15^( 1/2))^(1/2)*15^(1/2)-24857236960157204698513148202889704149473711/5330 053365085582121860253687936602606933517868*(495+66*15^(1/2))^(1/2), a[ 13,12] = 6443329452303863103711607826787077523918806208419166908252875 /48186908177657952343922139838789383990650083655460342201310944-171491 601151866911759544690813927486314725154315627741296625/903504528331086 6064485401219773009498246890685398814162745802*15^(1/2)-70902020152377 73474138169907437581683377756583694650388556125/2385251954794068641024 145922020074507537179140945286938964891728*(495+66*15^(1/2))^(1/2)+119 929995185051271338440264753881499366865434058651556229025/144560724532 973857031766419516368151971950250966381026603932832*(495+66*15^(1/2))^ (1/2)*15^(1/2), a[11,1] = 36250222198273693341131051026706723455692594 208/4999430483339723613905130972844241982587217763497*(495+66*15^(1/2) )^(1/2)*15^(1/2)+4861393204748134765917698036236910531265683193125/222 1969103706543828402280432375218658927652339332+14844281419468240123270 6118820643700603232458928/55549227592663595710057010809380466473191308 4833*15^(1/2)+143068868482411022237818049543538495099770530576/1666476 827779907871301710324281413994195739254499*(495+66*15^(1/2))^(1/2), a[ 12,8] = -3406598039589848782099314380271461341386607746012495410255775 46738764928298414303784311737346180734782282094165416156250/3082807063 3629320680961962978966603106322399038887894512718700999171421270070039 408336730423089458130745933155208375541+411948325656738822039997775092 1192364212933819981045324437683038272268098097268606484648262927673203 2063112500000/35142517508098583816059600080898513623932604948403377121 9847713817031123765375550406811559375414180212098687404767*(495+66*15^ (1/2))^(1/2)*15^(1/2)+267192777390828594436663466788077170914794954921 9658896717126925052525708719967923978519767792683025169075208750000/66 7707832653873092505132401537071758854719494019664165317710656252359135 1542135457729419628132869424029875060690573*(495+66*15^(1/2))^(1/2)-20 8680305765271225902781340429224706810764563450989964806472300765563647 61794880724541437909825056194904756420000000/6677078326538730925051324 0153707175885471949401966416531771065625235913515421354577294196281328 69424029875060690573*15^(1/2), a[11,10] = -377685957124001752132518096 635999239114085793540/408066831170924993283384570103245857808084168365 71*(495+66*15^(1/2))^(1/2)*15^(1/2)-1817814809245296229960335698010451 499156358365559525/652906929873479989253415312165193372492934669385136 -14248117750070402721956352075060738058061271231816/408066831170924993 28338457010324585780808416836571*15^(1/2)-4844266659916699015885626279 835272453019208362650/408066831170924993283384570103245857808084168365 71*(495+66*15^(1/2))^(1/2), a[11,8] = -2754138826372250841538793956331 51790451174339761235000000000/1736570677629765561734121675040476983622 7513968915818501163621*(495+66*15^(1/2))^(1/2)*15^(1/2)-90307060671915 00671304838647417536518671082752129344726562500/1929522975144183957482 357416711641092914168218768424277907069-113106637991389701974884748601 2312835922421062622057500000000/19295229751441839574823574167116410929 14168218768424277907069*15^(1/2)-1097562530284460395919027812165520839 910017377896612500000000/578856892543255187244707225013492327874250465 6305272833721207*(495+66*15^(1/2))^(1/2), a[12,11] = 1/5499, a[12,10] \+ = -11/342, a[2,1] = 39/88, c[4] = 101/152, c[8] = 17727/277750, c[6] = 25/152, c[5] = 8225/76912, c[3] = 101/228, c[7] = 2013/3445\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "We now have values for all the linking coefficients in rows 2 to \+ 13 of the Butcher tableau for the scheme." }}{PARA 0 "" 0 "" {TEXT -1 68 "The coefficients are expressed in terms of the radical expressions " }{XPPEDIT 18 0 "sqrt(15)" "6#-%%sqrtG6#\"#:" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "sqrt(495+66*sqrt(15)))" "6#-%%sqrtG6#,&\"$&\\\"\"\"* &\"#mF(-F$6#\"#:F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a[13,9] = subs(e15,a[13,9]);\n``;\nevalf[65](%%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*,**(\"9D')>:I)QK_UQi \"\"\"\"\";%y_LgZT/%fH4l5!\"\"\"#:#F,\"\"#F.**\"9Da6oe:#)**y$QA\"F,\"< '\\Ava%\\Uz8-4ko%F.,&\"$&\\F,*&\"#mF,F/F0F,F0F/F0F,#\":D,ojc:Os:]]E#\" ;7P!y8Ibse%R7?9F,*(\"9v)p\\_>n6nK'47F,\";sM(feQ,LucgNg#F.F5F0F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#8\"\"*$\"inzBi2,&)H!\\S>Z=?-VIjD$4&z6e#4))fr-!p)!#l" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The coeff icients in the rows 2 to 10 of the Butcher tableau are all rational." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 386 "subs(e15,matrix([[c[2],a[ 2,1],``$3],\n[c[3],a[3,1],a[3,2],``$2],\n[c[4],a[4,1],a[4,2],a[4,3],`` ],[c[5],seq(a[5,i],i=1..4)],\n[c[6],seq(a[6,i],i=1..4)],[``$4,a[6,5]], \n[c[7],seq(a[7,i],i=1..4)],[``$3,a[7,5],a[7,6]],\n[c[8],seq(a[8,i],i= 1..4)],[``$2,seq(a[8,i],i=5..7)],\n[c[9],seq(a[9,i],i=1..4)],[``,seq(a [9,i],i=5..8)],\n[c[10],seq(a[10,i],i=1..4)],[``,seq(a[10,i],i=5..8)], [``$4,a[10,9]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#717' #\"#R\"#))F(%!GF+F+7'#\"$,\"\"$G##\"&y!G\"'6n7#\"'6A6\"'Wo]F+F+7'#F.\" $_\"#F.\"$3'\"\"!#\"$.$F:F+7'#\"%D#)\"&7p(#\".D*HTQ>()\"0;OOu-W+\"F;# \".voI/W%p\"0Kss[0)3?#!.D1XE4!HFG7'#\"#DF8#\"(Dq;\"\")[[IIF;F;#\"'D1R \"+sY&)\\R7'F+F+F+F+#\"*Dp&[S\"+s&*e;K7'#\"%8?\"%XM#\"78q6GOf3f'\\`\" \"7D\"y]NGww!>DHF;F;#\"5dOSY*f1SEk%\"6DJEw<@\\(=+h7'F+F+F+#!&7'#\"&Fx\"\"']xF#\":` i0(RWos[L\\tc\"<+](oa&=!pIy$p#)e\"F;F;F;7'F+F+#\">))=uNX/^%*RF/VuQ)\"@ D19%**=KRh[ASC'[#=#!<'p\")psW0uL\\O%*o)*\">v$4'*4-fX0l>[p#[&#\"@*o8\\N 31;quUTwJsc\"D+]7ylpFm*Hr6dZl=QS7'#\"\"\"\"\"&#\"+JdpED\"-]7p;Q8F;F;F; 7'F+F;#\"/_Fg')yYC\"0DJ)p!y$>H#\"/tbZ\"3W>\"\"4Io=H?'*yCK\"#\"4](=d$HO ]]4\"\"5hu6E5d$pa7\"7'#Fep\"\"$#!,*e*y*o<\"-]'zgcL%F;F;F;7'F+F;#!2G`Z$ *[87g%\"2D8lk!ehhw#\"4v!pWQc8s?c\"78*4GX'4X$)pXN#\";]7G)[lu+qx4O/\"\"; V'>*)4)e!*)p#)e![M7'F+F+F+F+#\"-Dp=R,6\"-'QP(zU;Q)pprint476\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The rows \+ 2 to 13 of the Butcher tableau are given approximately as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "evalf[20](subs(e15,matrix([s eq([c[i],seq(a[i,j],j=1..i-1),``$(13-i)],i=2..13)]))):\nevalf[5](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7/$\"&=V%!\"&F(%!GF+F+ F+F+F+F+F+F+F+F+7/$\"&)HWF*$\"&f@#F*$\"&R@#F*F+F+F+F+F+F+F+F+F+F+7/$\" &Zk'F*$\"&7m\"F*$\"\"!F9$\"&O)\\F*F+F+F+F+F+F+F+F+F+7/$\"&%p5F*$\"&7o) !\"'F8$\"&qX$FA$!&TW\"FAF+F+F+F+F+F+F+F+7/$\"&Zk\"F*$\"&5&QFAF8F8$\"&' *))*!\"*$\"&(e7F*F+F+F+F+F+F+F+7/$\"&L%eF*$\"&uC&F*F8F8$\"&2h(FA$!&b8# !\"%$\"&!>@FYF+F+F+F+F+F+7/$\"&CQ'FA$\"&@d$FAF8F8F8$\"&if%FA$!&+!=FA$ \"&ZS\"!\")F+F+F+F+F+7/$\"&++#F*$\"&#))=FAF8F8F8F8$\"&7Q)FA$\"&;.*!#5$ \"&(H(*FAF+F+F+F+7/$\"&LL$F*$!&,3%FAF8F8F8F8$!&b+'F*$\"&_e\"!\"($\"&n- $F*$\"&Wq'F*F+F+F+7/$\"&hW*F*$\"&VM'FYF8F8F8F8F8$\"&`(>FY$!&GQ\"!\"$$ \"&C[\"Feq$!&5P)FYF+F+7/$\"&'z^FA$!&\"\\VFAF8F8F8F8$!&4r&F*$!&'Q[Fao$ \"&AZ#F*$\"&i^%F*$!&k@$FA$\"&&==FaoF+7/$\"&))[)FA$\"&;t\"FAF8F8F8F8F8F 8F8$\"&+p)Fao$!&!)>*FM$\"&O$=!#6$\"&%zmFAQ)pprint246\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The following tablea u gives the coefficients in approximate rational form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "evalf[20](subs(e15,[seq([c[i],seq( a[i,j],j=1..i-1),``$(13-i)],i=2..13)])):\nconvert(convert(evalf[17](%) ,rational,14),matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7.7/#\"#R\"#))F(%!GF+F+F+F+F+F+F+F+F+F+7/#\"$,\"\"$G##\"&y!G\"'6n7#\"' 6A6\"'Wo]F+F+F+F+F+F+F+F+F+F+7/#F.\"$_\"#F.\"$3'\"\"!#\"$.$F:F+F+F+F+F +F+F+F+F+7/#\"%D#)\"&7p(#\"&4I%\"'Ha\\F;#\"'3fS\")(oT<\"#!'ZrK\")fRlAF +F+F+F+F+F+F+F+7/#\"#DF8#\"(7I/\"\")FX3FF;F;#\"&pN$\"*RsVR$#\"'t2Z\"($ HSPF+F+F+F+F+F+F+7/#\"%8?\"%XM#\"(,fd#\"(0*3\\F;F;#\"'E<[\"(H'Hj#!(ZF: %\"(!fW>#\"(N'pL\"()=!f\"F+F+F+F+F+F+7/#\"&Fx\"\"']xF#\"'.[f\")Z7l;F;F ;F;#\"(V04#\")0T[X#!'b?^\")KsWG#\"&s:$\"*4YwC#F+F+F+F+F+7/#\"\"\"\"\"& #\"'g1B\")D#\"(1!>k\"(r<, \"F;F;F;F;F;#\"(cg,%\"(vJ.##!)]I+7\"'6!o)#\"(TZ[*\"'8)R'#!(z*3J\"'*Rr$ F+F+7/#\"'oDo\")X!yJ\"#!(6ix\"\")_3%3%F;F;F;F;#!(&3&*G\"(A%p]#!&6U)\"* 8*QS<#\"(\"R^B\"(@8^*#\"(J)o9\"(fBD$#!#6\"$U$#Fcp\"%*\\&F+7/#\"'zJK\"( ?r!Q#\"'N$R%\")(4r`#F;F;F;F;F;F;F;#\"'zI[\"*\\/!fb#!&(RK\"*gh@_$#\"$%f \"+B!Q&RK#\"'(*G!)\")3/-7Q)pprint256\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 82 ": The rational values given for the l inking coefficients in rows 2 to 6 are exact." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "We specify " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2] = -11/100;" "6#/&%\"bG6#\" \"#,$*&\"#6\"\"\"\"$+\"!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[3] = -17/100;" "6#/&%\"bG6#\"\"$,$*&\"#<\"\"\"\"$+\"!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[6] = -19/100;" "6#/&%\"bG6#\"\"',$*&\"#>\"\" \"\"$+\"!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[7] = -21/100;" "6 #/&%\"bG6#\"\"(,$*&\"#@\"\"\"\"$+\"!\"\"F-" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "b[9] = -13/56;" "6#/&%\"bG6#\"\"*,$*&\"#8\"\"\"\"#c!\" \"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[10] = -27/100;" "6#/&%\"bG6# \"#5,$*&\"#F\"\"\"\"$+\"!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "b[1 1] = -29/100;" "6#/&%\"bG6#\"#6,$*&\"#H\"\"\"\"$+\"!\"\"F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[i,2]=0" "6#/&%\"aG6$%\"iG\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9" }{TEXT -1 15 " . . 20, " }{XPPEDIT 18 0 "a[i,3]=0" "6#/&%\"aG6$%\"iG\"\"$\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9" }{TEXT -1 16 " . . 20, " }{XPPEDIT 18 0 "a[i,4]=0" "6#/&%\"aG6$%\"iG\"\"%\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9" }{TEXT -1 8 " . . 20," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[i,5]=0" "6#/&%\"aG6$%\"iG\"\"&\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9" }{TEXT -1 15 " . . 20, " }{XPPEDIT 18 0 "a[i,6]=0" "6#/&%\"aG6$%\"iG\"\"'\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9" }{TEXT -1 16 " \+ . . 18, " }{XPPEDIT 18 0 "a[i,7]=0" "6#/&%\"aG6$%\"iG\"\"(\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9" }{TEXT -1 9 " . . 17, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[i,8]=0" "6#/&%\"aG6$%\"iG\"\")\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9" }{TEXT -1 9 " . \+ . 17, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[21,j]=0" "6#/&%\"aG6$\"#@%\"jG\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 29 ", 3, 7, \+ 8, 12 . . 17, " }{XPPEDIT 18 0 "a[22,j]=0" "6#/&%\"aG6$\"#A%\"j G\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=2" "6#/%\"jG\"\"#" }{TEXT -1 21 ", 3, 6, 8, 11 . . 18," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[23,j]=0" "6#/&%\"aG6$\"#B %\"jG\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=3" "6#/%\"jG\"\"$" } {TEXT -1 25 ", 4, 5, 8 . . 20, " }{XPPEDIT 18 0 "a[24,j]=0" "6# /&%\"aG6$\"#C%\"jG\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=2" "6#/% \"jG\"\"#" }{TEXT -1 19 ", 4 . . 22, " }{XPPEDIT 18 0 "a[25,j]= 0" "6#/&%\"aG6$\"#D%\"jG\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=4" "6#/%\"jG\"\"%" }{TEXT -1 7 ", 5, 8," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[19,14] = 21/32;" "6#/&%\"aG6$\"#>\"#9*&\"#@\"\"\"\"#K!\"\"" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "a[20,14] = -50/501;" "6#/&%\"aG6$\"#?\"#9,$*&\"#]\"\"\" \"$,&!\"\"F." }{TEXT -1 6 ", " }{XPPEDIT 18 0 "a[20,15] = 13/1306; " "6#/&%\"aG6$\"#?\"#:*&\"#8\"\"\"\"%18!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "We make u se of the following symmetry conditions to specify additional nodes an d weights." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2]=c[ 24]" "6#/&%\"cG6#\"\"#&F%6#\"#C" }{TEXT -1 12 ", " } {XPPEDIT 18 0 "c[3]=c[23]" "6#/&%\"cG6#\"\"$&F%6#\"#B" }{TEXT -1 12 ", " }{XPPEDIT 18 0 "c[6] = c[22];" "6#/&%\"cG6#\"\"'&F%6#\"#A " }{TEXT -1 12 ", " }{XPPEDIT 18 0 "c[7]=c[21]" "6#/&%\"cG6# \"\"(&F%6#\"#@" }{TEXT -1 12 ", " }{XPPEDIT 18 0 "c[9] = c[2 0];" "6#/&%\"cG6#\"\"*&F%6#\"#?" }{TEXT -1 12 ", " } {XPPEDIT 18 0 "c[10]=c[19]" "6#/&%\"cG6#\"#5&F%6#\"#>" }{TEXT -1 12 ", " }{XPPEDIT 18 0 "c[11]=c[18]" "6#/&%\"cG6#\"#6&F%6#\"#=" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[2]+b[24]=0" "6#/,&&%\"bG6#\"\"#\"\"\" &F&6#\"#CF)\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "b[3]+b[23]=0" " 6#/,&&%\"bG6#\"\"$\"\"\"&F&6#\"#BF)\"\"!" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "b[6]+b[22] = 0;" "6#/,&&%\"bG6#\"\"'\"\"\"&F&6#\"#AF)\" \"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "b[7]+b[21]=0" "6#/,&&%\"bG6# \"\"(\"\"\"&F&6#\"#@F)\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "b[9] +b[20] = 0;" "6#/,&&%\"bG6#\"\"*\"\"\"&F&6#\"#?F)\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "b[10]+b[19]=0" "6#/,&&%\"bG6#\"#5\"\"\"&F&6#\"# >F)\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "b[11]+b[18]=0" "6#/,&&% \"bG6#\"#6\"\"\"&F&6#\"#=F)\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 763 "e16 := \{b[ 2]=-11/100,b[3]=-17/100,b[6]=-19/100,b[7]=-21/100,b[9]=-13/56,b[10]=-2 7/100,b[11]=-29/100,\n seq(a[i,2]=0,i=[$14..20]),seq(a[i,3]=0,i =[$14..20]),seq(a[i,4]=0,i=[$14..20]),\n seq(a[i,5]=0,i=[$14..2 0]),seq(a[i,6]=0,i=[$14..18]),seq(a[i,7]=0,i=[$14..17]),\n seq( a[i,8]=0,i=[$14..17]),seq(a[21,j]=0,j=[2,3,7,8,$12..17]),seq(a[22,j]=0 ,j=[2,3,6,8,$11..18]),\n seq(a[23,j]=0,j=[3,4,5,$8..20]),seq(a[2 4,j]=0,j=[2,$4..22]),seq(a[25,j]=0,j=[4,5,8]),\n a[19,14]=21/32, a[20,14]=-50/501,a[20,15]=13/1036\}:\ne17 := `union`(e15,e16):\ne18 := `union`(e17,subs(e17,\{c[18]=c[11],c[20]=c[9],c[21]=c[7],c[19]=c[10], c[22]=c[6],c[23]=c[3],\n c[24]=c[2],b[19]=-b[10],b[22]=-b[6],b [23]=-b[3],b[24]=-b[2],b[18]=-b[11],b[20]=-b[9],b[21]=-b[7]\})):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We form a system of equations i nvolving the following " }{TEXT 260 29 "column simplifying conditions " }{TEXT -1 80 " which we can use to calculate certain linking coeffic ients in columns 15 to 24." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*a[i,j],i=j+1..25)=b[j]*(1-c[j])" "6#/-%$SumG6$ *&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#D*&&F)6#F0F,,&F ,F,&%\"cG6#F0!\"\"F," }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 14;" "6# /%\"jG\"#9" }{TEXT -1 18 " . . 24 " }{XPPEDIT 18 0 "Sum(b[i] *c[i]*a[i,j],i=j+1..25)=1/2" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6 #F+F,&%\"aG6$F+%\"jGF,/F+;,&F3F,F,F,\"#D*&F,F,\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "b[j]*(1-c[j]^2)" "6#*&&%\"bG6#%\"jG\"\"\",&F(F(*$& %\"cG6#F'\"\"#!\"\"F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 14;" "6 #/%\"jG\"#9" }{TEXT -1 9 " . . 23, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i, j],i=j+1..25)=1/3" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"# F,&%\"aG6$F+%\"jGF,/F+;,&F5F,F,F,\"#D*&F,F,\"\"$!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "b[j]*(1-c[j]^3)" "6#*&&%\"bG6#%\"jG\"\"\",&F(F(*$&% \"cG6#F'\"\"$!\"\"F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 14;" "6# /%\"jG\"#9" }{TEXT -1 24 " . . 21, " }{XPPEDIT 18 0 "Su m(b[i]*c[i]^3*a[i,j],i=j+1..25)=1/4" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\" \"*$&%\"cG6#F+\"\"$F,&%\"aG6$F+%\"jGF,/F+;,&F5F,F,F,\"#D*&F,F,\"\"%!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "b[j]*(1-c[j]^4)" "6#*&&%\"bG6#%\"j G\"\"\",&F(F(*$&%\"cG6#F'\"\"%!\"\"F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 14;" "6#/%\"jG\"#9" }{TEXT -1 9 " . . 19, " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum (b[i]*c[i]^4*a[i,j],i=j+1..25)=1/5" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\" *$&%\"cG6#F+\"\"%F,&%\"aG6$F+%\"jGF,/F+;,&F5F,F,F,\"#D*&F,F,\"\"&!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "b[j]*(1-c[j]^5)" "6#*&&%\"bG6#%\"jG \"\"\",&F(F(*$&%\"cG6#F'\"\"&!\"\"F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "j = 14;" "6#/%\"jG\"#9" }{TEXT -1 9 " . . 16. " }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 415 "cdns2 := [seq(add(b[i]*a[i,j],i=j+1..25)=b[j]*(1-c[j]),j=[$14..24]),\n \+ seq(add(b[i]*c[i]*a[i,j],i=j+1..25)=b[j]/2*(1-c[j]^2),j=[$14..23]),\n \+ seq(add(b[i]*c[i]^2*a[i,j],i=j+1..25)=b[j]/3*(1-c[j]^3),j=[$14. .21]),#omitting 22\n seq(add(b[i]*c[i]^3*a[i,j],i=j+1..25)=b[j] /4*(1-c[j]^4),j=[$14..19]),#omitting 20\n seq(add(b[i]*c[i]^4*a [i,j],i=j+1..25)=b[j]/5*(1-c[j]^5),j=[$14..16])]:#omitting 17" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "There are 38 equations and 38 unknown coefficients." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 76 "eqns2 := simplify(expand(subs(e18,cdns2))):\nnops(% );\nindets(eqns2);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#Q" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#&F%6$F'\" #?&F%6$F'F+&F%6$F'F/&F%6$F'F,&F%6$F'\"#9&F%6$F5F2&F%6$F2F8&F%6$F2F;&F% 6$F8F;&F%6$F8FA&F%6$F8F>&F%6$F;F>&F%6$F;FA&F%6$F;F+&F%6$FAF+&F%6$FAF>& F%6$FAF(&F%6$FAF,&F%6$F>F+&F%6$F>F,&F%6$F>F(&F%6$F>F/&F%6$F+F(&F%6$F+F /&F%6$F+FJ&F%6$F(F,&F%6$F(FJ&F%6$F,F/&F%6$F,FJ&F%6$F/FJ" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#Q" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e19 := solve(\{op(eqns2)\}):\ne20 : = `union`(e18,e19):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "indets(map(rhs,e20 ));" }}{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 3 "e20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24891 "e20 := \{a[23,20] = 0, a[ 23,19] = 0, c[12] = -48827895154/211479617260065*(495+66*15^(1/2))^(1/ 2)*15^(1/2)-1080276228947/70493205753355*(495+66*15^(1/2))^(1/2)+25236 14583531/6408473250305+33872310083/1281694650061*15^(1/2), a[23,18] = \+ 0, a[25,5] = 0, a[23,17] = 0, a[25,4] = 0, a[25,8] = 0, a[24,15] = 0, \+ a[24,14] = 0, a[23,4] = 0, a[23,16] = 0, a[23,14] = 0, a[23,15] = 0, a [23,13] = 0, a[23,12] = 0, a[23,11] = 0, a[23,10] = 0, a[23,9] = 0, a[ 24,10] = 0, a[23,8] = 0, a[23,3] = 0, a[23,5] = 0, a[24,12] = 0, a[24, 13] = 0, a[24,22] = 0, a[24,21] = 0, a[24,20] = 0, a[24,18] = 0, a[24, 19] = 0, a[24,17] = 0, a[24,16] = 0, a[24,11] = 0, b[8] = 0, b[5] = 0, b[4] = 0, b[16] = 31/175+1/100*15^(1/2), b[25] = 1/42, b[13] = 31/175 -1/100*15^(1/2), b[1] = 1/42, c[16] = 1/2+1/66*(495-66*15^(1/2))^(1/2) , c[14] = 1/2-1/66*(495-66*15^(1/2))^(1/2), c[25] = 1, b[12] = 0, b[17 ] = 31/175-1/100*15^(1/2), c[17] = 1/2+1/66*(495+66*15^(1/2))^(1/2), b [14] = 31/175+1/100*15^(1/2), c[15] = 1/2, c[13] = 1/2-1/66*(495+66*15 ^(1/2))^(1/2), b[15] = 128/525, a[11,6] = 0, a[24,5] = 0, c[11] = 5178 9075/64972747+333240/64972747*15^(1/2)+3057458/714700217*(495+66*15^(1 /2))^(1/2)+70408/714700217*(495+66*15^(1/2))^(1/2)*15^(1/2), a[13,5] = 0, a[12,5] = 0, a[11,5] = 0, a[10,5] = 0, a[9,5] = 0, a[12,3] = 0, a[ 11,3] = 0, a[10,3] = 0, a[9,3] = 0, a[8,3] = 0, a[7,3] = 0, a[6,3] = 0 , a[13,4] = 0, a[12,4] = 0, a[11,4] = 0, a[10,4] = 0, a[9,4] = 0, a[8, 4] = 0, a[10,2] = 0, a[9,2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a [5,2] = 0, a[4,2] = 0, a[13,3] = 0, a[13,2] = 0, a[12,2] = 0, a[11,2] \+ = 0, a[13,8] = 0, a[13,7] = 0, a[13,6] = 0, a[24,8] = 0, a[24,9] = 0, \+ a[24,4] = 0, a[19,3] = 0, a[18,3] = 0, a[17,3] = 0, a[16,3] = 0, a[15, 3] = 0, a[14,3] = 0, a[17,2] = 0, a[16,2] = 0, a[15,2] = 0, a[14,2] = \+ 0, a[20,3] = 0, a[19,2] = 0, a[18,2] = 0, a[15,5] = 0, a[14,5] = 0, a[ 20,2] = 0, a[20,5] = 0, a[19,5] = 0, a[18,5] = 0, a[17,5] = 0, a[16,5] = 0, a[14,4] = 0, a[19,4] = 0, a[18,4] = 0, a[17,4] = 0, a[16,4] = 0, a[15,4] = 0, a[16,8] = 0, a[15,8] = 0, a[14,8] = 0, a[20,4] = 0, a[17 ,7] = 0, a[16,7] = 0, a[15,7] = 0, a[14,7] = 0, a[17,8] = 0, a[18,6] = 0, a[17,6] = 0, b[9] = -13/56, a[24,2] = 0, a[16,6] = 0, a[15,6] = 0, a[14,6] = 0, a[22,2] = 0, a[22,13] = 0, a[22,12] = 0, a[22,11] = 0, a [22,8] = 0, a[22,6] = 0, a[22,3] = 0, a[22,18] = 0, a[22,17] = 0, a[22 ,16] = 0, a[22,15] = 0, a[22,14] = 0, a[21,2] = 0, a[21,13] = 0, a[21, 12] = 0, a[21,8] = 0, a[24,6] = 0, a[24,7] = 0, a[21,7] = 0, a[21,3] = 0, a[21,17] = 0, a[21,16] = 0, a[21,15] = 0, a[21,14] = 0, c[9] = 1/5 , c[10] = 1/3, a[10,7] = 5620721356384469075/3545698345096452809913, a [10,1] = -17689789589/433566079650, c[2] = 39/88, a[10,8] = 1043609777 0007465488281250/34480588269890588098919643, a[8,1] = 5673493348726844 397056253/158826937830690185546875000, a[8,5] = 8387443042739945104453 5741888/1824862440224861393218994140625, a[10,6] = -46012134893475328/ 76616158064651325, a[8,6] = -986894364933740544726981696/5482694819650 5455902099609375, a[9,1] = 2526695731/133816691250, a[9,6] = 244678866 02752/291937806983125, a[7,4] = 46426400659946403657/61001874921177626 3125, a[7,1] = 1534965908593628117013/2925190767628355078125, a[7,6] = 11005843237812473721/5193844391648046875, a[8,7] = 567231764142747016 0608354913689/40381865475711712996627696578125000, a[9,8] = 1095050362 935718750/11254693571026117461, a[10,9] = 110139186925/164279737386, a [7,5] = -156479396217570600333298656/73273972406867506457421875, a[5,1 ] = 8719384129925/100440274363616, a[5,4] = -2900926450625/20088054872 7232, a[5,3] = 6944404306875/200880548727232, a[9,7] = 11944081475573/ 1322478962029186830, a[6,1] = 1167025/30304848, a[6,4] = 390625/394985 4672, a[6,5] = 404856925/3216589572, a[3,1] = 28078/126711, a[3,2] = 1 12211/506844, a[4,1] = 101/608, a[4,3] = 303/608, a[12,9] = -281211230 9092422771819313341069996795037876904500378145129471863425104791510387 56222724747330456172743380029/1777219414844154523916564814426187150039 1656432256173921137707234507471866579929942450646636509884128799622+67 6374087730775429952449408713302591862747979275682175058845619651319936 9023804229800976446366900597141/38492948123113591594467507351660973576 763388417275663680176970401792228430972341222548509067597756397660*(49 5+66*15^(1/2))^(1/2)*15^(1/2)+3647611941783523639106839681293674145014 752901446257471407287711249070189350233573351451468961784464641/641549 1353852265265744584558610162262793898069545943946696161733632038071828 723537091418177932959399610*(495+66*15^(1/2))^(1/2)-594585803588270123 2660115801569296816414941113900029074561383838890594834114310825436409 5422742138234247/12830982707704530531489169117220324525587796139091887 893392323467264076143657447074182836355865918799220*15^(1/2), a[11,7] \+ = 18993974973485227904952341910291889381118708193023824550/12060034665 080522994492944014191159522924996678137100092619*(495+66*15^(1/2))^(1/ 2)*15^(1/2)+3736499968282194034052421389626604730923328883163812355187 5/42880123253619637313752689828235233859288877077820800329312+83870865 016481720496920100044907813868655001976949702225/134000385167561366605 4771557132351058102777408681900010291*15^(1/2)+10188221773942597558585 0562279744393868968529126539562125/40200115550268409981643146713970531 74308332226045700030873*(495+66*15^(1/2))^(1/2), a[12,7] = 12039560526 3827674443959816224372224515900903963473583877146503450993695151583738 412984486608167592412726499223108027/295063494385794976532558803599500 9291446938757618744973811305715669070205151124895945261251585564237742 29187413128270-4574290713578312637346253714366174333264683073453621195 92334365398748049518495290294028445214608755002615341437/1278161119280 0302210637158483842362102867397693821723949799894804717653043756226536 4750324954973542895485894482620*(495+66*15^(1/2))^(1/2)*15^(1/2)-19409 1304446298395030544188331511201788876606372749180626268993467082976330 239046596838838518978497037496895389/127816111928003022106371584838423 6210286739769382172394979989480471765304375622653647503249549735428954 8589448262*(495+66*15^(1/2))^(1/2)+12780472568351676678614075153447418 1571563952278241986019318882448258044336581201019101995055480939112571 04546897/1278161119280030221063715848384236210286739769382172394979989 48047176530437562265364750324954973542895485894482620*15^(1/2), a[12,6 ] = 251085521083237108777248115694608871632336099720342563852044971041 903837740861882751285544579556435153690263824896/109059581580566357404 4725317279254749804567879870625150516917051380618003386804945891914477 3539359368893655280625-17224203165087627219883985321280669596094555813 3287845511282815171786743955334377646964770699567651065020416/69046901 9186871525194507956492089110354268996435976670159491643799061730539287 715031284885947411166121788875*(495+66*15^(1/2))^(1/2)*15^(1/2)-248326 5050966602026265671372002316578111995428215292500398470208821705013352 242278156416063385897318005911552/299203241647644327584286781146571947 8201832317889232237357797123129267499003580098468901172438781719861085 125*(495+66*15^(1/2))^(1/2)+989414013347979104246599002035391609046115 74525495141245531026292768496305967031119826379359226270888023523328/1 4960162082382216379214339057328597391009161589446161186788985615646337 495017900492344505862193908599305425625*15^(1/2), a[12,1] = 4536078586 4184293465355049313006780143265609187196347220787941439290058648676039 91192705032200412298285836937/1172606656536926424103513708860616450309 534511569707813821852440031214272905699213661387392662314110822638750- 5364435895516573669420621604632495515121639476730674940078684840492375 5537638953128683448772183267658/13367151790715393045193549113238448870 98633780843915294531482553071844639268719963591518065572670919625*(495 +66*15^(1/2))^(1/2)*15^(1/2)-35279136548504838526715366339693933451439 37128361020345607398270646119601220687075114545149449903002764/2539758 8402359246785867743315153052854874041836034390596098168508365048146105 679308238843245880747472875*(495+66*15^(1/2))^(1/2)+136142098290337291 8496309760282958026582387432115444780575878975425366396745823033150091 11953003119707396/1269879420117962339293387165757652642743702091801719 52980490842541825240730528396541194216229403737364375*15^(1/2), a[13,1 ] = 49032552280925998159/1175735428193602866744+1095947848209481903/17 63603142290404300116*15^(1/2)-7475873553226772611/64665448550648157670 92*(495+66*15^(1/2))^(1/2)+5352672249559690835/11639780739116668380765 6*(495+66*15^(1/2))^(1/2)*15^(1/2), a[13,10] = 10368744210236692483336 89/19655651658225505790722624+587392382348212140203319/393113033164510 11581445248*15^(1/2)-259903711964372646898395/432424336480961127395897 728*(495+66*15^(1/2))^(1/2)*15^(1/2)-185211952023363244825707/10810608 4120240281848974432*(495+66*15^(1/2))^(1/2), a[13,9] = -16238425232388 3015198625/10650929594044147603352784*15^(1/2)+12238378998215586811542 5/468640902137942494547522496*(495+66*15^(1/2))^(1/2)*15^(1/2)+2265050 157236155663680125/14201239458725530137803712-120963267116719524969875 /26035605674330138585973472*(495+66*15^(1/2))^(1/2), a[11,9] = 2874397 4673994042985226536878679230542480801150/17539834082278474517747485443 48597377943356553311*(495+66*15^(1/2))^(1/2)*15^(1/2)+2919917156154994 49706434755611025178229310468246875/5612746906329111845679195341915511 6094187409705952+1071091749312998743815087165568469189531652126075/175 3983408227847451774748544348597377943356553311*15^(1/2)+35325978470435 1651896312581655773178339865859125/17539834082278474517747485443485973 77943356553311*(495+66*15^(1/2))^(1/2), a[13,11] = 1451483888900070743 38014877677070795676982463/1292134149111656271966122106166449116832367 968+289384776308176063330366809130393474082987849/15505609789339875263 593465273997389401988415616*15^(1/2)-274115022589237178977231727148612 453501574585/511685123048215883698584354041913850265617715328*(495+66* 15^(1/2))^(1/2)*15^(1/2)-24857236960157204698513148202889704149473711/ 5330053365085582121860253687936602606933517868*(495+66*15^(1/2))^(1/2) , a[13,12] = 644332945230386310371160782678707752391880620841916690825 2875/48186908177657952343922139838789383990650083655460342201310944-17 1491601151866911759544690813927486314725154315627741296625/90350452833 10866064485401219773009498246890685398814162745802*15^(1/2)-7090202015 237773474138169907437581683377756583694650388556125/238525195479406864 1024145922020074507537179140945286938964891728*(495+66*15^(1/2))^(1/2) +119929995185051271338440264753881499366865434058651556229025/14456072 4532973857031766419516368151971950250966381026603932832*(495+66*15^(1/ 2))^(1/2)*15^(1/2), a[11,1] = 3625022219827369334113105102670672345569 2594208/4999430483339723613905130972844241982587217763497*(495+66*15^( 1/2))^(1/2)*15^(1/2)+4861393204748134765917698036236910531265683193125 /2221969103706543828402280432375218658927652339332+1484428141946824012 32706118820643700603232458928/5554922759266359571005701080938046647319 13084833*15^(1/2)+143068868482411022237818049543538495099770530576/166 6476827779907871301710324281413994195739254499*(495+66*15^(1/2))^(1/2) , a[12,8] = -340659803958984878209931438027146134138660774601249541025 577546738764928298414303784311737346180734782282094165416156250/308280 7063362932068096196297896660310632239903888789451271870099917142127007 0039408336730423089458130745933155208375541+41194832565673882203999777 5092119236421293381998104532443768303827226809809726860648464826292767 32032063112500000/3514251750809858381605960008089851362393260494840337 71219847713817031123765375550406811559375414180212098687404767*(495+66 *15^(1/2))^(1/2)*15^(1/2)+26719277739082859443666346678807717091479495 4921965889671712692505252570871996792397851976779268302516907520875000 0/66770783265387309250513240153707175885471949401966416531771065625235 91351542135457729419628132869424029875060690573*(495+66*15^(1/2))^(1/2 )-20868030576527122590278134042922470681076456345098996480647230076556 364761794880724541437909825056194904756420000000/667707832653873092505 1324015370717588547194940196641653177106562523591351542135457729419628 132869424029875060690573*15^(1/2), a[11,10] = -37768595712400175213251 8096635999239114085793540/40806683117092499328338457010324585780808416 836571*(495+66*15^(1/2))^(1/2)*15^(1/2)-181781480924529622996033569801 0451499156358365559525/65290692987347998925341531216519337249293466938 5136-14248117750070402721956352075060738058061271231816/40806683117092 499328338457010324585780808416836571*15^(1/2)-484426665991669901588562 6279835272453019208362650/40806683117092499328338457010324585780808416 836571*(495+66*15^(1/2))^(1/2), a[11,8] = -275413882637225084153879395 633151790451174339761235000000000/173657067762976556173412167504047698 36227513968915818501163621*(495+66*15^(1/2))^(1/2)*15^(1/2)-9030706067 191500671304838647417536518671082752129344726562500/192952297514418395 7482357416711641092914168218768424277907069-11310663799138970197488474 86012312835922421062622057500000000/1929522975144183957482357416711641 092914168218768424277907069*15^(1/2)-109756253028446039591902781216552 0839910017377896612500000000/57885689254325518724470722501349232787425 04656305272833721207*(495+66*15^(1/2))^(1/2), a[12,11] = 1/5499, b[2] \+ = -11/100, b[11] = -29/100, b[10] = -27/100, b[7] = -21/100, b[6] = -1 9/100, b[3] = -17/100, a[20,15] = 13/1036, a[20,14] = -50/501, a[19,14 ] = 21/32, a[12,10] = -11/342, a[24,23] = 274193/636804, a[20,16] = 45 73876925/1493463420644832*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1 /2)+7986761337425/7582198904812224*15^(1/2)*(495-66*15^(1/2))^(1/2)+37 543791335/995642280429888*15^(1/2)+78669144834545/995642280429888-6267 380675/14360225198508*(495+66*15^(1/2))^(1/2)*15^(1/2)+5387414718575/3 2856195254186304*(495-66*15^(1/2))^(1/2)-204724622275/8960780523868992 *(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-150291531475 /62227642526868*(495+66*15^(1/2))^(1/2), a[17,14] = -86343889643783/14 23361572162200*(495+66*15^(1/2))^(1/2)*15^(1/2)-4875329406099/71886948 0890-47503534664513/28754779235600*15^(1/2)-46830515603137/18978154295 4960*(495+66*15^(1/2))^(1/2)-56659550856457/355840393040550*15^(1/2)*( 495-66*15^(1/2))^(1/2)-21856354431557/869832071876900*(495+66*15^(1/2) )^(1/2)*(495-66*15^(1/2))^(1/2)-125819226953867/189781542954960*(495-6 6*15^(1/2))^(1/2)-34670051930677/6262790917513680*(495+66*15^(1/2))^(1 /2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[19,16] = 23318577/238751570176 *(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-19320841301/236364054 47424*15^(1/2)*(495-66*15^(1/2))^(1/2)-150970273/59687892544*15^(1/2)+ 5937215723/119375785088+3596875/5595739926*(495+66*15^(1/2))^(1/2)*15^ (1/2)-18335887105/2626267271936*(495-66*15^(1/2))^(1/2)-29492131/21487 64131584*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+4937 61427/358127355264*(495+66*15^(1/2))^(1/2), a[18,15] = 199906782229440 16799007968810994718650113/3272381042960700814201444438358736068377200 *(495+66*15^(1/2))^(1/2)*15^(1/2)-883939112584854206781344717816584649 32823/6326603349724021574122792580826889732195920*(495+66*15^(1/2))^(1 /2)+87051869427126857828192491903627294702861/958576265109700238503453 421337407535181200+4396995932514786753230371514007047435591/4564648881 4747630404926353397019406437200*15^(1/2)+45290633122123836192269606511 4103183779/7083075850564287476626503113330597550600*(495-66*15^(1/2))^ (1/2)+2225327884102816028172189393256449625076/10783982982484127683163 8509900458347707885*15^(1/2)*(495-66*15^(1/2))^(1/2)+11070456080194641 2846083674130999701608833/47449525122930161805920944356201672991469400 *(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+101010698855628268538 55484798548336576859/11862381280732540451480236089050418247867350*(495 +66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[16,15] = 41832 0022266565899999989/730900563458308064970225-119156065868215739462012/ 2192701690374924194910675*15^(1/2)+3495695795945596671104/146180112691 661612994045*(495+66*15^(1/2))^(1/2)-65831485804562645782499/109635084 51874620974553375*(495+66*15^(1/2))^(1/2)*15^(1/2)-4641549371926342688 144/32890525355623862923660125*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-6 6*15^(1/2))^(1/2)+7610546603899839370279/6578105071124772584732025*(49 5+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+17714250989146270037792/7 30900563458308064970225*(495-66*15^(1/2))^(1/2)-1502905439956391946034 9/10963508451874620974553375*15^(1/2)*(495-66*15^(1/2))^(1/2), a[2,1] \+ = 39/88, a[25,14] = -49414375111/1558819416000*(495+66*15^(1/2))^(1/2) *(495-66*15^(1/2))^(1/2)-165545310587/584557281000*15^(1/2)*(495-66*15 ^(1/2))^(1/2)-2733053737/843517000*15^(1/2)-16561593209/3374068000-247 43106247/2338229124000*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/ 2))^(1/2)-1363545178441/1558819416000*(495-66*15^(1/2))^(1/2)-47217730 03/27836061000*(495+66*15^(1/2))^(1/2)-3651713009/27836061000*(495+66* 15^(1/2))^(1/2)*15^(1/2), a[18,16] = -436840620567893961487535263885/2 696403426422943408076694358192*15^(1/2)-369545114918022904188418990316 329/625565594930122870673793091100544-18257270145419395739617682201851 /625565594930122870673793091100544*(495+66*15^(1/2))^(1/2)-59834444034 37426102960354262185/938348392395184306010689636650816*(495+66*15^(1/2 ))^(1/2)*15^(1/2)+107005288406322056654355448927205/375339356958073722 4042758546603264*(495-66*15^(1/2))^(1/2)+84055437892642261541931143840 615/11260180708742211672128275639809792*15^(1/2)*(495-66*15^(1/2))^(1/ 2)+1012338871884896232541346539753/3753393569580737224042758546603264* (495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+4578034999918 124747163891877361/3753393569580737224042758546603264*(495+66*15^(1/2) )^(1/2)*(495-66*15^(1/2))^(1/2), a[18,14] = 42300428275816845237517521 04701705965843/222105102055991070418278663608820791814660*(495+66*15^( 1/2))^(1/2)*15^(1/2)+87527611851436237649772046468066688557713/1794788 7034827561245921508170409760954720+51251514377913613082352617012218154 85791/6730457638060335467220565563903660358020*15^(1/2)+25588360673017 85957586054852230517925601/15864650146856505029877047400630056558190*( 495+66*15^(1/2))^(1/2)+72311136213172294095616390215153042369611/84803 7662395602268869791261051861205110520*15^(1/2)*(495-66*15^(1/2))^(1/2) +477343459314808814429839892297318102330687/11307168831941363584930550 14735814940147360*(495-66*15^(1/2))^(1/2)+4852212171487031154990368461 6247887004701/18656828572703249915135407743140946512431440*(495+66*15^ (1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+205838828141144106856071 75778197029916327/1381987301681722215935956129121551593513440*(495+66* 15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[17,16] = -3399762661/169041 9432009*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-21792326747/16 90419432009*15^(1/2)*(495-66*15^(1/2))^(1/2)+15024834410/51224831273*1 5^(1/2)+843590793/80496163429*(495+66*15^(1/2))^(1/2)*15^(1/2)-1674700 20833/3380838864018*(495-66*15^(1/2))^(1/2)-656911562/1690419432009*(4 95+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+23471698999/482 976980574*(495+66*15^(1/2))^(1/2)+61299209505/51224831273, a[25,16] = \+ 571468963/226284800+120370717/226284800*15^(1/2)-1751767/2036563200*(4 95+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-590329/16971360 0*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-519210887/2240219520 0*15^(1/2)*(495-66*15^(1/2))^(1/2)-28396679/298695936*(495-66*15^(1/2) )^(1/2)+5795203/339427200*(495+66*15^(1/2))^(1/2)*15^(1/2)+7675999/113 142400*(495+66*15^(1/2))^(1/2), a[16,14] = -14939172465658402889149/89 76679102595142257770+219312724397669057743089/359067164103805690310800 *15^(1/2)+659570008376205270318919/17773824623138381670384600*(495+66* 15^(1/2))^(1/2)*15^(1/2)-11242117128844970311919/157989552205674503736 752*(495+66*15^(1/2))^(1/2)-1749823831034832192631/4308805969245668283 7296*(495-66*15^(1/2))^(1/2)+44646359022841677628361/16158022384671256 06398600*15^(1/2)*(495-66*15^(1/2))^(1/2)-3070220299360933125107/19748 69402570931296709400*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+1 8545262695413350953267/11849216415425587780256400*(495+66*15^(1/2))^(1 /2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[25,15] = -3460846507/130820900 00+48037163/3270522500*15^(1/2)-108037591/64756345500*(495+66*15^(1/2) )^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-412156237/46254532500*(495+66 *15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-2890671767/58869405000*15^(1/ 2)*(495-66*15^(1/2))^(1/2)-4618638191/19623135000*(495-66*15^(1/2))^(1 /2)-848300537/15418177500*(495+66*15^(1/2))^(1/2)+429589031/1295126910 00*(495+66*15^(1/2))^(1/2)*15^(1/2), a[19,15] = 1908413061857/28985932 8166800-2931986748899/14348036744256600*(495+66*15^(1/2))^(1/2)*(495-6 6*15^(1/2))^(1/2)-1254438533/217394496125100*15^(1/2)*(495-66*15^(1/2) )^(1/2)-857188607021/289859328166800*15^(1/2)+5379357542731/9565357829 504400*(495+66*15^(1/2))^(1/2)*15^(1/2)+136382862089/144929664083400*( 495-66*15^(1/2))^(1/2)-1935518588/85404980620575*(495+66*15^(1/2))^(1/ 2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+34225020981/21690153808400*(495+66 *15^(1/2))^(1/2), a[17,15] = -4863804455866/35119141675695*15^(1/2)-62 267936631107/9657763960816125*(495+66*15^(1/2))^(1/2)*15^(1/2)-5561334 169199/58531902792825+2301512129984/643850930721075*(495+66*15^(1/2))^ (1/2)-331440117351157/9657763960816125*15^(1/2)*(495-66*15^(1/2))^(1/2 )-27173454387092/214616976907025*(495-66*15^(1/2))^(1/2)-3798661375472 2/28973291882448375*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2)) ^(1/2)-25832795272667/5794658376489675*(495+66*15^(1/2))^(1/2)*(495-66 *15^(1/2))^(1/2), a[15,14] = 115988619653/292402973068800*(495+66*15^( 1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-2683538373203/877208919206400*15^( 1/2)*(495-66*15^(1/2))^(1/2)-1968290835/88606961536*15^(1/2)+496725643 /7310074326720*(495+66*15^(1/2))^(1/2)*15^(1/2)-69268797311/7310074326 720*(495-66*15^(1/2))^(1/2)-124493701507/877208919206400*(495+66*15^(1 /2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-52034279227/11696118922752 *(495+66*15^(1/2))^(1/2)-74896827627/88606961536, c[4] = 101/152, c[8] = 17727/277750, a[20,19] = -110845/348543, a[21,19] = 81770692250/117 233567091, a[22,19] = 828786688/2261607237, a[23,21] = 253319312163456 /659439747650125, a[22,21] = -75739731820032/659439747650125, a[25,19] = 3601857/2273300, a[21,20] = 21260379985/34709636178, a[25,21] = 303 34395520464588/16485993691253125, a[22,20] = 76890112/586342617, a[25, 23] = 753491/379050, a[25,20] = 3071471/2273300, a[23,22] = 381/544, a [25,22] = 2667/1600, a[25,24] = 1029/400, a[21,18] = -3904228286721143 70457150585026889349975/6959540664768832104103104710308649125712826*(4 95+66*15^(1/2))^(1/2)*15^(1/2)+447940435326263177949331771032201927620 17625/167028975954451970498474513047407579017107824-294414422593464543 99120790160135926728275/4639693776512554736068736473539099417141884*15 ^(1/2)-107835852142499187656942652161788723246225/13919081329537664208 206209420617298251425652*(495+66*15^(1/2))^(1/2), a[25,17] = 112245633 /226284800-1120087/28285600*15^(1/2)-68125601/7467398400*(495+66*15^(1 /2))^(1/2)+993373/4480439040*(495+66*15^(1/2))^(1/2)*15^(1/2), a[25,18 ] = -6212380489353402482048992934870163/551398114240467508084664077927 9519700*(495+66*15^(1/2))^(1/2)*15^(1/2)+56874537829985536110363501647 573078867/44111849139237400646773126234236157600-477467113712319024289 213588759633869/11027962284809350161693281558559039400*15^(1/2)-168276 384308509585366452969427482141/11027962284809350161693281558559039400* (495+66*15^(1/2))^(1/2), a[20,17] = 229884359555/8366741852352+4673904 7805/4183370926176*15^(1/2)-20855494525/25100225557056*(495+66*15^(1/2 ))^(1/2)-34057541825/75300676671168*(495+66*15^(1/2))^(1/2)*15^(1/2), \+ a[20,18] = -52601571650923977687781073690186/1325511779651805411892910 350460322529*(495+66*15^(1/2))^(1/2)*15^(1/2)+146135103700320839898210 8006046541205/15906141355821664942714924205523870348-31257463047785142 9199757289182427/101962444588600416299454642343101733*15^(1/2)-3126986 093611720912749441466631523/1325511779651805411892910350460322529*(495 +66*15^(1/2))^(1/2), a[19,18] = 199389002800788393634510722754176/2796 728200664360059162382772909427897*(495+66*15^(1/2))^(1/2)*15^(1/2)-102 5465248156834803353840916920180809/55934564013287201183247655458188557 94+16445717956872940821116448130678578/2796728200664360059162382772909 427897*15^(1/2)+13437632256176229690482072355537756/279672820066436005 9162382772909427897*(495+66*15^(1/2))^(1/2), a[19,17] = 2466730563/341 07367168-46877457/4872481024*15^(1/2)-417081443/102322101504*(495+66*1 5^(1/2))^(1/2)+221689729/306966304512*(495+66*15^(1/2))^(1/2)*15^(1/2) , a[18,17] = 61673043751462390569674581/351685648446307151385663744-49 15309705585387914994781/351685648446307151385663744*15^(1/2)-469554899 57737999234808063/11605626398728135995726903552*(495+66*15^(1/2))^(1/2 )+6751918613532367335628313/34816879196184407987180710656*(495+66*15^( 1/2))^(1/2)*15^(1/2), c[6] = 25/152, c[5] = 8225/76912, c[3] = 101/228 , c[7] = 2013/3445, b[20] = 13/56, b[22] = 19/100, b[21] = 21/100, b[2 3] = 17/100, b[24] = 11/100, c[21] = 2013/3445, b[18] = 29/100, c[24] \+ = 39/88, c[23] = 101/228, c[22] = 25/152, c[19] = 1/3, c[18] = 5178907 5/64972747+333240/64972747*15^(1/2)+3057458/714700217*(495+66*15^(1/2) )^(1/2)+70408/714700217*(495+66*15^(1/2))^(1/2)*15^(1/2), b[19] = 27/1 00, c[20] = 1/5\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "We now have values for all the \+ linking coefficients in columns 14 to 25 in the Butcher tableau." }} {PARA 0 "" 0 "" {TEXT -1 68 "The coefficients are expressed in terms o f the radical expressions " }{XPPEDIT 18 0 "sqrt(15)" "6#-%%sqrtG6#\" #:" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "sqrt(495+66*sqrt(15)))" "6#-%%sq rtG6#,&\"$&\\\"\"\"*&\"#mF(-F$6#\"#:F(F(" }{TEXT -1 8 " and " } {XPPEDIT 18 0 "sqrt(495-66*sqrt(15));" "6#-%%sqrtG6#,&\"$&\\\"\"\"*&\" #mF(-F$6#\"#:F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a[15,14]=subs(e20,a[15,14]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"#9,2**\"-`'>'))f6\"\"\"\"0+)oI(HS#H!\"\" ,&\"$&\\F,*&\"#mF,F'#F,\"\"#F,F3,&F0F,*&F2F,F'F3F.F3F,**\"..KPQNo#F,\" 0+k?>*3s()F.F'F3F5F3F.*(\"+N3Ho>F,\",O:'pg))F.F'F3F.**\"*Vcs'\\F,\".?n Ku+J(F.F/F3F'F3F,*(\",6tzo#pF,F?F.F5F3F.*,\"-2:q$\\C\"F,F9F.F/F3F'F3F5 F3F.*(\",F#zU._F,\"/_F#*=hp6F.F/F3F.#\",Fw#o*[(F " 0 "" {MPLTEXT 1 0 92 "evalf[20]( subs(e20,matrix([seq([seq(a[i,j],j=14..i-1),``$(25-i)],i=15..25)]))): \nevalf[7](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7-7-$!(_ 2W\"!\"'%!GF+F+F+F+F+F+F+F+F+7-$\"(Eti&F*$\"(JZJ*!\"(F+F+F+F+F+F+F+F+F +7-$!(-;f'!\"&$!(=_F*F*$\"(-nu\"F*F+F+F+F+F+F+F+F+7-$\"(W!\\OF5$\"(7@M &F*$!(P=Y)F1$\"(Nj4$!\")F+F+F+F+F+F+F+7-$\"(+Dc'F1$!(+z:\"FC$\"(hPy(! \"*$\"(I3o\"!#6$!(\\b8#FCF+F+F+F+F+F+7-$!(S+)**FC$\"(E[D\"FC$!(mP(QFK$ !(\\K.*!#7$\"()p:6FC$!(R-=$F1F+F+F+F+F+7-$\"\"!F[oFjnFjnFjn$\"(_&RDFC$ \"(C](pF1$\"(5_7'F1F+F+F+F+7-FjnFjnFjnFjnFjn$\"(#fkOF1$\"(^8J\"F1$!(Z& [6F1F+F+F+7-FjnFjnFjnFjnFjnFjnFjn$\"(K9%QF1$\"(wO+(F1F+F+7-FjnFjnFjnFj nFjnFjnFjnFjnFjn$\"(odI%F1F+7-$!(nQt*F5$!(DjW\"F5$\"(;(4DF*$\"(wC;\"F1 $\"(G.%eF1$\"(=We\"F*$\"(26N\"F*$\"(5+%=F*$\"(vom\"F*$\"(Ty)>F*$\"(+Dd #F*Q)pprint286\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The following tableau gives the coefficients in approxima te rational form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "evalf[ 20](subs(e20,[seq([seq(a[i,j],j=14..i-1),``$(25-i)],i=15..25)])):\ncon vert(convert(evalf[19](%),rational,16),matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7-7-#!)Ph\">$\"):C:A%!GF+F+F+F+F+F+F+F+F+7- #\")zkp^\"((o'=*#\")o,cC\")R_)\"*51Hv#F+F+F+F+F+F+F+7-#\"#@\"#K#!( H'Qb\"*+SLy%#\"(,[[\"\"*Oiv!>#\"'K2<\",Jyfd,\"#!()3ly\"*fMHo$F+F+F+F+F +F+7-#!#]\"$,&#\"#8\"%O5#!((z**G\"*'=t&[(#!&:/)\"+CA6-*)#\"(wb;)\"**>! )=t#!'X36\"'V&[$F+F+F+F+F+7-\"\"!FhoFhoFho#\"(U*=H\"*V#R\\6#\")*[*f=\" )VemE#\")Aq0=\")V)z%HF+F+F+F+7-FhoFhoFhoFhoFho#\")`B(\\\"\")7o&3%#\"(< D^(\")m$)Gd#!)jd()z\"*Y2X&pF+F+F+7-FhoFhoFhoFhoFhoFhoFho#\")b>45\")M8F E#\"$\"Q\"$W&F+F+7-FhoFhoFhoFhoFhoFhoFhoFhoFho#\"'$>u#\"'/ojF+7-#!*N)4 8Z\"(f>%[#!*!3]/7\"(6!G$)#\")4g)\\#\"(5d&**#\"(A)\\!)\") " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The following " }{TEXT 260 18 "symmetry relations" }{TEXT -1 54 " \+ can be used determine some more linking coefficients." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[18,7] = a[11,7];" "6#/&%\"aG6$\" #=\"\"(&F%6$\"#6F(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "a[18,8] = a[1 1,8];" "6#/&%\"aG6$\"#=\"\")&F%6$\"#6F(" }{TEXT -1 6 ", " } {XPPEDIT 18 0 "a[19, 6] = a[10,6];" "6#/&%\"aG6$\"#>\"\"'&F%6$\"#5F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "a[19,7] = a[10,7];" "6#/&%\"aG6$\" #>\"\"(&F%6$\"#5F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "a[19,8] = a[10 ,8];" "6#/&%\"aG6$\"#>\"\")&F%6$\"#5F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "a[20,6] = a[9,6];" "6#/&%\"aG6$\"#?\"\"'&F%6$\"\"*F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "a[20,7] = a[9,7];" "6#/&%\"aG6$\"#?\"\"(&F%6 $\"\"*F(" }{TEXT -1 4 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[20,8]=a[9,8]" "6#/&%\"aG6$\"#?\"\")&F%6$\"\"*F(" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "a[21,4] = a[7,4];" "6#/&%\"aG6$\"#@ \"\"%&F%6$\"\"(F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "a[21,5] = a[7,5 ];" "6#/&%\"aG6$\"#@\"\"&&F%6$\"\"(F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "a[21,6] = a[7,6];" "6#/&%\"aG6$\"#@\"\"'&F%6$\"\"(F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "a[23,2]=a[3,2]" "6#/&%\"aG6$\"#B\"\"#&F%6$\" \"$F(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "a[22,4] = a[6,4];" "6#/&% \"aG6$\"#A\"\"%&F%6$\"\"'F(" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "a[22, 5]=a[6,5]" "6#/&%\"aG6$\"#A\"\"&&F%6$\"\"'F(" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[25,2]+a[25, 24] = 0;" "6#/,&&%\"aG6$\"#D\"\"#\"\"\"&F &6$F(\"#CF*\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "a[25,3]+a[25,23 ] = 0;" "6#/,&&%\"aG6$\"#D\"\"$\"\"\"&F&6$F(\"#BF*\"\"!" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "a[25,6]+a[25,22] = 0;" "6#/,&&%\"aG6$\"#D\"\"' \"\"\"&F&6$F(\"#AF*\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "a[25,7] +a[25,21] = 0;" "6#/,&&%\"aG6$\"#D\"\"(\"\"\"&F&6$F(\"#@F*\"\"!" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[2 4,3]+a[24,23] = 0;" "6#/,&&%\"aG6$\"#C\"\"$\"\"\"&F&6$F(\"#BF*\"\"!" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "a[23,7]+a[23,21] = 0;" "6#/,&&%\"a G6$\"#B\"\"(\"\"\"&F&6$F(\"#@F*\"\"!" }{TEXT -1 7 ", " }{XPPEDIT 18 0 "a[23,6]+a[23,22] = 0;" "6#/,&&%\"aG6$\"#B\"\"'\"\"\"&F&6$F(\"#AF *\"\"!" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "a[22, 7]+a[22, 21] = 0;" "6#/,&&%\"aG6$\"#A\"\"(\"\"\"&F&6$F(\"#@F*\"\"!" }{TEXT -1 7 ", \+ " }{XPPEDIT 18 0 "a[22,9]+a[22,20] = 0;" "6#/,&&%\"aG6$\"#A\"\"*\"\"\" &F&6$F(\"#?F*\"\"!" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 4 " \+ " }{XPPEDIT 18 0 "a[22,10]+a[22,19]=0" "6#/,&&%\"aG6$\"#A\"#5\"\"\"& F&6$F(\"#>F*\"\"!" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "a[21,9]+a[21,20 ] = 0;" "6#/,&&%\"aG6$\"#@\"\"*\"\"\"&F&6$F(\"#?F*\"\"!" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "a[21,10]+a[21,19] = 0;" "6#/,&&%\"aG6$\"#@\"#5 \"\"\"&F&6$F(\"#>F*\"\"!" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "a[21,11] +a[21,18]=0" "6#/,&&%\"aG6$\"#@\"#6\"\"\"&F&6$F(\"#=F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "We also incorporate the " }{TEXT 260 18 "row-sum conditions" } {TEXT -1 53 " for rows 21 to 24 into the next system of equations." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 604 "cdns3 := [a[18,7]=a[11,7],a[18,8]=a[11,8],a[19,6]=a[10,6],a[19,7] =a[10,7],\n a[19,8]=a[10,8],a[20,6]=a[9,6],a[20,7]=a[9,7],a[2 0,8]=a[9,8],a[21,4]=a[7,4],\n a[21,5]=a[7,5],a[21,6]=a[7,6],a [23,2]=a[3,2],a[22,4]=a[6,4],a[22,5]=a[6,5],\n \n a[ 25,2]+a[25,24]=0,a[25,3]+a[25,23]=0,a[25,6]+a[25,22]=0,a[25,7]+a[25,21 ]=0,\n a[24,3]+a[24,23]=0,a[23,7]+a[23,21]=0,a[23,6]+a[23,22] =0,\n a[22,7]+a[22,21]=0,a[22,9]+a[22,20]=0,a[22,10]+a[22,19] =0,\n a[21,9]+a[21,20]=0,a[21,10]+a[21,19]=0,a[21,11]+a[21,18 ]=0,\n seq(add(a[i,j],j=1..i-1)=c[i],i=21..24)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "eqns 3 := subs(e20,cdns3):\nnops(%);\nindets(eqns3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#J" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#F_o&F%6$FfoF+&F%6$FfoF(&F%6$\"#=F_o&F%6$F]pF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#J" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 31 equations and 31 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e21 := solve(\{op(eqns3)\}):\ne22 : = `union`(e20,e21):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "indets(map(rhs,e22 ));" }}{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 3 "e22" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27520 "e22 := \{a[23,20] = 0, a[ 23,19] = 0, c[12] = -48827895154/211479617260065*(495+66*15^(1/2))^(1/ 2)*15^(1/2)-1080276228947/70493205753355*(495+66*15^(1/2))^(1/2)+25236 14583531/6408473250305+33872310083/1281694650061*15^(1/2), a[23,18] = \+ 0, a[25,5] = 0, a[23,17] = 0, a[25,4] = 0, a[25,8] = 0, a[24,15] = 0, \+ a[24,14] = 0, a[23,4] = 0, a[23,16] = 0, a[23,14] = 0, a[23,15] = 0, a [23,13] = 0, a[23,12] = 0, a[23,11] = 0, a[23,10] = 0, a[23,9] = 0, a[ 24,10] = 0, a[23,8] = 0, a[23,3] = 0, a[23,5] = 0, a[24,12] = 0, a[24, 13] = 0, a[24,22] = 0, a[24,21] = 0, a[24,20] = 0, a[24,18] = 0, a[24, 19] = 0, a[24,17] = 0, a[24,16] = 0, a[25,3] = -753491/379050, a[24,11 ] = 0, b[8] = 0, b[5] = 0, b[4] = 0, b[16] = 31/175+1/100*15^(1/2), b[ 25] = 1/42, b[13] = 31/175-1/100*15^(1/2), b[1] = 1/42, c[16] = 1/2+1/ 66*(495-66*15^(1/2))^(1/2), c[14] = 1/2-1/66*(495-66*15^(1/2))^(1/2), \+ c[25] = 1, b[12] = 0, b[17] = 31/175-1/100*15^(1/2), c[17] = 1/2+1/66* (495+66*15^(1/2))^(1/2), b[14] = 31/175+1/100*15^(1/2), c[15] = 1/2, c [13] = 1/2-1/66*(495+66*15^(1/2))^(1/2), b[15] = 128/525, a[11,6] = 0, a[24,5] = 0, a[24,1] = 39/88, c[11] = 51789075/64972747+333240/649727 47*15^(1/2)+3057458/714700217*(495+66*15^(1/2))^(1/2)+70408/714700217* (495+66*15^(1/2))^(1/2)*15^(1/2), a[13,5] = 0, a[12,5] = 0, a[11,5] = \+ 0, a[10,5] = 0, a[9,5] = 0, a[12,3] = 0, a[11,3] = 0, a[10,3] = 0, a[9 ,3] = 0, a[8,3] = 0, a[7,3] = 0, a[6,3] = 0, a[13,4] = 0, a[12,4] = 0, a[11,4] = 0, a[10,4] = 0, a[9,4] = 0, a[8,4] = 0, a[10,2] = 0, a[9,2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[5,2] = 0, a[4,2] = 0, a[13 ,3] = 0, a[13,2] = 0, a[12,2] = 0, a[11,2] = 0, a[13,8] = 0, a[13,7] = 0, a[13,6] = 0, a[24,8] = 0, a[24,9] = 0, a[24,4] = 0, a[19,3] = 0, a [18,3] = 0, a[17,3] = 0, a[16,3] = 0, a[15,3] = 0, a[14,3] = 0, a[17,2 ] = 0, a[16,2] = 0, a[15,2] = 0, a[14,2] = 0, a[20,3] = 0, a[19,2] = 0 , a[18,2] = 0, a[15,5] = 0, a[14,5] = 0, a[20,2] = 0, a[20,5] = 0, a[1 9,5] = 0, a[18,5] = 0, a[17,5] = 0, a[16,5] = 0, a[14,4] = 0, a[19,4] \+ = 0, a[18,4] = 0, a[17,4] = 0, a[16,4] = 0, a[15,4] = 0, a[16,8] = 0, \+ a[15,8] = 0, a[14,8] = 0, a[20,4] = 0, a[17,7] = 0, a[16,7] = 0, a[15, 7] = 0, a[14,7] = 0, a[17,8] = 0, a[18,6] = 0, a[17,6] = 0, b[9] = -13 /56, a[24,2] = 0, a[16,6] = 0, a[15,6] = 0, a[14,6] = 0, a[22,2] = 0, \+ a[22,13] = 0, a[22,12] = 0, a[22,11] = 0, a[22,8] = 0, a[22,6] = 0, a[ 22,3] = 0, a[22,18] = 0, a[22,17] = 0, a[22,16] = 0, a[22,15] = 0, a[2 2,14] = 0, a[21,2] = 0, a[21,13] = 0, a[21,12] = 0, a[21,8] = 0, a[24, 6] = 0, a[24,7] = 0, a[21,7] = 0, a[21,3] = 0, a[21,17] = 0, a[21,16] \+ = 0, a[21,15] = 0, a[21,14] = 0, c[9] = 1/5, a[22,9] = -76890112/58634 2617, c[10] = 1/3, a[22,10] = -828786688/2261607237, a[21,9] = -212603 79985/34709636178, a[21,10] = -81770692250/117233567091, a[10,7] = 562 0721356384469075/3545698345096452809913, a[10,1] = -17689789589/433566 079650, c[2] = 39/88, a[10,8] = 10436097770007465488281250/34480588269 890588098919643, a[22,7] = 75739731820032/659439747650125, a[8,1] = 56 73493348726844397056253/158826937830690185546875000, a[8,5] = 83874430 427399451044535741888/1824862440224861393218994140625, a[10,6] = -4601 2134893475328/76616158064651325, a[8,6] = -986894364933740544726981696 /54826948196505455902099609375, a[9,1] = 2526695731/133816691250, a[9, 6] = 24467886602752/291937806983125, a[7,4] = 46426400659946403657/610 018749211776263125, a[7,1] = 1534965908593628117013/292519076762835507 8125, a[7,6] = 11005843237812473721/5193844391648046875, a[8,7] = 5672 317641427470160608354913689/40381865475711712996627696578125000, a[23, 7] = -253319312163456/659439747650125, a[9,8] = 1095050362935718750/11 254693571026117461, a[10,9] = 110139186925/164279737386, a[7,5] = -156 479396217570600333298656/73273972406867506457421875, a[5,1] = 87193841 29925/100440274363616, a[5,4] = -2900926450625/200880548727232, a[5,3] = 6944404306875/200880548727232, a[9,7] = 11944081475573/132247896202 9186830, a[6,1] = 1167025/30304848, a[6,4] = 390625/3949854672, a[6,5] = 404856925/3216589572, a[3,1] = 28078/126711, a[3,2] = 112211/506844 , a[4,1] = 101/608, a[4,3] = 303/608, a[12,9] = -281211230909242277181 9313341069996795037876904500378145129471863425104791510387562227247473 30456172743380029/1777219414844154523916564814426187150039165643225617 3921137707234507471866579929942450646636509884128799622+67637408773077 5429952449408713302591862747979275682175058845619651319936902380422980 0976446366900597141/38492948123113591594467507351660973576763388417275 663680176970401792228430972341222548509067597756397660*(495+66*15^(1/2 ))^(1/2)*15^(1/2)+3647611941783523639106839681293674145014752901446257 471407287711249070189350233573351451468961784464641/641549135385226526 5744584558610162262793898069545943946696161733632038071828723537091418 177932959399610*(495+66*15^(1/2))^(1/2)-594585803588270123266011580156 9296816414941113900029074561383838890594834114310825436409542274213823 4247/12830982707704530531489169117220324525587796139091887893392323467 264076143657447074182836355865918799220*15^(1/2), a[11,7] = 1899397497 3485227904952341910291889381118708193023824550/12060034665080522994492 944014191159522924996678137100092619*(495+66*15^(1/2))^(1/2)*15^(1/2)+ 37364999682821940340524213896266047309233288831638123551875/4288012325 3619637313752689828235233859288877077820800329312+83870865016481720496 920100044907813868655001976949702225/134000385167561366605477155713235 1058102777408681900010291*15^(1/2)+10188221773942597558585056227974439 3868968529126539562125/40200115550268409981643146713970531743083322260 45700030873*(495+66*15^(1/2))^(1/2), a[12,7] = 12039560526382767444395 9816224372224515900903963473583877146503450993695151583738412984486608 167592412726499223108027/295063494385794976532558803599500929144693875 7618744973811305715669070205151124895945261251585564237742291874131282 70-4574290713578312637346253714366174333264683073453621195923343653987 48049518495290294028445214608755002615341437/1278161119280030221063715 8483842362102867397693821723949799894804717653043756226536475032495497 3542895485894482620*(495+66*15^(1/2))^(1/2)*15^(1/2)-19409130444629839 5030544188331511201788876606372749180626268993467082976330239046596838 838518978497037496895389/127816111928003022106371584838423621028673976 93821723949799894804717653043756226536475032495497354289548589448262*( 495+66*15^(1/2))^(1/2)+12780472568351676678614075153447418157156395227 824198601931888244825804433658120101910199505548093911257104546897/127 8161119280030221063715848384236210286739769382172394979989480471765304 37562265364750324954973542895485894482620*15^(1/2), a[12,6] = 25108552 1083237108777248115694608871632336099720342563852044971041903837740861 882751285544579556435153690263824896/109059581580566357404472531727925 4749804567879870625150516917051380618003386804945891914477353935936889 3655280625-17224203165087627219883985321280669596094555813328784551128 2815171786743955334377646964770699567651065020416/69046901918687152519 4507956492089110354268996435976670159491643799061730539287715031284885 947411166121788875*(495+66*15^(1/2))^(1/2)*15^(1/2)-248326505096660202 6265671372002316578111995428215292500398470208821705013352242278156416 063385897318005911552/299203241647644327584286781146571947820183231788 9232237357797123129267499003580098468901172438781719861085125*(495+66* 15^(1/2))^(1/2)+989414013347979104246599002035391609046115745254951412 45531026292768496305967031119826379359226270888023523328/1496016208238 2216379214339057328597391009161589446161186788985615646337495017900492 344505862193908599305425625*15^(1/2), a[12,1] = 4536078586418429346535 5049313006780143265609187196347220787941439290058648676039911927050322 00412298285836937/1172606656536926424103513708860616450309534511569707 813821852440031214272905699213661387392662314110822638750-536443589551 6573669420621604632495515121639476730674940078684840492375553763895312 8683448772183267658/13367151790715393045193549113238448870986337808439 15294531482553071844639268719963591518065572670919625*(495+66*15^(1/2) )^(1/2)*15^(1/2)-35279136548504838526715366339693933451439371283610203 45607398270646119601220687075114545149449903002764/2539758840235924678 5867743315153052854874041836034390596098168508365048146105679308238843 245880747472875*(495+66*15^(1/2))^(1/2)+136142098290337291849630976028 2958026582387432115444780575878975425366396745823033150091119530031197 07396/1269879420117962339293387165757652642743702091801719529804908425 41825240730528396541194216229403737364375*15^(1/2), a[13,1] = 49032552 280925998159/1175735428193602866744+1095947848209481903/17636031422904 04300116*15^(1/2)-7475873553226772611/6466544855064815767092*(495+66*1 5^(1/2))^(1/2)+5352672249559690835/116397807391166683807656*(495+66*15 ^(1/2))^(1/2)*15^(1/2), a[13,10] = 1036874421023669248333689/196556516 58225505790722624+587392382348212140203319/39311303316451011581445248* 15^(1/2)-259903711964372646898395/432424336480961127395897728*(495+66* 15^(1/2))^(1/2)*15^(1/2)-185211952023363244825707/10810608412024028184 8974432*(495+66*15^(1/2))^(1/2), a[13,9] = -162384252323883015198625/1 0650929594044147603352784*15^(1/2)+122383789982155868115425/4686409021 37942494547522496*(495+66*15^(1/2))^(1/2)*15^(1/2)+2265050157236155663 680125/14201239458725530137803712-120963267116719524969875/26035605674 330138585973472*(495+66*15^(1/2))^(1/2), a[11,9] = 2874397467399404298 5226536878679230542480801150/17539834082278474517747485443485973779433 56553311*(495+66*15^(1/2))^(1/2)*15^(1/2)+2919917156154994497064347556 11025178229310468246875/5612746906329111845679195341915511609418740970 5952+1071091749312998743815087165568469189531652126075/175398340822784 7451774748544348597377943356553311*15^(1/2)+35325978470435165189631258 1655773178339865859125/17539834082278474517747485443485973779433565533 11*(495+66*15^(1/2))^(1/2), a[13,11] = 1451483888900070743380148776770 70795676982463/1292134149111656271966122106166449116832367968+28938477 6308176063330366809130393474082987849/15505609789339875263593465273997 389401988415616*15^(1/2)-274115022589237178977231727148612453501574585 /511685123048215883698584354041913850265617715328*(495+66*15^(1/2))^(1 /2)*15^(1/2)-24857236960157204698513148202889704149473711/533005336508 5582121860253687936602606933517868*(495+66*15^(1/2))^(1/2), a[13,12] = 6443329452303863103711607826787077523918806208419166908252875/4818690 8177657952343922139838789383990650083655460342201310944-17149160115186 6911759544690813927486314725154315627741296625/90350452833108660644854 01219773009498246890685398814162745802*15^(1/2)-7090202015237773474138 169907437581683377756583694650388556125/238525195479406864102414592202 0074507537179140945286938964891728*(495+66*15^(1/2))^(1/2)+11992999518 5051271338440264753881499366865434058651556229025/14456072453297385703 1766419516368151971950250966381026603932832*(495+66*15^(1/2))^(1/2)*15 ^(1/2), a[11,1] = 36250222198273693341131051026706723455692594208/4999 430483339723613905130972844241982587217763497*(495+66*15^(1/2))^(1/2)* 15^(1/2)+4861393204748134765917698036236910531265683193125/22219691037 06543828402280432375218658927652339332+1484428141946824012327061188206 43700603232458928/555492275926635957100570108093804664731913084833*15^ (1/2)+143068868482411022237818049543538495099770530576/166647682777990 7871301710324281413994195739254499*(495+66*15^(1/2))^(1/2), a[12,8] = \+ -340659803958984878209931438027146134138660774601249541025577546738764 928298414303784311737346180734782282094165416156250/308280706336293206 8096196297896660310632239903888789451271870099917142127007003940833673 0423089458130745933155208375541+41194832565673882203999777509211923642 1293381998104532443768303827226809809726860648464826292767320320631125 00000/3514251750809858381605960008089851362393260494840337712198477138 17031123765375550406811559375414180212098687404767*(495+66*15^(1/2))^( 1/2)*15^(1/2)+26719277739082859443666346678807717091479495492196588967 17126925052525708719967923978519767792683025169075208750000/6677078326 5387309250513240153707175885471949401966416531771065625235913515421354 57729419628132869424029875060690573*(495+66*15^(1/2))^(1/2)-2086803057 6527122590278134042922470681076456345098996480647230076556364761794880 724541437909825056194904756420000000/667707832653873092505132401537071 7588547194940196641653177106562523591351542135457729419628132869424029 875060690573*15^(1/2), a[11,10] = -37768595712400175213251809663599923 9114085793540/40806683117092499328338457010324585780808416836571*(495+ 66*15^(1/2))^(1/2)*15^(1/2)-181781480924529622996033569801045149915635 8365559525/652906929873479989253415312165193372492934669385136-1424811 7750070402721956352075060738058061271231816/40806683117092499328338457 010324585780808416836571*15^(1/2)-484426665991669901588562627983527245 3019208362650/40806683117092499328338457010324585780808416836571*(495+ 66*15^(1/2))^(1/2), a[11,8] = -275413882637225084153879395633151790451 174339761235000000000/173657067762976556173412167504047698362275139689 15818501163621*(495+66*15^(1/2))^(1/2)*15^(1/2)-9030706067191500671304 838647417536518671082752129344726562500/192952297514418395748235741671 1641092914168218768424277907069-11310663799138970197488474860123128359 22421062622057500000000/1929522975144183957482357416711641092914168218 768424277907069*15^(1/2)-109756253028446039591902781216552083991001737 7896612500000000/57885689254325518724470722501349232787425046563052728 33721207*(495+66*15^(1/2))^(1/2), a[25,7] = -30334395520464588/1648599 3691253125, a[12,11] = 1/5499, b[2] = -11/100, b[11] = -29/100, b[10] \+ = -27/100, b[7] = -21/100, b[6] = -19/100, b[3] = -17/100, a[20,15] = \+ 13/1036, a[20,14] = -50/501, a[19,14] = 21/32, a[12,10] = -11/342, a[2 3,1] = 28078/126711, a[24,23] = 274193/636804, a[22,1] = 1167025/30304 848, a[20,16] = 4573876925/1493463420644832*(495+66*15^(1/2))^(1/2)*(4 95-66*15^(1/2))^(1/2)+7986761337425/7582198904812224*15^(1/2)*(495-66* 15^(1/2))^(1/2)+37543791335/995642280429888*15^(1/2)+78669144834545/99 5642280429888-6267380675/14360225198508*(495+66*15^(1/2))^(1/2)*15^(1/ 2)+5387414718575/32856195254186304*(495-66*15^(1/2))^(1/2)-20472462227 5/8960780523868992*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^ (1/2)-150291531475/62227642526868*(495+66*15^(1/2))^(1/2), a[17,14] = \+ -86343889643783/1423361572162200*(495+66*15^(1/2))^(1/2)*15^(1/2)-4875 329406099/718869480890-47503534664513/28754779235600*15^(1/2)-46830515 603137/189781542954960*(495+66*15^(1/2))^(1/2)-56659550856457/35584039 3040550*15^(1/2)*(495-66*15^(1/2))^(1/2)-21856354431557/86983207187690 0*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-125819226953867/1897 81542954960*(495-66*15^(1/2))^(1/2)-34670051930677/6262790917513680*(4 95+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[19,16] = 233 18577/238751570176*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-193 20841301/23636405447424*15^(1/2)*(495-66*15^(1/2))^(1/2)-150970273/596 87892544*15^(1/2)+5937215723/119375785088+3596875/5595739926*(495+66*1 5^(1/2))^(1/2)*15^(1/2)-18335887105/2626267271936*(495-66*15^(1/2))^(1 /2)-29492131/2148764131584*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15 ^(1/2))^(1/2)+493761427/358127355264*(495+66*15^(1/2))^(1/2), a[18,15] = 19990678222944016799007968810994718650113/3272381042960700814201444 438358736068377200*(495+66*15^(1/2))^(1/2)*15^(1/2)-883939112584854206 78134471781658464932823/6326603349724021574122792580826889732195920*(4 95+66*15^(1/2))^(1/2)+87051869427126857828192491903627294702861/958576 265109700238503453421337407535181200+439699593251478675323037151400704 7435591/45646488814747630404926353397019406437200*15^(1/2)+45290633122 1238361922696065114103183779/7083075850564287476626503113330597550600* (495-66*15^(1/2))^(1/2)+2225327884102816028172189393256449625076/10783 9829824841276831638509900458347707885*15^(1/2)*(495-66*15^(1/2))^(1/2) +110704560801946412846083674130999701608833/47449525122930161805920944 356201672991469400*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+101 01069885562826853855484798548336576859/1186238128073254045148023608905 0418247867350*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2) , a[16,15] = 418320022266565899999989/730900563458308064970225-1191560 65868215739462012/2192701690374924194910675*15^(1/2)+34956957959455966 71104/146180112691661612994045*(495+66*15^(1/2))^(1/2)-658314858045626 45782499/10963508451874620974553375*(495+66*15^(1/2))^(1/2)*15^(1/2)-4 641549371926342688144/32890525355623862923660125*(495+66*15^(1/2))^(1/ 2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+7610546603899839370279/65781050711 24772584732025*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+1771425 0989146270037792/730900563458308064970225*(495-66*15^(1/2))^(1/2)-1502 9054399563919460349/10963508451874620974553375*15^(1/2)*(495-66*15^(1/ 2))^(1/2), a[2,1] = 39/88, a[25,14] = -49414375111/1558819416000*(495+ 66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-165545310587/584557281000*1 5^(1/2)*(495-66*15^(1/2))^(1/2)-2733053737/843517000*15^(1/2)-16561593 209/3374068000-24743106247/2338229124000*(495+66*15^(1/2))^(1/2)*15^(1 /2)*(495-66*15^(1/2))^(1/2)-1363545178441/1558819416000*(495-66*15^(1/ 2))^(1/2)-4721773003/27836061000*(495+66*15^(1/2))^(1/2)-3651713009/27 836061000*(495+66*15^(1/2))^(1/2)*15^(1/2), a[18,16] = -43684062056789 3961487535263885/2696403426422943408076694358192*15^(1/2)-369545114918 022904188418990316329/625565594930122870673793091100544-18257270145419 395739617682201851/625565594930122870673793091100544*(495+66*15^(1/2)) ^(1/2)-5983444403437426102960354262185/9383483923951843060106896366508 16*(495+66*15^(1/2))^(1/2)*15^(1/2)+107005288406322056654355448927205/ 3753393569580737224042758546603264*(495-66*15^(1/2))^(1/2)+84055437892 642261541931143840615/11260180708742211672128275639809792*15^(1/2)*(49 5-66*15^(1/2))^(1/2)+1012338871884896232541346539753/37533935695807372 24042758546603264*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^( 1/2)+4578034999918124747163891877361/375339356958073722404275854660326 4*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[18,14] = 42300428 27581684523751752104701705965843/2221051020559910704182786636088207918 14660*(495+66*15^(1/2))^(1/2)*15^(1/2)+8752761185143623764977204646806 6688557713/17947887034827561245921508170409760954720+51251514377913613 08235261701221815485791/6730457638060335467220565563903660358020*15^(1 /2)+2558836067301785957586054852230517925601/1586465014685650502987704 7400630056558190*(495+66*15^(1/2))^(1/2)+72311136213172294095616390215 153042369611/848037662395602268869791261051861205110520*15^(1/2)*(495- 66*15^(1/2))^(1/2)+477343459314808814429839892297318102330687/11307168 83194136358493055014735814940147360*(495-66*15^(1/2))^(1/2)+4852212171 4870311549903684616247887004701/18656828572703249915135407743140946512 431440*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+205838 82814114410685607175778197029916327/1381987301681722215935956129121551 593513440*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[17,16] = \+ -3399762661/1690419432009*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1 /2)-21792326747/1690419432009*15^(1/2)*(495-66*15^(1/2))^(1/2)+1502483 4410/51224831273*15^(1/2)+843590793/80496163429*(495+66*15^(1/2))^(1/2 )*15^(1/2)-167470020833/3380838864018*(495-66*15^(1/2))^(1/2)-65691156 2/1690419432009*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/ 2)+23471698999/482976980574*(495+66*15^(1/2))^(1/2)+61299209505/512248 31273, a[25,16] = 571468963/226284800+120370717/226284800*15^(1/2)-175 1767/2036563200*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/ 2)-590329/169713600*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-51 9210887/22402195200*15^(1/2)*(495-66*15^(1/2))^(1/2)-28396679/29869593 6*(495-66*15^(1/2))^(1/2)+5795203/339427200*(495+66*15^(1/2))^(1/2)*15 ^(1/2)+7675999/113142400*(495+66*15^(1/2))^(1/2), a[16,14] = -14939172 465658402889149/8976679102595142257770+219312724397669057743089/359067 164103805690310800*15^(1/2)+659570008376205270318919/17773824623138381 670384600*(495+66*15^(1/2))^(1/2)*15^(1/2)-11242117128844970311919/157 989552205674503736752*(495+66*15^(1/2))^(1/2)-1749823831034832192631/4 3088059692456682837296*(495-66*15^(1/2))^(1/2)+44646359022841677628361 /1615802238467125606398600*15^(1/2)*(495-66*15^(1/2))^(1/2)-3070220299 360933125107/1974869402570931296709400*(495+66*15^(1/2))^(1/2)*(495-66 *15^(1/2))^(1/2)+18545262695413350953267/11849216415425587780256400*(4 95+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[25,15] = -34 60846507/13082090000+48037163/3270522500*15^(1/2)-108037591/6475634550 0*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-412156237/4 6254532500*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-2890671767/ 58869405000*15^(1/2)*(495-66*15^(1/2))^(1/2)-4618638191/19623135000*(4 95-66*15^(1/2))^(1/2)-848300537/15418177500*(495+66*15^(1/2))^(1/2)+42 9589031/129512691000*(495+66*15^(1/2))^(1/2)*15^(1/2), a[19,15] = 1908 413061857/289859328166800-2931986748899/14348036744256600*(495+66*15^( 1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-1254438533/217394496125100*15^(1/2 )*(495-66*15^(1/2))^(1/2)-857188607021/289859328166800*15^(1/2)+537935 7542731/9565357829504400*(495+66*15^(1/2))^(1/2)*15^(1/2)+136382862089 /144929664083400*(495-66*15^(1/2))^(1/2)-1935518588/85404980620575*(49 5+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+34225020981/2169 0153808400*(495+66*15^(1/2))^(1/2), a[17,15] = -4863804455866/35119141 675695*15^(1/2)-62267936631107/9657763960816125*(495+66*15^(1/2))^(1/2 )*15^(1/2)-5561334169199/58531902792825+2301512129984/643850930721075* (495+66*15^(1/2))^(1/2)-331440117351157/9657763960816125*15^(1/2)*(495 -66*15^(1/2))^(1/2)-27173454387092/214616976907025*(495-66*15^(1/2))^( 1/2)-37986613754722/28973291882448375*(495+66*15^(1/2))^(1/2)*15^(1/2) *(495-66*15^(1/2))^(1/2)-25832795272667/5794658376489675*(495+66*15^(1 /2))^(1/2)*(495-66*15^(1/2))^(1/2), a[15,14] = 115988619653/2924029730 68800*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-2683538373203/87 7208919206400*15^(1/2)*(495-66*15^(1/2))^(1/2)-1968290835/88606961536* 15^(1/2)+496725643/7310074326720*(495+66*15^(1/2))^(1/2)*15^(1/2)-6926 8797311/7310074326720*(495-66*15^(1/2))^(1/2)-124493701507/87720891920 6400*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-52034279 227/11696118922752*(495+66*15^(1/2))^(1/2)-74896827627/88606961536, a[ 21,1] = 1534965908593628117013/2925190767628355078125, a[25,2] = -1029 /400, a[25,6] = -2667/1600, c[4] = 101/152, a[23,6] = -381/544, a[21,1 1] = -44794043532626317794933177103220192762017625/1670289759544519704 98474513047407579017107824+390422828672114370457150585026889349975/695 9540664768832104103104710308649125712826*(495+66*15^(1/2))^(1/2)*15^(1 /2)+29441442259346454399120790160135926728275/463969377651255473606873 6473539099417141884*15^(1/2)+10783585214249918765694265216178872324622 5/13919081329537664208206209420617298251425652*(495+66*15^(1/2))^(1/2) , c[8] = 17727/277750, a[20,19] = -110845/348543, a[21,19] = 817706922 50/117233567091, a[22,19] = 828786688/2261607237, a[23,21] = 253319312 163456/659439747650125, a[22,21] = -75739731820032/659439747650125, a[ 25,19] = 3601857/2273300, a[21,20] = 21260379985/34709636178, a[25,21] = 30334395520464588/16485993691253125, a[22,20] = 76890112/586342617, a[25,23] = 753491/379050, a[25,20] = 3071471/2273300, a[23,22] = 381/ 544, a[25,22] = 2667/1600, a[25,24] = 1029/400, a[21,18] = -3904228286 72114370457150585026889349975/6959540664768832104103104710308649125712 826*(495+66*15^(1/2))^(1/2)*15^(1/2)+447940435326263177949331771032201 92762017625/167028975954451970498474513047407579017107824-294414422593 46454399120790160135926728275/4639693776512554736068736473539099417141 884*15^(1/2)-107835852142499187656942652161788723246225/13919081329537 664208206209420617298251425652*(495+66*15^(1/2))^(1/2), a[25,17] = 112 245633/226284800-1120087/28285600*15^(1/2)-68125601/7467398400*(495+66 *15^(1/2))^(1/2)+993373/4480439040*(495+66*15^(1/2))^(1/2)*15^(1/2), a [25,18] = -6212380489353402482048992934870163/551398114240467508084664 0779279519700*(495+66*15^(1/2))^(1/2)*15^(1/2)+56874537829985536110363 501647573078867/44111849139237400646773126234236157600-477467113712319 024289213588759633869/11027962284809350161693281558559039400*15^(1/2)- 168276384308509585366452969427482141/110279622848093501616932815585590 39400*(495+66*15^(1/2))^(1/2), a[20,17] = 229884359555/8366741852352+4 6739047805/4183370926176*15^(1/2)-20855494525/25100225557056*(495+66*1 5^(1/2))^(1/2)-34057541825/75300676671168*(495+66*15^(1/2))^(1/2)*15^( 1/2), a[20,18] = -52601571650923977687781073690186/1325511779651805411 892910350460322529*(495+66*15^(1/2))^(1/2)*15^(1/2)+146135103700320839 8982108006046541205/15906141355821664942714924205523870348-31257463047 7851429199757289182427/101962444588600416299454642343101733*15^(1/2)-3 126986093611720912749441466631523/132551177965180541189291035046032252 9*(495+66*15^(1/2))^(1/2), a[19,18] = 19938900280078839363451072275417 6/2796728200664360059162382772909427897*(495+66*15^(1/2))^(1/2)*15^(1/ 2)-1025465248156834803353840916920180809/55934564013287201183247655458 18855794+16445717956872940821116448130678578/2796728200664360059162382 772909427897*15^(1/2)+13437632256176229690482072355537756/279672820066 4360059162382772909427897*(495+66*15^(1/2))^(1/2), a[19,17] = 24667305 63/34107367168-46877457/4872481024*15^(1/2)-417081443/102322101504*(49 5+66*15^(1/2))^(1/2)+221689729/306966304512*(495+66*15^(1/2))^(1/2)*15 ^(1/2), a[18,17] = 61673043751462390569674581/351685648446307151385663 744-4915309705585387914994781/351685648446307151385663744*15^(1/2)-469 55489957737999234808063/11605626398728135995726903552*(495+66*15^(1/2) )^(1/2)+6751918613532367335628313/34816879196184407987180710656*(495+6 6*15^(1/2))^(1/2)*15^(1/2), c[6] = 25/152, a[21,5] = -1564793962175706 00333298656/73273972406867506457421875, a[21,6] = 11005843237812473721 /5193844391648046875, a[23,2] = 112211/506844, a[22,4] = 390625/394985 4672, a[22,5] = 404856925/3216589572, c[5] = 8225/76912, c[3] = 101/22 8, c[7] = 2013/3445, a[18,7] = 189939749734852279049523419102918893811 18708193023824550/1206003466508052299449294401419115952292499667813710 0092619*(495+66*15^(1/2))^(1/2)*15^(1/2)+37364999682821940340524213896 266047309233288831638123551875/428801232536196373137526898282352338592 88877077820800329312+8387086501648172049692010004490781386865500197694 9702225/1340003851675613666054771557132351058102777408681900010291*15^ (1/2)+101882217739425975585850562279744393868968529126539562125/402001 1555026840998164314671397053174308332226045700030873*(495+66*15^(1/2)) ^(1/2), a[18,8] = -275413882637225084153879395633151790451174339761235 000000000/173657067762976556173412167504047698362275139689158185011636 21*(495+66*15^(1/2))^(1/2)*15^(1/2)-9030706067191500671304838647417536 518671082752129344726562500/192952297514418395748235741671164109291416 8218768424277907069-11310663799138970197488474860123128359224210626220 57500000000/1929522975144183957482357416711641092914168218768424277907 069*15^(1/2)-109756253028446039591902781216552083991001737789661250000 0000/5788568925432551872447072250134923278742504656305272833721207*(49 5+66*15^(1/2))^(1/2), a[19,6] = -46012134893475328/76616158064651325, \+ a[19,7] = 5620721356384469075/3545698345096452809913, a[19,8] = 104360 97770007465488281250/34480588269890588098919643, a[20,6] = 24467886602 752/291937806983125, a[20,7] = 11944081475573/1322478962029186830, a[2 0,8] = 1095050362935718750/11254693571026117461, a[21,4] = 46426400659 946403657/610018749211776263125, b[20] = 13/56, b[22] = 19/100, b[21] \+ = 21/100, b[23] = 17/100, b[24] = 11/100, c[21] = 2013/3445, b[18] = 2 9/100, c[24] = 39/88, c[23] = 101/228, c[22] = 25/152, c[19] = 1/3, c[ 18] = 51789075/64972747+333240/64972747*15^(1/2)+3057458/714700217*(49 5+66*15^(1/2))^(1/2)+70408/714700217*(495+66*15^(1/2))^(1/2)*15^(1/2), b[19] = 27/100, c[20] = 1/5, a[24,3] = -274193/636804\}:" }{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 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 51 "#------- -------------------------------------------" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 37 "construction of the scheme .. part 3 " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e22" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27520 "e22 := \{a[23,20] = \+ 0, a[23,19] = 0, c[12] = -48827895154/211479617260065*(495+66*15^(1/2) )^(1/2)*15^(1/2)-1080276228947/70493205753355*(495+66*15^(1/2))^(1/2)+ 2523614583531/6408473250305+33872310083/1281694650061*15^(1/2), a[23,1 8] = 0, a[25,5] = 0, a[23,17] = 0, a[25,4] = 0, a[25,8] = 0, a[24,15] \+ = 0, a[24,14] = 0, a[23,4] = 0, a[23,16] = 0, a[23,14] = 0, a[23,15] = 0, a[23,13] = 0, a[23,12] = 0, a[23,11] = 0, a[23,10] = 0, a[23,9] = \+ 0, a[24,10] = 0, a[23,8] = 0, a[23,3] = 0, a[23,5] = 0, a[24,12] = 0, \+ a[24,13] = 0, a[24,22] = 0, a[24,21] = 0, a[24,20] = 0, a[24,18] = 0, \+ a[24,19] = 0, a[24,17] = 0, a[24,16] = 0, a[25,3] = -753491/379050, a[ 24,11] = 0, b[8] = 0, b[5] = 0, b[4] = 0, b[16] = 31/175+1/100*15^(1/2 ), b[25] = 1/42, b[13] = 31/175-1/100*15^(1/2), b[1] = 1/42, c[16] = 1 /2+1/66*(495-66*15^(1/2))^(1/2), c[14] = 1/2-1/66*(495-66*15^(1/2))^(1 /2), c[25] = 1, b[12] = 0, b[17] = 31/175-1/100*15^(1/2), c[17] = 1/2+ 1/66*(495+66*15^(1/2))^(1/2), b[14] = 31/175+1/100*15^(1/2), c[15] = 1 /2, c[13] = 1/2-1/66*(495+66*15^(1/2))^(1/2), b[15] = 128/525, a[11,6] = 0, a[24,5] = 0, a[24,1] = 39/88, c[11] = 51789075/64972747+333240/6 4972747*15^(1/2)+3057458/714700217*(495+66*15^(1/2))^(1/2)+70408/71470 0217*(495+66*15^(1/2))^(1/2)*15^(1/2), a[13,5] = 0, a[12,5] = 0, a[11, 5] = 0, a[10,5] = 0, a[9,5] = 0, a[12,3] = 0, a[11,3] = 0, a[10,3] = 0 , a[9,3] = 0, a[8,3] = 0, a[7,3] = 0, a[6,3] = 0, a[13,4] = 0, a[12,4] = 0, a[11,4] = 0, a[10,4] = 0, a[9,4] = 0, a[8,4] = 0, a[10,2] = 0, a [9,2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[5,2] = 0, a[4,2] = 0, a[13,3] = 0, a[13,2] = 0, a[12,2] = 0, a[11,2] = 0, a[13,8] = 0, a[13 ,7] = 0, a[13,6] = 0, a[24,8] = 0, a[24,9] = 0, a[24,4] = 0, a[19,3] = 0, a[18,3] = 0, a[17,3] = 0, a[16,3] = 0, a[15,3] = 0, a[14,3] = 0, a [17,2] = 0, a[16,2] = 0, a[15,2] = 0, a[14,2] = 0, a[20,3] = 0, a[19,2 ] = 0, a[18,2] = 0, a[15,5] = 0, a[14,5] = 0, a[20,2] = 0, a[20,5] = 0 , a[19,5] = 0, a[18,5] = 0, a[17,5] = 0, a[16,5] = 0, a[14,4] = 0, a[1 9,4] = 0, a[18,4] = 0, a[17,4] = 0, a[16,4] = 0, a[15,4] = 0, a[16,8] \+ = 0, a[15,8] = 0, a[14,8] = 0, a[20,4] = 0, a[17,7] = 0, a[16,7] = 0, \+ a[15,7] = 0, a[14,7] = 0, a[17,8] = 0, a[18,6] = 0, a[17,6] = 0, b[9] \+ = -13/56, a[24,2] = 0, a[16,6] = 0, a[15,6] = 0, a[14,6] = 0, a[22,2] \+ = 0, a[22,13] = 0, a[22,12] = 0, a[22,11] = 0, a[22,8] = 0, a[22,6] = \+ 0, a[22,3] = 0, a[22,18] = 0, a[22,17] = 0, a[22,16] = 0, a[22,15] = 0 , a[22,14] = 0, a[21,2] = 0, a[21,13] = 0, a[21,12] = 0, a[21,8] = 0, \+ a[24,6] = 0, a[24,7] = 0, a[21,7] = 0, a[21,3] = 0, a[21,17] = 0, a[21 ,16] = 0, a[21,15] = 0, a[21,14] = 0, c[9] = 1/5, a[22,9] = -76890112/ 586342617, c[10] = 1/3, a[22,10] = -828786688/2261607237, a[21,9] = -2 1260379985/34709636178, a[21,10] = -81770692250/117233567091, a[10,7] \+ = 5620721356384469075/3545698345096452809913, a[10,1] = -17689789589/4 33566079650, c[2] = 39/88, a[10,8] = 10436097770007465488281250/344805 88269890588098919643, a[22,7] = 75739731820032/659439747650125, a[8,1] = 5673493348726844397056253/158826937830690185546875000, a[8,5] = 838 74430427399451044535741888/1824862440224861393218994140625, a[10,6] = \+ -46012134893475328/76616158064651325, a[8,6] = -9868943649337405447269 81696/54826948196505455902099609375, a[9,1] = 2526695731/133816691250, a[9,6] = 24467886602752/291937806983125, a[7,4] = 4642640065994640365 7/610018749211776263125, a[7,1] = 1534965908593628117013/2925190767628 355078125, a[7,6] = 11005843237812473721/5193844391648046875, a[8,7] = 5672317641427470160608354913689/40381865475711712996627696578125000, \+ a[23,7] = -253319312163456/659439747650125, a[9,8] = 10950503629357187 50/11254693571026117461, a[10,9] = 110139186925/164279737386, a[7,5] = -156479396217570600333298656/73273972406867506457421875, a[5,1] = 871 9384129925/100440274363616, a[5,4] = -2900926450625/200880548727232, a [5,3] = 6944404306875/200880548727232, a[9,7] = 11944081475573/1322478 962029186830, a[6,1] = 1167025/30304848, a[6,4] = 390625/3949854672, a [6,5] = 404856925/3216589572, a[3,1] = 28078/126711, a[3,2] = 112211/5 06844, a[4,1] = 101/608, a[4,3] = 303/608, a[12,9] = -2812112309092422 7718193133410699967950378769045003781451294718634251047915103875622272 4747330456172743380029/17772194148441545239165648144261871500391656432 256173921137707234507471866579929942450646636509884128799622+676374087 7307754299524494087133025918627479792756821750588456196513199369023804 229800976446366900597141/384929481231135915944675073516609735767633884 17275663680176970401792228430972341222548509067597756397660*(495+66*15 ^(1/2))^(1/2)*15^(1/2)+36476119417835236391068396812936741450147529014 46257471407287711249070189350233573351451468961784464641/6415491353852 2652657445845586101622627938980695459439466961617336320380718287235370 91418177932959399610*(495+66*15^(1/2))^(1/2)-5945858035882701232660115 8015692968164149411139000290745613838388905948341143108254364095422742 138234247/128309827077045305314891691172203245255877961390918878933923 23467264076143657447074182836355865918799220*15^(1/2), a[11,7] = 18993 974973485227904952341910291889381118708193023824550/120600346650805229 94492944014191159522924996678137100092619*(495+66*15^(1/2))^(1/2)*15^( 1/2)+37364999682821940340524213896266047309233288831638123551875/42880 123253619637313752689828235233859288877077820800329312+838708650164817 20496920100044907813868655001976949702225/1340003851675613666054771557 132351058102777408681900010291*15^(1/2)+101882217739425975585850562279 744393868968529126539562125/402001155502684099816431467139705317430833 2226045700030873*(495+66*15^(1/2))^(1/2), a[12,7] = 120395605263827674 4439598162243722245159009039634735838771465034509936951515837384129844 86608167592412726499223108027/2950634943857949765325588035995009291446 9387576187449738113057156690702051511248959452612515855642377422918741 3128270-45742907135783126373462537143661743332646830734536211959233436 5398748049518495290294028445214608755002615341437/12781611192800302210 6371584838423621028673976938217239497998948047176530437562265364750324 954973542895485894482620*(495+66*15^(1/2))^(1/2)*15^(1/2)-194091304446 2983950305441883315112017888766063727491806262689934670829763302390465 96838838518978497037496895389/1278161119280030221063715848384236210286 7397693821723949799894804717653043756226536475032495497354289548589448 262*(495+66*15^(1/2))^(1/2)+127804725683516766786140751534474181571563 9522782419860193188824482580443365812010191019950554809391125710454689 7/12781611192800302210637158483842362102867397693821723949799894804717 6530437562265364750324954973542895485894482620*15^(1/2), a[12,6] = 251 0855210832371087772481156946088716323360997203425638520449710419038377 40861882751285544579556435153690263824896/1090595815805663574044725317 2792547498045678798706251505169170513806180033868049458919144773539359 368893655280625-172242031650876272198839853212806695960945558133287845 511282815171786743955334377646964770699567651065020416/690469019186871 5251945079564920891103542689964359766701594916437990617305392877150312 84885947411166121788875*(495+66*15^(1/2))^(1/2)*15^(1/2)-2483265050966 6020262656713720023165781119954282152925003984702088217050133522422781 56416063385897318005911552/2992032416476443275842867811465719478201832 317889232237357797123129267499003580098468901172438781719861085125*(49 5+66*15^(1/2))^(1/2)+9894140133479791042465990020353916090461157452549 5141245531026292768496305967031119826379359226270888023523328/14960162 0823822163792143390573285973910091615894461611867889856156463374950179 00492344505862193908599305425625*15^(1/2), a[12,1] = 45360785864184293 4653550493130067801432656091871963472207879414392900586486760399119270 5032200412298285836937/11726066565369264241035137088606164503095345115 69707813821852440031214272905699213661387392662314110822638750-5364435 8955165736694206216046324955151216394767306749400786848404923755537638 953128683448772183267658/133671517907153930451935491132384488709863378 0843915294531482553071844639268719963591518065572670919625*(495+66*15^ (1/2))^(1/2)*15^(1/2)-352791365485048385267153663396939334514393712836 1020345607398270646119601220687075114545149449903002764/25397588402359 2467858677433151530528548740418360343905960981685083650481461056793082 38843245880747472875*(495+66*15^(1/2))^(1/2)+1361420982903372918496309 7602829580265823874321154447805758789754253663967458230331500911195300 3119707396/12698794201179623392933871657576526427437020918017195298049 0842541825240730528396541194216229403737364375*15^(1/2), a[13,1] = 490 32552280925998159/1175735428193602866744+1095947848209481903/176360314 2290404300116*15^(1/2)-7475873553226772611/6466544855064815767092*(495 +66*15^(1/2))^(1/2)+5352672249559690835/116397807391166683807656*(495+ 66*15^(1/2))^(1/2)*15^(1/2), a[13,10] = 1036874421023669248333689/1965 5651658225505790722624+587392382348212140203319/3931130331645101158144 5248*15^(1/2)-259903711964372646898395/432424336480961127395897728*(49 5+66*15^(1/2))^(1/2)*15^(1/2)-185211952023363244825707/108106084120240 281848974432*(495+66*15^(1/2))^(1/2), a[13,9] = -162384252323883015198 625/10650929594044147603352784*15^(1/2)+122383789982155868115425/46864 0902137942494547522496*(495+66*15^(1/2))^(1/2)*15^(1/2)+22650501572361 55663680125/14201239458725530137803712-120963267116719524969875/260356 05674330138585973472*(495+66*15^(1/2))^(1/2), a[11,9] = 28743974673994 042985226536878679230542480801150/175398340822784745177474854434859737 7943356553311*(495+66*15^(1/2))^(1/2)*15^(1/2)+29199171561549944970643 4755611025178229310468246875/56127469063291118456791953419155116094187 409705952+1071091749312998743815087165568469189531652126075/1753983408 227847451774748544348597377943356553311*15^(1/2)+353259784704351651896 312581655773178339865859125/175398340822784745177474854434859737794335 6553311*(495+66*15^(1/2))^(1/2), a[13,11] = 14514838889000707433801487 7677070795676982463/1292134149111656271966122106166449116832367968+289 384776308176063330366809130393474082987849/155056097893398752635934652 73997389401988415616*15^(1/2)-2741150225892371789772317271486124535015 74585/511685123048215883698584354041913850265617715328*(495+66*15^(1/2 ))^(1/2)*15^(1/2)-24857236960157204698513148202889704149473711/5330053 365085582121860253687936602606933517868*(495+66*15^(1/2))^(1/2), a[13, 12] = 6443329452303863103711607826787077523918806208419166908252875/48 186908177657952343922139838789383990650083655460342201310944-171491601 151866911759544690813927486314725154315627741296625/903504528331086606 4485401219773009498246890685398814162745802*15^(1/2)-70902020152377734 74138169907437581683377756583694650388556125/2385251954794068641024145 922020074507537179140945286938964891728*(495+66*15^(1/2))^(1/2)+119929 995185051271338440264753881499366865434058651556229025/144560724532973 857031766419516368151971950250966381026603932832*(495+66*15^(1/2))^(1/ 2)*15^(1/2), a[11,1] = 36250222198273693341131051026706723455692594208 /4999430483339723613905130972844241982587217763497*(495+66*15^(1/2))^( 1/2)*15^(1/2)+4861393204748134765917698036236910531265683193125/222196 9103706543828402280432375218658927652339332+14844281419468240123270611 8820643700603232458928/55549227592663595710057010809380466473191308483 3*15^(1/2)+143068868482411022237818049543538495099770530576/1666476827 779907871301710324281413994195739254499*(495+66*15^(1/2))^(1/2), a[12, 8] = -3406598039589848782099314380271461341386607746012495410255775467 38764928298414303784311737346180734782282094165416156250/3082807063362 9320680961962978966603106322399038887894512718700999171421270070039408 336730423089458130745933155208375541+411948325656738822039997775092119 2364212933819981045324437683038272268098097268606484648262927673203206 3112500000/35142517508098583816059600080898513623932604948403377121984 7713817031123765375550406811559375414180212098687404767*(495+66*15^(1/ 2))^(1/2)*15^(1/2)+267192777390828594436663466788077170914794954921965 8896717126925052525708719967923978519767792683025169075208750000/66770 7832653873092505132401537071758854719494019664165317710656252359135154 2135457729419628132869424029875060690573*(495+66*15^(1/2))^(1/2)-20868 0305765271225902781340429224706810764563450989964806472300765563647617 94880724541437909825056194904756420000000/6677078326538730925051324015 3707175885471949401966416531771065625235913515421354577294196281328694 24029875060690573*15^(1/2), a[11,10] = -377685957124001752132518096635 999239114085793540/40806683117092499328338457010324585780808416836571* (495+66*15^(1/2))^(1/2)*15^(1/2)-1817814809245296229960335698010451499 156358365559525/652906929873479989253415312165193372492934669385136-14 248117750070402721956352075060738058061271231816/408066831170924993283 38457010324585780808416836571*15^(1/2)-4844266659916699015885626279835 272453019208362650/40806683117092499328338457010324585780808416836571* (495+66*15^(1/2))^(1/2), a[11,8] = -2754138826372250841538793956331517 90451174339761235000000000/1736570677629765561734121675040476983622751 3968915818501163621*(495+66*15^(1/2))^(1/2)*15^(1/2)-90307060671915006 71304838647417536518671082752129344726562500/1929522975144183957482357 416711641092914168218768424277907069-113106637991389701974884748601231 2835922421062622057500000000/19295229751441839574823574167116410929141 68218768424277907069*15^(1/2)-1097562530284460395919027812165520839910 017377896612500000000/578856892543255187244707225013492327874250465630 5272833721207*(495+66*15^(1/2))^(1/2), a[25,7] = -30334395520464588/16 485993691253125, a[12,11] = 1/5499, b[2] = -11/100, b[11] = -29/100, b [10] = -27/100, b[7] = -21/100, b[6] = -19/100, b[3] = -17/100, a[20,1 5] = 13/1036, a[20,14] = -50/501, a[19,14] = 21/32, a[12,10] = -11/342 , a[23,1] = 28078/126711, a[24,23] = 274193/636804, a[22,1] = 1167025/ 30304848, a[20,16] = 4573876925/1493463420644832*(495+66*15^(1/2))^(1/ 2)*(495-66*15^(1/2))^(1/2)+7986761337425/7582198904812224*15^(1/2)*(49 5-66*15^(1/2))^(1/2)+37543791335/995642280429888*15^(1/2)+786691448345 45/995642280429888-6267380675/14360225198508*(495+66*15^(1/2))^(1/2)*1 5^(1/2)+5387414718575/32856195254186304*(495-66*15^(1/2))^(1/2)-204724 622275/8960780523868992*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1 /2))^(1/2)-150291531475/62227642526868*(495+66*15^(1/2))^(1/2), a[17,1 4] = -86343889643783/1423361572162200*(495+66*15^(1/2))^(1/2)*15^(1/2) -4875329406099/718869480890-47503534664513/28754779235600*15^(1/2)-468 30515603137/189781542954960*(495+66*15^(1/2))^(1/2)-56659550856457/355 840393040550*15^(1/2)*(495-66*15^(1/2))^(1/2)-21856354431557/869832071 876900*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-125819226953867 /189781542954960*(495-66*15^(1/2))^(1/2)-34670051930677/62627909175136 80*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[19,16] \+ = 23318577/238751570176*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2 )-19320841301/23636405447424*15^(1/2)*(495-66*15^(1/2))^(1/2)-15097027 3/59687892544*15^(1/2)+5937215723/119375785088+3596875/5595739926*(495 +66*15^(1/2))^(1/2)*15^(1/2)-18335887105/2626267271936*(495-66*15^(1/2 ))^(1/2)-29492131/2148764131584*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495- 66*15^(1/2))^(1/2)+493761427/358127355264*(495+66*15^(1/2))^(1/2), a[1 8,15] = 19990678222944016799007968810994718650113/32723810429607008142 01444438358736068377200*(495+66*15^(1/2))^(1/2)*15^(1/2)-8839391125848 5420678134471781658464932823/63266033497240215741227925808268897321959 20*(495+66*15^(1/2))^(1/2)+87051869427126857828192491903627294702861/9 58576265109700238503453421337407535181200+4396995932514786753230371514 007047435591/45646488814747630404926353397019406437200*15^(1/2)+452906 331221238361922696065114103183779/708307585056428747662650311333059755 0600*(495-66*15^(1/2))^(1/2)+2225327884102816028172189393256449625076/ 107839829824841276831638509900458347707885*15^(1/2)*(495-66*15^(1/2))^ (1/2)+110704560801946412846083674130999701608833/474495251229301618059 20944356201672991469400*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2 )+10101069885562826853855484798548336576859/11862381280732540451480236 089050418247867350*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^ (1/2), a[16,15] = 418320022266565899999989/730900563458308064970225-11 9156065868215739462012/2192701690374924194910675*15^(1/2)+349569579594 5596671104/146180112691661612994045*(495+66*15^(1/2))^(1/2)-6583148580 4562645782499/10963508451874620974553375*(495+66*15^(1/2))^(1/2)*15^(1 /2)-4641549371926342688144/32890525355623862923660125*(495+66*15^(1/2) )^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+7610546603899839370279/657810 5071124772584732025*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+17 714250989146270037792/730900563458308064970225*(495-66*15^(1/2))^(1/2) -15029054399563919460349/10963508451874620974553375*15^(1/2)*(495-66*1 5^(1/2))^(1/2), a[2,1] = 39/88, a[25,14] = -49414375111/1558819416000* (495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-165545310587/584557281 000*15^(1/2)*(495-66*15^(1/2))^(1/2)-2733053737/843517000*15^(1/2)-165 61593209/3374068000-24743106247/2338229124000*(495+66*15^(1/2))^(1/2)* 15^(1/2)*(495-66*15^(1/2))^(1/2)-1363545178441/1558819416000*(495-66*1 5^(1/2))^(1/2)-4721773003/27836061000*(495+66*15^(1/2))^(1/2)-36517130 09/27836061000*(495+66*15^(1/2))^(1/2)*15^(1/2), a[18,16] = -436840620 567893961487535263885/2696403426422943408076694358192*15^(1/2)-3695451 14918022904188418990316329/625565594930122870673793091100544-182572701 45419395739617682201851/625565594930122870673793091100544*(495+66*15^( 1/2))^(1/2)-5983444403437426102960354262185/93834839239518430601068963 6650816*(495+66*15^(1/2))^(1/2)*15^(1/2)+10700528840632205665435544892 7205/3753393569580737224042758546603264*(495-66*15^(1/2))^(1/2)+840554 37892642261541931143840615/11260180708742211672128275639809792*15^(1/2 )*(495-66*15^(1/2))^(1/2)+1012338871884896232541346539753/375339356958 0737224042758546603264*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/ 2))^(1/2)+4578034999918124747163891877361/3753393569580737224042758546 603264*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[18,14] = 423 0042827581684523751752104701705965843/22210510205599107041827866360882 0791814660*(495+66*15^(1/2))^(1/2)*15^(1/2)+87527611851436237649772046 468066688557713/17947887034827561245921508170409760954720+512515143779 1361308235261701221815485791/6730457638060335467220565563903660358020* 15^(1/2)+2558836067301785957586054852230517925601/15864650146856505029 877047400630056558190*(495+66*15^(1/2))^(1/2)+723111362131722940956163 90215153042369611/848037662395602268869791261051861205110520*15^(1/2)* (495-66*15^(1/2))^(1/2)+477343459314808814429839892297318102330687/113 0716883194136358493055014735814940147360*(495-66*15^(1/2))^(1/2)+48522 121714870311549903684616247887004701/186568285727032499151354077431409 46512431440*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+2 0583882814114410685607175778197029916327/13819873016817222159359561291 21551593513440*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[17,1 6] = -3399762661/1690419432009*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2 ))^(1/2)-21792326747/1690419432009*15^(1/2)*(495-66*15^(1/2))^(1/2)+15 024834410/51224831273*15^(1/2)+843590793/80496163429*(495+66*15^(1/2)) ^(1/2)*15^(1/2)-167470020833/3380838864018*(495-66*15^(1/2))^(1/2)-656 911562/1690419432009*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2) )^(1/2)+23471698999/482976980574*(495+66*15^(1/2))^(1/2)+61299209505/5 1224831273, a[25,16] = 571468963/226284800+120370717/226284800*15^(1/2 )-1751767/2036563200*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2) )^(1/2)-590329/169713600*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/ 2)-519210887/22402195200*15^(1/2)*(495-66*15^(1/2))^(1/2)-28396679/298 695936*(495-66*15^(1/2))^(1/2)+5795203/339427200*(495+66*15^(1/2))^(1/ 2)*15^(1/2)+7675999/113142400*(495+66*15^(1/2))^(1/2), a[16,14] = -149 39172465658402889149/8976679102595142257770+219312724397669057743089/3 59067164103805690310800*15^(1/2)+659570008376205270318919/177738246231 38381670384600*(495+66*15^(1/2))^(1/2)*15^(1/2)-1124211712884497031191 9/157989552205674503736752*(495+66*15^(1/2))^(1/2)-1749823831034832192 631/43088059692456682837296*(495-66*15^(1/2))^(1/2)+446463590228416776 28361/1615802238467125606398600*15^(1/2)*(495-66*15^(1/2))^(1/2)-30702 20299360933125107/1974869402570931296709400*(495+66*15^(1/2))^(1/2)*(4 95-66*15^(1/2))^(1/2)+18545262695413350953267/118492164154255877802564 00*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[25,15] \+ = -3460846507/13082090000+48037163/3270522500*15^(1/2)-108037591/64756 345500*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-412156 237/46254532500*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-289067 1767/58869405000*15^(1/2)*(495-66*15^(1/2))^(1/2)-4618638191/196231350 00*(495-66*15^(1/2))^(1/2)-848300537/15418177500*(495+66*15^(1/2))^(1/ 2)+429589031/129512691000*(495+66*15^(1/2))^(1/2)*15^(1/2), a[19,15] = 1908413061857/289859328166800-2931986748899/14348036744256600*(495+66 *15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-1254438533/217394496125100*15 ^(1/2)*(495-66*15^(1/2))^(1/2)-857188607021/289859328166800*15^(1/2)+5 379357542731/9565357829504400*(495+66*15^(1/2))^(1/2)*15^(1/2)+1363828 62089/144929664083400*(495-66*15^(1/2))^(1/2)-1935518588/8540498062057 5*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+34225020981 /21690153808400*(495+66*15^(1/2))^(1/2), a[17,15] = -4863804455866/351 19141675695*15^(1/2)-62267936631107/9657763960816125*(495+66*15^(1/2)) ^(1/2)*15^(1/2)-5561334169199/58531902792825+2301512129984/64385093072 1075*(495+66*15^(1/2))^(1/2)-331440117351157/9657763960816125*15^(1/2) *(495-66*15^(1/2))^(1/2)-27173454387092/214616976907025*(495-66*15^(1/ 2))^(1/2)-37986613754722/28973291882448375*(495+66*15^(1/2))^(1/2)*15^ (1/2)*(495-66*15^(1/2))^(1/2)-25832795272667/5794658376489675*(495+66* 15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[15,14] = 115988619653/29240 2973068800*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-26835383732 03/877208919206400*15^(1/2)*(495-66*15^(1/2))^(1/2)-1968290835/8860696 1536*15^(1/2)+496725643/7310074326720*(495+66*15^(1/2))^(1/2)*15^(1/2) -69268797311/7310074326720*(495-66*15^(1/2))^(1/2)-124493701507/877208 919206400*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-520 34279227/11696118922752*(495+66*15^(1/2))^(1/2)-74896827627/8860696153 6, a[21,1] = 1534965908593628117013/2925190767628355078125, a[25,2] = \+ -1029/400, a[25,6] = -2667/1600, c[4] = 101/152, a[23,6] = -381/544, a [21,11] = -44794043532626317794933177103220192762017625/16702897595445 1970498474513047407579017107824+39042282867211437045715058502688934997 5/6959540664768832104103104710308649125712826*(495+66*15^(1/2))^(1/2)* 15^(1/2)+29441442259346454399120790160135926728275/4639693776512554736 068736473539099417141884*15^(1/2)+107835852142499187656942652161788723 246225/13919081329537664208206209420617298251425652*(495+66*15^(1/2))^ (1/2), c[8] = 17727/277750, a[20,19] = -110845/348543, a[21,19] = 8177 0692250/117233567091, a[22,19] = 828786688/2261607237, a[23,21] = 2533 19312163456/659439747650125, a[22,21] = -75739731820032/65943974765012 5, a[25,19] = 3601857/2273300, a[21,20] = 21260379985/34709636178, a[2 5,21] = 30334395520464588/16485993691253125, a[22,20] = 76890112/58634 2617, a[25,23] = 753491/379050, a[25,20] = 3071471/2273300, a[23,22] = 381/544, a[25,22] = 2667/1600, a[25,24] = 1029/400, a[21,18] = -39042 2828672114370457150585026889349975/69595406647688321041031047103086491 25712826*(495+66*15^(1/2))^(1/2)*15^(1/2)+4479404353262631779493317710 3220192762017625/167028975954451970498474513047407579017107824-2944144 2259346454399120790160135926728275/46396937765125547360687364735390994 17141884*15^(1/2)-107835852142499187656942652161788723246225/139190813 29537664208206209420617298251425652*(495+66*15^(1/2))^(1/2), a[25,17] \+ = 112245633/226284800-1120087/28285600*15^(1/2)-68125601/7467398400*(4 95+66*15^(1/2))^(1/2)+993373/4480439040*(495+66*15^(1/2))^(1/2)*15^(1/ 2), a[25,18] = -6212380489353402482048992934870163/5513981142404675080 846640779279519700*(495+66*15^(1/2))^(1/2)*15^(1/2)+568745378299855361 10363501647573078867/44111849139237400646773126234236157600-4774671137 12319024289213588759633869/11027962284809350161693281558559039400*15^( 1/2)-168276384308509585366452969427482141/1102796228480935016169328155 8559039400*(495+66*15^(1/2))^(1/2), a[20,17] = 229884359555/8366741852 352+46739047805/4183370926176*15^(1/2)-20855494525/25100225557056*(495 +66*15^(1/2))^(1/2)-34057541825/75300676671168*(495+66*15^(1/2))^(1/2) *15^(1/2), a[20,18] = -52601571650923977687781073690186/13255117796518 05411892910350460322529*(495+66*15^(1/2))^(1/2)*15^(1/2)+1461351037003 208398982108006046541205/15906141355821664942714924205523870348-312574 630477851429199757289182427/101962444588600416299454642343101733*15^(1 /2)-3126986093611720912749441466631523/1325511779651805411892910350460 322529*(495+66*15^(1/2))^(1/2), a[19,18] = 199389002800788393634510722 754176/2796728200664360059162382772909427897*(495+66*15^(1/2))^(1/2)*1 5^(1/2)-1025465248156834803353840916920180809/559345640132872011832476 5545818855794+16445717956872940821116448130678578/27967282006643600591 62382772909427897*15^(1/2)+13437632256176229690482072355537756/2796728 200664360059162382772909427897*(495+66*15^(1/2))^(1/2), a[19,17] = 246 6730563/34107367168-46877457/4872481024*15^(1/2)-417081443/10232210150 4*(495+66*15^(1/2))^(1/2)+221689729/306966304512*(495+66*15^(1/2))^(1/ 2)*15^(1/2), a[18,17] = 61673043751462390569674581/3516856484463071513 85663744-4915309705585387914994781/351685648446307151385663744*15^(1/2 )-46955489957737999234808063/11605626398728135995726903552*(495+66*15^ (1/2))^(1/2)+6751918613532367335628313/34816879196184407987180710656*( 495+66*15^(1/2))^(1/2)*15^(1/2), c[6] = 25/152, a[21,5] = -15647939621 7570600333298656/73273972406867506457421875, a[21,6] = 110058432378124 73721/5193844391648046875, a[23,2] = 112211/506844, a[22,4] = 390625/3 949854672, a[22,5] = 404856925/3216589572, c[5] = 8225/76912, c[3] = 1 01/228, c[7] = 2013/3445, a[18,7] = 1899397497348522790495234191029188 9381118708193023824550/12060034665080522994492944014191159522924996678 137100092619*(495+66*15^(1/2))^(1/2)*15^(1/2)+373649996828219403405242 13896266047309233288831638123551875/4288012325361963731375268982823523 3859288877077820800329312+83870865016481720496920100044907813868655001 976949702225/134000385167561366605477155713235105810277740868190001029 1*15^(1/2)+101882217739425975585850562279744393868968529126539562125/4 020011555026840998164314671397053174308332226045700030873*(495+66*15^( 1/2))^(1/2), a[18,8] = -2754138826372250841538793956331517904511743397 61235000000000/1736570677629765561734121675040476983622751396891581850 1163621*(495+66*15^(1/2))^(1/2)*15^(1/2)-90307060671915006713048386474 17536518671082752129344726562500/1929522975144183957482357416711641092 914168218768424277907069-113106637991389701974884748601231283592242106 2622057500000000/19295229751441839574823574167116410929141682187684242 77907069*15^(1/2)-1097562530284460395919027812165520839910017377896612 500000000/578856892543255187244707225013492327874250465630527283372120 7*(495+66*15^(1/2))^(1/2), a[19,6] = -46012134893475328/76616158064651 325, a[19,7] = 5620721356384469075/3545698345096452809913, a[19,8] = 1 0436097770007465488281250/34480588269890588098919643, a[20,6] = 244678 86602752/291937806983125, a[20,7] = 11944081475573/1322478962029186830 , a[20,8] = 1095050362935718750/11254693571026117461, a[21,4] = 464264 00659946403657/610018749211776263125, b[20] = 13/56, b[22] = 19/100, b [21] = 21/100, b[23] = 17/100, b[24] = 11/100, c[21] = 2013/3445, b[18 ] = 29/100, c[24] = 39/88, c[23] = 101/228, c[22] = 25/152, c[19] = 1/ 3, c[18] = 51789075/64972747+333240/64972747*15^(1/2)+3057458/71470021 7*(495+66*15^(1/2))^(1/2)+70408/714700217*(495+66*15^(1/2))^(1/2)*15^( 1/2), b[19] = 27/100, c[20] = 1/5, a[24,3] = -274193/636804\}:" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 101 "We form a system of equations involving the follo wing stage-order conditions for the stages 14 to 25." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j]*c[j],j=2..i-1)=1/2" "6# /-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\"\"&%\"cG6#F,F-/F,;\"\"#,&F+F-F-!\"\" *&F-F-F3F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^2" "6#*$&%\"cG6#%\"iG \"\"#" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9" } {TEXT -1 20 " . . 20, 25, " }{XPPEDIT 18 0 "Sum(a[i,j]*c[j]^2,j =2..i-1)=1/3" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\"\"*$&%\"cG6#F,\"\"# F-/F,;F2,&F+F-F-!\"\"*&F-F-\"\"$F6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[ i]^3" "6#*$&%\"cG6#%\"iG\"\"$" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "i=1 4" "6#/%\"iG\"#9" }{TEXT -1 13 " . . 20, 25, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a [i,j]*c[j]^3,j=2..i-1)=1/4" "6#/-%$SumG6$*&&%\"aG6$%\"iG%\"jG\"\"\"*$& %\"cG6#F,\"\"$F-/F,;\"\"#,&F+F-F-!\"\"*&F-F-\"\"%F7" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[i]^4" "6#*$&%\"cG6#%\"iG\"\"%" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9" }{TEXT -1 17 " . . 20, 25, " } {XPPEDIT 18 0 "Sum(a[i,j]*c[j]^4,j=2..i-1)=1/5" "6#/-%$SumG6$*&&%\"aG6 $%\"iG%\"jG\"\"\"*$&%\"cG6#F,\"\"%F-/F,;\"\"#,&F+F-F-!\"\"*&F-F-\"\"&F 7" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^5" "6#*$&%\"cG6#%\"iG\"\"&" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9" }{TEXT -1 13 " . . 20, 25, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i,j]*c[j]^5,j=2..i-1)=1/6" "6#/-% $SumG6$*&&%\"aG6$%\"iG%\"jG\"\"\"*$&%\"cG6#F,\"\"&F-/F,;\"\"#,&F+F-F-! \"\"*&F-F-\"\"'F7" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[i]^6" "6#*$&%\"cG 6#%\"iG\"\"'" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "i=14" "6#/%\"iG\"#9 " }{TEXT -1 13 " . . 17, 25. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 353 "SOeqs2 := [seq(add(a[i,j]*c [j],j=2..i-1)=1/2*c[i]^2,i=[$14..20,25]),\n seq(add(a[i,j]*c [j]^2,j=2..i-1)=1/3*c[i]^3,i=[$14..20,25]),\n seq(add(a[i,j] *c[j]^3,j=2..i-1)=1/4*c[i]^4,i=[$14..20,25]),\n seq(add(a[i, j]*c[j]^4,j=2..i-1)=1/5*c[i]^5,i=[$14..20,25]),\n seq(add(a[ i,j]*c[j]^5,j=2..i-1)=1/6*c[i]^6,i=[$14..17,25])]:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "There are 37 equations \+ involving 40 unknown linking coefficients." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 77 "eqns4 := simplify(expand(subs(e22,SOeqs2))):\nnops( %);\nindets(eqns4);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#F,&F%6$FJF/&F%6$FJF(&F%6 $FJF7&F%6$FJF<&F%6$\"#=F/&F%6$FUF<&F%6$FUF(&F%6$FUF7&F%6$F'F/&F%6$F'F, &F%6$F'F<&F%6$F'F7&F%6$\"#;F,&F%6$F`oF/&F%6$F`oF(&F%6$F`oF7&F%6$F`oF<& F%6$\"#:F/&F%6$F[pF,&F%6$F[pF7&F%6$F[pF(&F%6$F[pF<&F%6$F+F(&F%6$F+F7&F %6$F+F<&F%6$FUF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "We obtain a solut ion in which coefficients are expressed in terms of 3 specific coeffic ients as parameters." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "info level[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "e23 \+ := simplify(expand(rationalize(solve(\{op(eqns4)\},indets(eqns4) minus \{a[18,11],a[19,11],a[20,11]\})))):\ne24 := `union`(simplify(subs(e23 ,e22)),e23):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "indets(map(rhs,e24));\nn ops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%&%\"aG6$\"#?\"#6&F%6$\"#> F(&F%6$\"#=F(" }}{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 3 "e24" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81300 "e24 := \{a[23,20] = \+ 0, a[23,19] = 0, c[12] = -48827895154/211479617260065*(495+66*15^(1/2) )^(1/2)*15^(1/2)-1080276228947/70493205753355*(495+66*15^(1/2))^(1/2)+ 2523614583531/6408473250305+33872310083/1281694650061*15^(1/2), a[23,1 8] = 0, a[25,5] = 0, a[23,17] = 0, a[25,4] = 0, a[25,8] = 0, a[18,9] = 2223543721376062484038978612772140819847550375/3932223085546832966488 89007567566605040336593*a[18,11]*(495+66*15^(1/2))^(1/2)*15^(1/2)-1505 4898190632470035569468555260352443683969375889427304041938301610213165 36065416353644245763144435640153184309633610775/2475732320180481031367 9102573972844303024017682286782831936288590511980849107917043976063351 3386719213386280996323579904*(495+66*15^(1/2))^(1/2)*15^(1/2)+16616441 472597949084948490344554059307420567000/393222308554683296648889007567 566605040336593*a[18,11]+812425101340191515378783451093554657674435692 73470226403692659616283771302835/2736575336117975318107720938542678599 68355240340338747926969688839365655386112*(495+66*15^(1/2))^(1/2)*15^( 1/2)*(495-66*15^(1/2))^(1/2)+55809967504215124855113644269300941373361 312409660450334322262273336365295665/273657533611797531810772093854267 859968355240340338747926969688839365655386112*(495+66*15^(1/2))^(1/2)* (495-66*15^(1/2))^(1/2)+8314183124992573937204002756033966371548884004 7730581635616236023920507650843466824834764362390115862050029428366377 15/6691168432920219003697054749722390352168653427645076441063861781219 454283542680282155692797659100519280710297197934592*(495+66*15^(1/2))^ (1/2)+9274846226872910719416421799714804790481155050794373112777184769 0142534214205/82926525336908342972961240561899351505562194042526893311 20293601192898648064*(495-66*15^(1/2))^(1/2)-2892170168183140553790252 0597674170358354993788768717140886871753331527379925152720608429406860 89176859287335207879062805/2250665745618619119425372961270258573002183 4256624348029032989627738164408279924585432784864853338110307843726938 507264*15^(1/2)-1627449597893346072079604782545865369205399250/3932223 08554683296648889007567566605040336593*a[18,11]*(495+66*15^(1/2))^(1/2 )+45216479318404965835919476599765765471888831375/39322230855468329664 8889007567566605040336593*a[18,11]*15^(1/2)+18450699021452315200498376 269046317057675775268520897653081514757460269996225/276421751123027809 9098708018729978383518739801417563110373431200397632882688*15^(1/2)*(4 95-66*15^(1/2))^(1/2)-692191377741139046539438644460353851529305421911 6510176523331651198499990423929330378774074722284792467329347679595289 5/60828803935638354579064134088385366837896849342227967646035107101995 0389412970934741426617969009138116428208836175872, a[24,15] = 0, a[24, 14] = 0, a[23,4] = 0, a[23,16] = 0, a[23,14] = 0, a[23,15] = 0, a[23,1 3] = 0, a[23,12] = 0, a[23,11] = 0, a[23,10] = 0, a[23,9] = 0, a[24,10 ] = 0, a[23,8] = 0, a[23,3] = 0, a[23,5] = 0, a[24,12] = 0, a[24,13] = 0, a[24,22] = 0, a[24,21] = 0, a[24,20] = 0, a[24,18] = 0, a[24,19] = 0, a[24,17] = 0, a[24,16] = 0, a[25,3] = -753491/379050, a[24,11] = 0 , b[8] = 0, b[5] = 0, b[4] = 0, b[16] = 31/175+1/100*15^(1/2), b[25] = 1/42, b[13] = 31/175-1/100*15^(1/2), b[1] = 1/42, c[16] = 1/2+1/66*(4 95-66*15^(1/2))^(1/2), c[14] = 1/2-1/66*(495-66*15^(1/2))^(1/2), c[25] = 1, b[12] = 0, b[17] = 31/175-1/100*15^(1/2), c[17] = 1/2+1/66*(495+ 66*15^(1/2))^(1/2), b[14] = 31/175+1/100*15^(1/2), c[15] = 1/2, c[13] \+ = 1/2-1/66*(495+66*15^(1/2))^(1/2), b[15] = 128/525, a[11,6] = 0, a[24 ,5] = 0, a[24,1] = 39/88, c[11] = 51789075/64972747+333240/64972747*15 ^(1/2)+3057458/714700217*(495+66*15^(1/2))^(1/2)+70408/714700217*(495+ 66*15^(1/2))^(1/2)*15^(1/2), a[25,13] = -55890504392518564023586021030 994240051/5930958253122427625046584152986240000*(495+66*15^(1/2))^(1/2 )*15^(1/2)-89110091060970255586136770621176187511/19769860843741425416 82194717662080000*(495+66*15^(1/2))^(1/2)+4316820120185247256314338955 5061879503/65240540784346703875512425682848640000*(495+66*15^(1/2))^(1 /2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+605158707918666886682487761532823 01603/21746846928115567958504141894282880000*(495+66*15^(1/2))^(1/2)*( 495-66*15^(1/2))^(1/2)+1332897194407130790492306856492982943/178106854 44812094970109862321280000*(495-66*15^(1/2))^(1/2)+7459264014975951670 53150707146538419/40346654783145766156779484033920000*15^(1/2)*(495-66 *15^(1/2))^(1/2)-16195323390898774863275935548635966849/59908669223458 864899460445989760000*15^(1/2)-10162782157776673605214552785788829801/ 8558381317636980699922920855680000, a[13,5] = 0, a[12,5] = 0, a[11,5] \+ = 0, a[10,5] = 0, a[9,5] = 0, a[12,3] = 0, a[11,3] = 0, a[10,3] = 0, a [9,3] = 0, a[8,3] = 0, a[7,3] = 0, a[6,3] = 0, a[13,4] = 0, a[12,4] = \+ 0, a[11,4] = 0, a[10,4] = 0, a[9,4] = 0, a[8,4] = 0, a[10,2] = 0, a[9, 2] = 0, a[8,2] = 0, a[7,2] = 0, a[6,2] = 0, a[5,2] = 0, a[4,2] = 0, a[ 13,3] = 0, a[13,2] = 0, a[12,2] = 0, a[11,2] = 0, a[13,8] = 0, a[13,7] = 0, a[13,6] = 0, a[24,8] = 0, a[24,9] = 0, a[24,4] = 0, a[19,3] = 0, a[18,3] = 0, a[17,3] = 0, a[16,3] = 0, a[15,3] = 0, a[14,3] = 0, a[17 ,2] = 0, a[16,2] = 0, a[15,2] = 0, a[14,2] = 0, a[20,3] = 0, a[19,2] = 0, a[18,2] = 0, a[15,5] = 0, a[14,5] = 0, a[20,2] = 0, a[20,5] = 0, a [19,5] = 0, a[18,5] = 0, a[17,5] = 0, a[16,5] = 0, a[14,4] = 0, a[19,4 ] = 0, a[18,4] = 0, a[17,4] = 0, a[16,4] = 0, a[15,4] = 0, a[16,8] = 0 , a[15,8] = 0, a[14,8] = 0, a[20,4] = 0, a[17,7] = 0, a[16,7] = 0, a[1 5,7] = 0, a[14,7] = 0, a[17,8] = 0, a[18,6] = 0, a[17,6] = 0, b[9] = - 13/56, a[24,2] = 0, a[16,6] = 0, a[15,6] = 0, a[14,6] = 0, a[22,2] = 0 , a[22,13] = 0, a[22,12] = 0, a[22,11] = 0, a[22,8] = 0, a[22,6] = 0, \+ a[22,3] = 0, a[22,18] = 0, a[22,17] = 0, a[22,16] = 0, a[22,15] = 0, a [22,14] = 0, a[21,2] = 0, a[21,13] = 0, a[21,12] = 0, a[21,8] = 0, a[2 4,6] = 0, a[24,7] = 0, a[21,7] = 0, a[21,3] = 0, a[21,17] = 0, a[21,16 ] = 0, a[21,15] = 0, a[21,14] = 0, c[9] = 1/5, a[22,9] = -76890112/586 342617, c[10] = 1/3, a[14,13] = 1598444751079441171273/161592790849830 21920856*(495+66*15^(1/2))^(1/2)*15^(1/2)+3102710944829420080669/37704 984531627051148664*(495+66*15^(1/2))^(1/2)-6561603091532524625551/1244 264489543692687905912*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2 ))^(1/2)-35515474297055779886885/3732793468631078063717736*(495+66*15^ (1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-39711470248879690525337/113114953 594881153445992*(495-66*15^(1/2))^(1/2)-1929879449437757437867/1615927 9084983021920856*15^(1/2)*(495-66*15^(1/2))^(1/2)+22769936085086746243 8055/113114953594881153445992*15^(1/2)+180454599750998397262245/377049 84531627051148664, a[16,11] = 5987640790061104842043034716022995891054 0540624222967777878571466744857688055291/69906497981222548799810529299 137412339756182250363816410938135503053687098320928000*(495+66*15^(1/2 ))^(1/2)*15^(1/2)-4437320340144449356981964601102323025710119720019094 3252676868247630645717834437/13981299596244509759962105859827482467951 236450072763282187627100610737419664185600*(495+66*15^(1/2))^(1/2)+130 5317780023295179218831198837410926413680734515910378425133360264879721 56959197/3844857388967240183989579111452557678686590023770009902601597 4526679527904076510400*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2) -968487390760391514438394728499291777274160071205048044579302980535087 052391884449/115345721669017205519687373343576730360597700713100297078 0479235800385837122295312000*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66* 15^(1/2))^(1/2)-756923750214600389221522573860117240432945020810456862 043431960429821151852359123/139812995962445097599621058598274824679512 36450072763282187627100610737419664185600*(495-66*15^(1/2))^(1/2)+2636 2907124838052880221729290475175397175999883388888367829617492349170306 81433653/2097194939436676463994315878974122370192685467510914492328144 06509161061294962784000*15^(1/2)*(495-66*15^(1/2))^(1/2)-6621032159576 2015124957409326988268616589156444676249824790888301449276524367/23537 5414078190399999362051512247179595138660775635745491374193612975377435 4240*15^(1/2)+14432106093050505738265298560822011759664831069245950328 191010631884195945409799/105918936335185679999712923180511230817812397 349036085471118387125838919845940800, a[17,12] = 206251002395070521505 3052801440659831794061496790498054401935194981336150529071932571468645 /352601731202086337510993798475719270897282160729607164560748995336005 930604885069735873664*(495+66*15^(1/2))^(1/2)*15^(1/2)+219500242335255 7720235216963967421439452557011458752267108404911179920138764268222984 5120625/10578051936062590125329813954271578126918464821888214936822469 86008017791814655209207620992*(495+66*15^(1/2))^(1/2)-5378897234444436 3179336234916588886175202390539232560967375301097331667949957462101192 73075/3878619043222949712620931783232911979870103768025678810168238948 696065236653735767094610304*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^ (1/2)-4407945088539854365081626684641509421824352825576746473116241593 509983795684422570864451955/116358571296688491378627953496987359396103 11304077036430504716846088195709961207301283830912*(495+66*15^(1/2))^( 1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-13589366832942464938357413120373 806395353933604717963748637460799469178688902194663998720875/352601731 2020863375109937984757192708972821607296071645607489953360059306048850 69735873664*(495-66*15^(1/2))^(1/2)-1083110157372727762182119803076536 0331327460472318599780773058565618350566813041820876945915/10578051936 0625901253298139542715781269184648218882149368224698600801779181465520 9207620992*15^(1/2)*(495-66*15^(1/2))^(1/2)+50353393311493821386944511 18606326994182012611675267644687789729295278859341198095736492525/3205 4702836553303410090345315974479172480196429964287687340817757818720964 080460885079424*15^(1/2)+186184681316279680157231291363445269507215764 81827215389915566861437225213489812831788545975/3205470283655330341009 0345315974479172480196429964287687340817757818720964080460885079424, a [14,12] = -28980415024149608470250567748006844874496969165876629344425 090037471180265975/556025912731270395525696435008504195893379880002316 479053295626653067982544432*(495+66*15^(1/2))^(1/2)*15^(1/2)-997657195 78606126127875725205455669533869458956454230864234408168976968137375/3 7068394182084693035046429000566946392891992000154431936886375110204532 1696288*(495+66*15^(1/2))^(1/2)+45774857980725657001711782735014862936 15281976114508992388556678726688329375/2780129563656351977628482175042 52097946689940001158239526647813326533991272216*(495+66*15^(1/2))^(1/2 )*(495-66*15^(1/2))^(1/2)+20084177182331026053542378772015369699101564 28731711178806635228757150383625/5560259127312703955256964350085041958 93379880002316479053295626653067982544432*(495+66*15^(1/2))^(1/2)*15^( 1/2)*(495-66*15^(1/2))^(1/2)+39206073957228074427597296141539854400232 84311589623651473153296272407874375/9190510954235874306209858429892631 337080659173592007918236291349637487314784*(495-66*15^(1/2))^(1/2)+318 5731292368166872733884152439309898933750225009103026067878223749145681 25/3063503651411958102069952809964210445693553057864002639412097116545 829104928*15^(1/2)*(495-66*15^(1/2))^(1/2)-441568756593832302293880625 8212942840937272391217708621242288549218667135125/28082116804609615935 64123409133859575219090303042002419461089023500343346184*15^(1/2)-7508 6602679703111016362746685512631347185273591884428191176525250463168342 875/112328467218438463742564936365354383008763612121680096778443560940 01373384736, a[25,12] = 1168618125430076591583129397248101071100533837 792663395374644503559384618275734363510485/104706140440824104208089263 717873109209311757882731505580006210957228071535080121028608*(495+66*1 5^(1/2))^(1/2)*15^(1/2)+1911638993195319573979763709852104653027798380 7510930912377426235711720030722182461760131/47117763198370846893640168 6730428991441902910472291775110027949307526321907860544628736*(495+66* 15^(1/2))^(1/2)-422450567104915323317140679936989137727162275027603309 99553990676479270452454394452453399/1554886185546237947490125566210415 6717582796045585628578630922327148368622959397972748288*(495+66*15^(1/ 2))^(1/2)*(495-66*15^(1/2))^(1/2)-222652452734645343694396717210205353 38092592305627692535776859090192037286750097575722049/3109772371092475 8949802511324208313435165592091171257157261844654296737245918795945496 576*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-587411227 5601023652742697812783234371407712138069788447755816170789214870469497 570158013/785296053306180781560669477884048319069838184120486291850046 58217921053651310090771456*(495-66*15^(1/2))^(1/2)-1839461788352365064 3775763346675053858197603549923151729948827893542264877001589687787601 /942355263967416937872803373460857982883805820944583550220055898615052 643815721089257472*15^(1/2)*(495-66*15^(1/2))^(1/2)+862644011307087121 1344608667737577916410295170478395381011602754761902195759989492385519 /285562201202247556931152537412381206934486612407449560672744211701531 10418658214825984*15^(1/2)+1607291908558767841044832217100789605316818 8265680104594443690210134967338994484694598237/14278110060112377846557 626870619060346724330620372478033637210585076555209329107412992, a[22, 10] = -828786688/2261607237, a[17,13] = -42638164139831532001924485887 60470591/932433116203822377006723358441251000*(495+66*15^(1/2))^(1/2)* 15^(1/2)-1781056779349424692794297435444687417/72522575704741740433856 261212097300*(495+66*15^(1/2))^(1/2)+489040967189085159685878691326554 0229/3190993331008636579089675493332281200*(495+66*15^(1/2))^(1/2)*(49 5-66*15^(1/2))^(1/2)+15404935302574800991602822823564436629/4786489996 5129548686345132399984218000*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66* 15^(1/2))^(1/2)+5799267746539103233473792702417045641/1450451514094834 80867712522424194600*(495-66*15^(1/2))^(1/2)+6094452916961920934289980 9856869816113/6527031813426756639047063509088757000*15^(1/2)*(495-66*1 5^(1/2))^(1/2)-1547691621063187838577670935413803321/11302219590349362 145536040708378800*15^(1/2)-16554919228184262713720331805086482661/263 71845710815178339584094986217200, a[16,12] = -230205365767349335929124 1828547481887451972053081524746564839578045009423200985468614339798479 742045/134310435693599165879859890528421668020150319908775971624935248 16916542463738666427206376710067109504*(495+66*15^(1/2))^(1/2)*15^(1/2 )-79231932929126561638146323105707037331151487950072772519934253135399 3768154289597302767998992852575/44770145231199721959953296842807222673 38343996959199054164508272305514154579555475735458903355703168*(495+66 *15^(1/2))^(1/2)+73742945216210880242573096398564631121096006850047056 2404024388067646212119166941988572601067769075/21105925608994154638263 6970830376906888807645570933669696326818551545667287321900998957348301 05457792*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+2687522664216 7283632160207382721093418537784921540130098539284263330183275302880755 33725009135737155/4432244377888772474035376387437915044664960556989607 06362286318958245901303375992097810431432214613632*(495+66*15^(1/2))^( 1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+12307994526617750241741297704088 129050255226551479731223561846113908118953900485191828337179017106925/ 1343104356935991658798598905284216680201503199087759716249352481691654 2463738666427206376710067109504*(495-66*15^(1/2))^(1/2)+23770980844264 8749118964629510298945007958835186412556078186620126638165105466677324 9314021201985405/13431043569359916587985989052842166802015031990877597 162493524816916542463738666427206376710067109504*15^(1/2)*(495-66*15^( 1/2))^(1/2)-1828047218937844428922860217574845796717180069053489339746 55452886877353149406481449951808917440175/4522236892040375955550838064 9300224922609535322820192466308164366722365197773287633691504074300032 *15^(1/2)-944435372872199721229862616551096719395365355537264360612686 4973867917446569402315152828775670675/12222261870379394474461724499810 87160070527981697843039629950388289793653993872638748419029035136, a[1 5,13] = -606603048332993922647492911922851/294787421833178214408244911 6871680*(495+66*15^(1/2))^(1/2)*15^(1/2)+16086839770332652392981039451 4891/57801455261407493021224492487680*(495+66*15^(1/2))^(1/2)-21222358 079398713889334503007015519/162133082008248017924534701427942400*(495+ 66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+244309798160992540855648174 85473/28611720354396709045506123781401600*(495+66*15^(1/2))^(1/2)*15^( 1/2)*(495-66*15^(1/2))^(1/2)-8662877116414312827094002696547313/294787 4218331782144082449116871680*(495-66*15^(1/2))^(1/2)-37839699184736525 3345251632120523/2601065486763337185955102161945600*15^(1/2)*(495-66*1 5^(1/2))^(1/2)-23786446617715796291634345596229/1786590435352595238837 8479496192*15^(1/2)+313704936756581279384432239924553/5254677751037044 820111317498880, a[18,12] = 326054422683533972382940944248612996897635 913850385781247371049011432957465096500/301784038385219524590571456153 54945999821385189319997810410043944835273879836333*a[18,11]*(495+66*15 ^(1/2))^(1/2)*15^(1/2)+15823682392989671569242458171015925195942220225 21414699230759737356169996754285750/3017840383852195245905714561535494 5999821385189319997810410043944835273879836333*a[18,11]*(495+66*15^(1/ 2))^(1/2)-747372671300498667539026196501318989969427341014191728143465 6926132708891148727738785582397414940304991763213055205/57302690508185 6998616317653266565326667317742689846699733586751025523045161663367264 082359969724796060595445746944*(495+66*15^(1/2))^(1/2)*15^(1/2)-139394 9871469822135612513882653260202003271415173293742196029841113970969012 66995065502954333755186242660563863738115/2292107620327427994465270613 0662613066692709707593867989343470041020921806466534690563294398788991 84242381782987776*(495+66*15^(1/2))^(1/2)+3770871490095090661864522663 446309313246689250056107716814369899196703391447184250/274349125804745 0223550649601395904181801944108119999800946367631348661261803303*a[18, 11]+857851650415272772312447769420893896501186398237361908471368108566 604429824662000/274349125804745022355064960139590418180194410811999980 0946367631348661261803303*a[18,11]*15^(1/2)+89577241738632593667996277 2727699291279998689821498125405022409225920292297248657757196363569571 46074363575238468395/2521318382360170793911797674372887437336198067835 3254788277817045123013987113188159619623838667891026666199612865536*(4 95+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+386039128904569930738269 1637401966572811767002023644789966087965401873789100315881459134576783 834026970961692240515/472747196692532023858462063944916394500537137719 1235272802090695960565122583722779928679469750229567499912427412288*(4 95+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+725744490237562 1507123662587679111860425352473870483067678171139642953367463106905004 0380942109437611103062332743035/76403587344247599815509020435542043555 6423656919795599644782334700697393548884489685443146626299728080793927 662592*(495-66*15^(1/2))^(1/2)+877566366322837042528601836596813475591 0534654207028086604938017624113411654851342763464152165397485138273289 477915/382017936721237999077545102177710217778211828459897799822391167 350348696774442244842721573313149864040396963831296*15^(1/2)*(495-66*1 5^(1/2))^(1/2)-3563439262122015078529897296713688429499115749870926921 9530377936467609724830866187235794433055420807684346524333485/95504484 1803094997693862755444275544445529571149744499555977918375871741936105 61210680393328287466010099240957824*15^(1/2)-1225684940449428584903341 1277575338678574479564644080452228333230964497405269076030571384577348 85820297927912430778355/7640358734424759981550902043554204355564236569 19795599644782334700697393548884489685443146626299728080793927662592, \+ a[17,11] = -8608942482554632631447666421142576735289428749541431498948 6346860821/24785617871568523890681535843832843134880948971288323747224 68389588000*(495+66*15^(1/2))^(1/2)*15^(1/2)+7036530330296001919254839 255948123706229511388021308973656944198447/660949809908493970418174289 16887581693015863923435529992599157055680*(495+66*15^(1/2))^(1/2)+3415 5013188321120860918992576383167411849157891882541686237264128309/12830 202192341353543411618554454648210997197114549249939739836369632000*(49 5+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-7576475876322330 809383185602169500822747595273068447080370266969753/855346812822756902 894107903630309880733146474303283329315989091308800*(495+66*15^(1/2))^ (1/2)*(495-66*15^(1/2))^(1/2)+1853988204647744802168040662958692701268 72623165675323129497604093/8931754187952621221867220123903727255812954 58424804459359448068320*(495-66*15^(1/2))^(1/2)-1385194297222359505923 79845748441288599837256004184505631117900226293/3304749049542469852090 871445844379084650793196171776499629957852784000*15^(1/2)*(495-66*15^( 1/2))^(1/2)+1035392710447999643103121176108421273457784120335352459294 571705397/160230256947513689798345282222757773801250579208328557557816 1383168*15^(1/2)-12791345310106667261521854084128062610260512651582528 1677549047401409/40057564236878422449586320555689443450312644802082139 389454034579200, a[21,9] = -21260379985/34709636178, a[19,12] = 326054 4226835339723829409442486129968976359138503857812473710490114329574650 96500/3017840383852195245905714561535494599982138518931999781041004394 4835273879836333*a[19,11]*(495+66*15^(1/2))^(1/2)*15^(1/2)+15823682392 9896715692424581710159251959422202252141469923075973735616999675428575 0/30178403838521952459057145615354945999821385189319997810410043944835 273879836333*a[19,11]*(495+66*15^(1/2))^(1/2)+416490435096412044152702 7531690561222672537470457972905441622190337716087392780194301289413900 4545755786053811720580699278915/11407192041181175193386273417755049185 5606443406873004142037489265993239284355475782668196805836983643884347 411780662826013184*(495+66*15^(1/2))^(1/2)*15^(1/2)+291519410868577582 1005962821339113645172532705968221806716116472166827992783166414658818 088013617493119630621406854073224305095/532335628588454842358026092828 5689619928300692320740193295082832413017833269922203191182517605725903 38126954588309759854728192*(495+66*15^(1/2))^(1/2)+3770871490095090661 864522663446309313246689250056107716814369899196703391447184250/274349 1258047450223550649601395904181801944108119999800946367631348661261803 303*a[19,11]+857851650415272772312447769420893896501186398237361908471 368108566604429824662000/274349125804745022355064960139590418180194410 8119999800946367631348661261803303*a[19,11]*15^(1/2)-36577317402799027 19742660499187265840940765529237617355699914688695/9579095807402172978 822471359286652411393373624128459627529810291712*(495+66*15^(1/2))^(1/ 2)*(495-66*15^(1/2))^(1/2)-1585095619059925454994978508230421047628467 668766233554250058535345/335268353259076054258786497575032834398768076 84449608696354336020992*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1 /2))^(1/2)-19072177869001468518107128000272158396389633926540023579300 470614265/203192941369137002581082725803050202665920046572421870886995 9758848*(495-66*15^(1/2))^(1/2)-40407656143706228150516695033569486107 2313072942339601783184085915/25399117671142125322635340725381275333240 0058215527338608744969856*15^(1/2)*(495-66*15^(1/2))^(1/2)+11489663893 8362372052338712888321237097149585906140815897785942983454090377563412 8773751879515886985272191989246473141008549245/72591222080243842139730 8308402594039081131912589191844540238568056320613627716664071524888764 41716864290039262042239980190208*15^(1/2)+6153256768298409382667369301 9564978114937759871897976009821679776052851205961192329669037772690861 71940027372193833456031134635/4839414805349589475982055389350626927207 5460839279456302682571203754707575181110938101659250961144576193359508 028159986793472, a[25,11] = 128920787551671343348376648677848873644596 31885161414654166871749310538128360356282255762947040291/2849740893022 2627879101099930305297509498618121353812422052614331119206690907876161 3200232238080000*(495+66*15^(1/2))^(1/2)*15^(1/2)-12480141995674691253 3228504120517562720675622369823626493412499565231034612699574402577387 154565829/664939541705194650512358998373790275221634422831588956514561 001059448156121183777097467208555520000*(495+66*15^(1/2))^(1/2)-269375 2889365901399339990992030284330802789275903520277745199948003/82581034 27088723934441915787121089288026551962529029816316338321920000*(495+66 *15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+86206037733181423374735221781 16724694615075400164606062275312778891/9909724112506468721330298944545 307145631862355034835779579605986304000*(495+66*15^(1/2))^(1/2)*15^(1/ 2)*(495-66*15^(1/2))^(1/2)+9415833269344714094602052035425875010781868 302770380527360876893541/834151861322093326711304624961726190709752723 48778078952690286080000*(495-66*15^(1/2))^(1/2)-2351260393591354352664 66723342816311162099922849416444217438064061/3128069479957849975167392 3436064732151615727130791779607258857280000*15^(1/2)*(495-66*15^(1/2)) ^(1/2)-511975650628347237449832088490580845648606388072013087013572919 932259111943700950216968199185543/816879043863875492029925059427260780 370558259006866039944178133979666039460913731077969543680000*15^(1/2)- 1290393536512988930091830278930902869460971301936267208599441737646119 5912450807078157049600542041/20149683081975595470071484799205765915807 103722169362318623060638165095640035872033256582077440000, a[15,12] = \+ -118673264953965898239568178186990539375320133228945328781256378008127 1262993935492821075/29859252522666041600461145767521730374799587990546 39506083291835447600675497430644543488*(495+66*15^(1/2))^(1/2)*15^(1/2 )-62787138053343564726123359273066353779726848092727005318663855357084 118475306201635375/124413552177775173335254774031340543228331616627276 646086803826476983361479059610189312*(495+66*15^(1/2))^(1/2)+206957924 4553928310941747084237830205961522603833836536321316258542043853599099 29293975/4478887878399906240069171865128259556219938198581959259124937 753171401013246145966815232*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^ (1/2)+5891979785833502419846398661311140104554853460303611448422530389 02651606518412193051375/2687332727039943744041503119076955733731962919 1491755554749626519028406079476875800891392*(495+66*15^(1/2))^(1/2)*15 ^(1/2)*(495-66*15^(1/2))^(1/2)+109931052493305895825340776604828162916 6843436849055712163350435869590310234999724676375/74648131306665104001 1528644188043259369989699763659876520822958861900168874357661135872*(4 95-66*15^(1/2))^(1/2)+491265714510885901435656358464046529229445991366 0535043858759590531421311858910677650975/89577757567998124801383437302 56519112439876397163918518249875506342802026492291933630464*15^(1/2)*( 495-66*15^(1/2))^(1/2)-26100793447835537272907704911698240760371277437 03573983625462105790320139194698784616725/2714477502060549236405558706 13833912498178072641330864189390166858872788681584604049408*15^(1/2)-1 2789425881975715683435041476352781333133614948791446840597433217053145 97196773722377875/6786193755151373091013896765345847812454451816033271 6047347541714718197170396151012352, a[18,13] = -7338698874424210336667 2422267080822593823256/4904585119217563989229997497115053593255737*a[1 8,11]*(495+66*15^(1/2))^(1/2)*15^(1/2)-1431786853008435229789758421753 12251707656968/4904585119217563989229997497115053593255737*a[18,11]*(4 95+66*15^(1/2))^(1/2)+885259944389574590565947355373573867573766335103 4522565338348494150549272886437/52265804274245950212290592845670859775 7956213019082008454535119984589725305600*(495+66*15^(1/2))^(1/2)*15^(1 /2)+112192714154624464316444273729347978808852908751618372069524439142 9213158050147/29036557930136639006828107136483810986553122945504556025 251951110254984739200*(495+66*15^(1/2))^(1/2)-433948807018695022378898 091347998031973552360/445871374474323999020908863374095781205067*a[18, 11]-163173591027274144770728064063666100479682008/44587137447432399902 0908863374095781205067*a[18,11]*15^(1/2)-12752290425153509785926176716 830303309190555030846981445451187176821286178053/483942632168943983447 1351189413968497758853824250759337541991851709164123200*(495+66*15^(1/ 2))^(1/2)*(495-66*15^(1/2))^(1/2)-493849939844753039813418249653592738 768576380182432508714776296551478809980913/522658042742459502122905928 456708597757956213019082008454535119984589725305600*(495+66*15^(1/2))^ (1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-3099636932261322582444751858964 01700915613401865539887876962819068971800358007/3959530626836814410022 014609520519679984516765296075821625266060489316100800*(495-66*15^(1/2 ))^(1/2)-1283244070555184860584462163783483530758183213054745899492653 71059421418920481/5279374169115752546696019479360692906646022353728101 095500354747319088134400*15^(1/2)*(495-66*15^(1/2))^(1/2)+675822862913 82262022138859144351988668143932124683523280481453549782328509703/1599 8103542775007717260665088971796686806128344630609380304105294906327680 0*15^(1/2)+14622882596121954640545255729133360930118270109705610723844 8077021891596518397/11998577657081255787945498816728847515104596258472 9570352280789711797457600, a[21,10] = -81770692250/117233567091, a[10, 7] = 5620721356384469075/3545698345096452809913, a[10,1] = -1768978958 9/433566079650, c[2] = 39/88, a[10,8] = 10436097770007465488281250/344 80588269890588098919643, a[22,7] = 75739731820032/659439747650125, a[8 ,1] = 5673493348726844397056253/158826937830690185546875000, a[8,5] = \+ 83874430427399451044535741888/1824862440224861393218994140625, a[10,6] = -46012134893475328/76616158064651325, a[8,6] = -9868943649337405447 26981696/54826948196505455902099609375, a[9,1] = 2526695731/1338166912 50, a[9,6] = 24467886602752/291937806983125, a[7,4] = 4642640065994640 3657/610018749211776263125, a[7,1] = 1534965908593628117013/2925190767 628355078125, a[7,6] = 11005843237812473721/5193844391648046875, a[8,7 ] = 5672317641427470160608354913689/4038186547571171299662769657812500 0, a[23,7] = -253319312163456/659439747650125, a[9,8] = 10950503629357 18750/11254693571026117461, a[10,9] = 110139186925/164279737386, a[7,5 ] = -156479396217570600333298656/73273972406867506457421875, a[5,1] = \+ 8719384129925/100440274363616, a[5,4] = -2900926450625/200880548727232 , a[16,9] = 50063971577508796989932533520141190241209119641155965/5704 99532233022530103277424880151739690209749047396608*(495+66*15^(1/2))^( 1/2)*15^(1/2)-335814103222680807523386900403987635039205386347311925/5 70499532233022530103277424880151739690209749047396608*(495+66*15^(1/2) )^(1/2)+25861692976938895204697334600993725104726707177012055/62754948 54563247831136051673681669136592307239521362688*(495+66*15^(1/2))^(1/2 )*15^(1/2)*(495-66*15^(1/2))^(1/2)-71974246200573822850376257540909842 207760493224762025/188264845636897434934081550210450074097769217185640 88064*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+3311568025001119 1448970159142023181041838885854451525/57049953223302253010327742488015 1739690209749047396608*(495-66*15^(1/2))^(1/2)+45063383612987448944940 602137484447778392516290975315/570499532233022530103277424880151739690 209749047396608*15^(1/2)*(495-66*15^(1/2))^(1/2)+683113728364719630288 4273272119561096079229960021975/74090848341950977935490574659759966193 53373364251904*15^(1/2)-2340489906258242610205580473767823534203739139 1480725/2469694944731699264516352488658665539784457788083968, a[5,3] = 6944404306875/200880548727232, a[9,7] = 11944081475573/13224789620291 86830, a[6,1] = 1167025/30304848, a[6,4] = 390625/3949854672, a[6,5] = 404856925/3216589572, a[3,1] = 28078/126711, a[3,2] = 112211/506844, \+ a[4,1] = 101/608, a[4,3] = 303/608, a[20,9] = 222354372137606248403897 8612772140819847550375/393222308554683296648889007567566605040336593*a [20,11]*(495+66*15^(1/2))^(1/2)*15^(1/2)-11367189637788770782538847112 5772895277822050699161966937789676107738635204003170257379674511806375 /253708121851439468792963280183795694527584074132703451256672973565869 8908672958290564117519179278848*(495+66*15^(1/2))^(1/2)*15^(1/2)+16616 441472597949084948490344554059307420567000/393222308554683296648889007 567566605040336593*a[20,11]+22001837263360670222750021550875/188019104 8504642391664616634131968*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^ (1/2))^(1/2)-55067278908856132674259135641875/188019104850464239166461 6634131968*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-16274495978 93346072079604782545865369205399250/3932223085546832966488890075675666 05040336593*a[20,11]*(495+66*15^(1/2))^(1/2)+1377859307451573540388659 2071036808175262294400447136816461323434720812660788442789884295220104 4125/12685406092571973439648164009189784726379203706635172562833648678 29349454336479145282058759589639424*(495+66*15^(1/2))^(1/2)-9979528979 41762377636363793151875/2905749802234447332572589343658496*(495-66*15^ (1/2))^(1/2)-221797517825917274822664192197208749367740253415150129125 88674266942392703786089595207882272158125/3252668228864608574268760002 3563550580459496683679929648291406867419216777858439622616891271529216 *15^(1/2)+45216479318404965835919476599765765471888831375/393222308554 683296648889007567566605040336593*a[20,11]*15^(1/2)+311047226319446480 8806111068125/15134113553304413190482236164888*15^(1/2)*(495-66*15^(1/ 2))^(1/2)+402044734921748905574474633611967505214814540529160682390734 811128081843642407134727690163951422625/422846869752399114654938800306 326157545973456887839085427788289276449818112159715094019586529879808, a[12,9] = -2812112309092422771819313341069996795037876904500378145129 47186342510479151038756222724747330456172743380029/1777219414844154523 9165648144261871500391656432256173921137707234507471866579929942450646 636509884128799622+676374087730775429952449408713302591862747979275682 1750588456196513199369023804229800976446366900597141/38492948123113591 5944675073516609735767633884172756636801769704017922284309723412225485 09067597756397660*(495+66*15^(1/2))^(1/2)*15^(1/2)+3647611941783523639 1068396812936741450147529014462574714072877112490701893502335733514514 68961784464641/6415491353852265265744584558610162262793898069545943946 696161733632038071828723537091418177932959399610*(495+66*15^(1/2))^(1/ 2)-5945858035882701232660115801569296816414941113900029074561383838890 5948341143108254364095422742138234247/12830982707704530531489169117220 3245255877961390918878933923234672640761436574470741828363558659187992 20*15^(1/2), a[11,7] = 18993974973485227904952341910291889381118708193 023824550/12060034665080522994492944014191159522924996678137100092619* (495+66*15^(1/2))^(1/2)*15^(1/2)+3736499968282194034052421389626604730 9233288831638123551875/42880123253619637313752689828235233859288877077 820800329312+83870865016481720496920100044907813868655001976949702225/ 1340003851675613666054771557132351058102777408681900010291*15^(1/2)+10 1882217739425975585850562279744393868968529126539562125/40200115550268 40998164314671397053174308332226045700030873*(495+66*15^(1/2))^(1/2), \+ a[12,7] = 120395605263827674443959816224372224515900903963473583877146 503450993695151583738412984486608167592412726499223108027/295063494385 7949765325588035995009291446938757618744973811305715669070205151124895 94526125158556423774229187413128270-4574290713578312637346253714366174 3332646830734536211959233436539874804951849529029402844521460875500261 5341437/12781611192800302210637158483842362102867397693821723949799894 8047176530437562265364750324954973542895485894482620*(495+66*15^(1/2)) ^(1/2)*15^(1/2)-194091304446298395030544188331511201788876606372749180 626268993467082976330239046596838838518978497037496895389/127816111928 0030221063715848384236210286739769382172394979989480471765304375622653 6475032495497354289548589448262*(495+66*15^(1/2))^(1/2)+12780472568351 6766786140751534474181571563952278241986019318882448258044336581201019 10199505548093911257104546897/1278161119280030221063715848384236210286 7397693821723949799894804717653043756226536475032495497354289548589448 2620*15^(1/2), a[12,6] = 251085521083237108777248115694608871632336099 720342563852044971041903837740861882751285544579556435153690263824896/ 1090595815805663574044725317279254749804567879870625150516917051380618 0033868049458919144773539359368893655280625-17224203165087627219883985 3212806695960945558133287845511282815171786743955334377646964770699567 651065020416/690469019186871525194507956492089110354268996435976670159 491643799061730539287715031284885947411166121788875*(495+66*15^(1/2))^ (1/2)*15^(1/2)-2483265050966602026265671372002316578111995428215292500 398470208821705013352242278156416063385897318005911552/299203241647644 3275842867811465719478201832317889232237357797123129267499003580098468 901172438781719861085125*(495+66*15^(1/2))^(1/2)+989414013347979104246 5990020353916090461157452549514124553102629276849630596703111982637935 9226270888023523328/14960162082382216379214339057328597391009161589446 161186788985615646337495017900492344505862193908599305425625*15^(1/2), a[12,1] = 45360785864184293465355049313006780143265609187196347220787 94143929005864867603991192705032200412298285836937/1172606656536926424 1035137088606164503095345115697078138218524400312142729056992136613873 92662314110822638750-5364435895516573669420621604632495515121639476730 6749400786848404923755537638953128683448772183267658/13367151790715393 0451935491132384488709863378084391529453148255307184463926871996359151 8065572670919625*(495+66*15^(1/2))^(1/2)*15^(1/2)-35279136548504838526 7153663396939334514393712836102034560739827064611960122068707511454514 9449903002764/25397588402359246785867743315153052854874041836034390596 098168508365048146105679308238843245880747472875*(495+66*15^(1/2))^(1/ 2)+1361420982903372918496309760282958026582387432115444780575878975425 36639674582303315009111953003119707396/1269879420117962339293387165757 6526427437020918017195298049084254182524073052839654119421622940373736 4375*15^(1/2), a[13,1] = 49032552280925998159/1175735428193602866744+1 095947848209481903/1763603142290404300116*15^(1/2)-7475873553226772611 /6466544855064815767092*(495+66*15^(1/2))^(1/2)+5352672249559690835/11 6397807391166683807656*(495+66*15^(1/2))^(1/2)*15^(1/2), a[13,10] = 10 36874421023669248333689/19655651658225505790722624+5873923823482121402 03319/39311303316451011581445248*15^(1/2)-259903711964372646898395/432 424336480961127395897728*(495+66*15^(1/2))^(1/2)*15^(1/2)-185211952023 363244825707/108106084120240281848974432*(495+66*15^(1/2))^(1/2), a[13 ,9] = -162384252323883015198625/10650929594044147603352784*15^(1/2)+12 2383789982155868115425/468640902137942494547522496*(495+66*15^(1/2))^( 1/2)*15^(1/2)+2265050157236155663680125/14201239458725530137803712-120 963267116719524969875/26035605674330138585973472*(495+66*15^(1/2))^(1/ 2), a[11,9] = 28743974673994042985226536878679230542480801150/17539834 08227847451774748544348597377943356553311*(495+66*15^(1/2))^(1/2)*15^( 1/2)+291991715615499449706434755611025178229310468246875/5612746906329 1118456791953419155116094187409705952+10710917493129987438150871655684 69189531652126075/1753983408227847451774748544348597377943356553311*15 ^(1/2)+353259784704351651896312581655773178339865859125/17539834082278 47451774748544348597377943356553311*(495+66*15^(1/2))^(1/2), a[13,11] \+ = 145148388890007074338014877677070795676982463/1292134149111656271966 122106166449116832367968+289384776308176063330366809130393474082987849 /15505609789339875263593465273997389401988415616*15^(1/2)-274115022589 237178977231727148612453501574585/511685123048215883698584354041913850 265617715328*(495+66*15^(1/2))^(1/2)*15^(1/2)-248572369601572046985131 48202889704149473711/5330053365085582121860253687936602606933517868*(4 95+66*15^(1/2))^(1/2), a[13,12] = 644332945230386310371160782678707752 3918806208419166908252875/48186908177657952343922139838789383990650083 655460342201310944-171491601151866911759544690813927486314725154315627 741296625/903504528331086606448540121977300949824689068539881416274580 2*15^(1/2)-70902020152377734741381699074375816833777565836946503885561 25/2385251954794068641024145922020074507537179140945286938964891728*(4 95+66*15^(1/2))^(1/2)+119929995185051271338440264753881499366865434058 651556229025/144560724532973857031766419516368151971950250966381026603 932832*(495+66*15^(1/2))^(1/2)*15^(1/2), a[11,1] = 3625022219827369334 1131051026706723455692594208/49994304833397236139051309728442419825872 17763497*(495+66*15^(1/2))^(1/2)*15^(1/2)+4861393204748134765917698036 236910531265683193125/222196910370654382840228043237521865892765233933 2+148442814194682401232706118820643700603232458928/5554922759266359571 00570108093804664731913084833*15^(1/2)+1430688684824110222378180495435 38495099770530576/1666476827779907871301710324281413994195739254499*(4 95+66*15^(1/2))^(1/2), a[12,8] = -340659803958984878209931438027146134 1386607746012495410255775467387649282984143037843117373461807347822820 94165416156250/3082807063362932068096196297896660310632239903888789451 2718700999171421270070039408336730423089458130745933155208375541+41194 8325656738822039997775092119236421293381998104532443768303827226809809 72686064846482629276732032063112500000/3514251750809858381605960008089 8513623932604948403377121984771381703112376537555040681155937541418021 2098687404767*(495+66*15^(1/2))^(1/2)*15^(1/2)+26719277739082859443666 3466788077170914794954921965889671712692505252570871996792397851976779 2683025169075208750000/66770783265387309250513240153707175885471949401 96641653177106562523591351542135457729419628132869424029875060690573*( 495+66*15^(1/2))^(1/2)-20868030576527122590278134042922470681076456345 098996480647230076556364761794880724541437909825056194904756420000000/ 6677078326538730925051324015370717588547194940196641653177106562523591 351542135457729419628132869424029875060690573*15^(1/2), a[11,10] = -37 7685957124001752132518096635999239114085793540/40806683117092499328338 457010324585780808416836571*(495+66*15^(1/2))^(1/2)*15^(1/2)-181781480 9245296229960335698010451499156358365559525/65290692987347998925341531 2165193372492934669385136-14248117750070402721956352075060738058061271 231816/40806683117092499328338457010324585780808416836571*15^(1/2)-484 4266659916699015885626279835272453019208362650/40806683117092499328338 457010324585780808416836571*(495+66*15^(1/2))^(1/2), a[11,8] = -275413 882637225084153879395633151790451174339761235000000000/173657067762976 55617341216750404769836227513968915818501163621*(495+66*15^(1/2))^(1/2 )*15^(1/2)-90307060671915006713048386474175365186710827521293447265625 00/1929522975144183957482357416711641092914168218768424277907069-11310 66379913897019748847486012312835922421062622057500000000/1929522975144 183957482357416711641092914168218768424277907069*15^(1/2)-109756253028 4460395919027812165520839910017377896612500000000/57885689254325518724 47072250134923278742504656305272833721207*(495+66*15^(1/2))^(1/2), a[1 7,9] = 612177400769958345783425955471231078477005/88101043752736150868 0786914250849809507584*(495+66*15^(1/2))^(1/2)*15^(1/2)+71682645856539 36119575060989007511226465125/8810104375273615086807869142508498095075 84*(495+66*15^(1/2))^(1/2)-140764211411413887441926681309100606007705/ 2357298197708345658362105527319841382195968*(495+66*15^(1/2))^(1/2)*15 ^(1/2)*(495-66*15^(1/2))^(1/2)-389980312509168424209729744263362231434 70825/87220033315208789359397904510834131141250816*(495+66*15^(1/2))^( 1/2)*(495-66*15^(1/2))^(1/2)-96804269325517284031085832400263184854696 25/881010437527361508680786914250849809507584*(495-66*15^(1/2))^(1/2)- 5184692133686600909167941296222900578262135/26430313125820845260423607 42752549428522752*15^(1/2)*(495-66*15^(1/2))^(1/2)+6559804249926315934 853561094124426777066675/240275573871098593276578249341140857138432*15 ^(1/2)+15296729363916364958642717710438043125084525/800918579570328644 25526083113713619046144, a[19,10] = 6898329747455131657467437204950065 29038916799/838355228023916751926128515866680316613341143*a[19,11]*(49 5+66*15^(1/2))^(1/2)*15^(1/2)+3382290577401928539609288423027542445454 0351702235847327817661677584118594602086584153901216979/84539605081903 8835075988682333859259901497662865543569935239866266666913726114838605 92767129600*(495+66*15^(1/2))^(1/2)*15^(1/2)-9823219876217135627910816 3507503711334331028565/838355228023916751926128515866680316613341143*a [19,11]-64756860023418543185545753880169/34495354169759231054625553558 01600*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+3283163 65056802986366711645757659/3449535416975923105462555355801600*(495+66* 15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-155785203917630454013486516841 973503639408133193975179119970540870036642710015242676841866793849/845 3960508190388350759886823338592599014976628655435699352398662666669137 2611483860592767129600*(495+66*15^(1/2))^(1/2)+10730457185122429885489 9234618737/62718825763198601917501006469120*(495-66*15^(1/2))^(1/2)+53 1095388179856377275020936589978856194899762584821485802927529740030332 984026935136420636513787/845396050819038835075988682333859259901497662 86554356993523986626666691372611483860592767129600*15^(1/2)-4827425744 40905150427531405059365444676866008/7621411163853788653873895598788002 8783031013*a[19,11]*(495+66*15^(1/2))^(1/2)+64352767296599509369014452 82737804392845587547/838355228023916751926128515866680316613341143*a[1 9,11]*15^(1/2)-88450806209080005923093304671847/3135941288159930095875 05032345600*15^(1/2)*(495-66*15^(1/2))^(1/2)-5437451178200795302139705 2651814533794371279042399511182814346023908225007910544345097190678661 7/16907921016380776701519773646677185198029953257310871398704797325333 338274522296772118553425920, a[18,10] = 689832974745513165746743720495 006529038916799/838355228023916751926128515866680316613341143*a[18,11] *(495+66*15^(1/2))^(1/2)*15^(1/2)-143582146669778144976760630208641011 3437559661191978447808467044840426658030842367651177080816494692505062 675190079858060607/236607649625207748594742908217425176358864817631457 3402446161467683072622604109027519780703109507437064566955840167155763 200*(495+66*15^(1/2))^(1/2)*15^(1/2)-982321987621713562791081635075037 11334331028565/838355228023916751926128515866680316613341143*a[18,11]+ 7049412024716373098016264422505342449671847954443270106576357606890162 126894541/129707694938672307741459107756198082189744102472615418531915 7790619570612953446400*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/ 2))^(1/2)-401784889047197232266023179607929131659861030090375731766154 979513551911904282211/129707694938672307741459107756198082189744102472 6154185319157790619570612953446400*(495+66*15^(1/2))^(1/2)*(495-66*15^ (1/2))^(1/2)+152539088916093205790125426268222236664032978155905401378 64161258242223330472799669945013654436152052101928538920633733733689/2 3660764962520774859474290821742517635886481763145734024461614676830726 22604109027519780703109507437064566955840167155763200*(495+66*15^(1/2) )^(1/2)-15334829592655513300908052053981641810712219343105566232693254 3844909164413794689/23583217261576783225719837773854196761771654995020 985187621050738537647508244480*(495-66*15^(1/2))^(1/2)-105594412045474 8941195521320071768900596153436261735734515627740542725963650842162646 486023474274373532835343635847781568073/338010928036011069421061297453 4645376555211680449390574923087810975818032291584325028258147299296338 66366707977166736537600*15^(1/2)-4827425744409051504275314050593654446 76866008/76214111638537886538738955987880028783031013*a[18,11]*(495+66 *15^(1/2))^(1/2)+6435276729659950936901445282737804392845587547/838355 228023916751926128515866680316613341143*a[18,11]*15^(1/2)-431818194985 60779330182330113517149057235628048677763563132887145174774632744749/1 1791608630788391612859918886927098380885827497510492593810525369268823 7541222400*15^(1/2)*(495-66*15^(1/2))^(1/2)+83331575640744124273516607 6990775926459860824361067465259036839374085720361121856858366337790088 3420143883365436975477623263/67602185607202213884212259490692907531104 2336089878114984617562195163606458316865005651629459859267732733415954 33347307520, a[17,10] = -523474846258041930813933069750681477537147/87 0992096833131738010148887671193233701120*(495+66*15^(1/2))^(1/2)*15^(1 /2)+453936754388698231805806756431700677913713/43549604841656586900507 44438355966168505600*(495+66*15^(1/2))^(1/2)+1275599892128102801175269 64548619108824367/3236794954447449026389066812291596476592000*(495+66* 15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+118587818616933908064 8273599898636788893537/23952282662911122795279094410957813926780800*(4 95+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+879825782340809362526927 5080577035600962103/4354960484165658690050744438355966168505600*(495-6 6*15^(1/2))^(1/2)+3952411632936359566769644071717315106143261/43549604 84165658690050744438355966168505600*15^(1/2)*(495-66*15^(1/2))^(1/2)-5 6323059143888272252165736895642219352623/44989261200058457541846533454 09055959200*15^(1/2)-2456212915091625339793591322898399091106073/19795 2749280257213184124747197998462204800, a[25,7] = -30334395520464588/16 485993691253125, a[12,11] = 1/5499, a[14,9] = -26179709957154870405433 6684175/6747960349884233978989776419904*(495+66*15^(1/2))^(1/2)*15^(1/ 2)+5934505114557474795791690125/210873760933882311843430513122*(495+66 *15^(1/2))^(1/2)+12519194985069398569031057125/67479603498842339789897 76419904*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+1109 5046594182150706691994375/20243881049652701936969329259712*(495+66*15^ (1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+8033012316259971085532118125/1533 62735224641681340676736816*(495-66*15^(1/2))^(1/2)+1425332275365304763 708338125/34080607827698151409039274848*15^(1/2)*(495-66*15^(1/2))^(1/ 2)-14777787443889351383199282875/18589422451471718950385059008*15^(1/2 )-46784760598689413532687625/1161838903216982434399066188, b[2] = -11/ 100, b[11] = -29/100, b[10] = -27/100, b[7] = -21/100, b[6] = -19/100, b[3] = -17/100, a[20,15] = 13/1036, a[20,14] = -50/501, a[19,14] = 21 /32, a[12,10] = -11/342, a[23,1] = 28078/126711, a[15,11] = -109730074 1383102888839027715904625360709712131221694471788967065599/22388395214 718747101577535278410507682903636619968393176006827622400*(495+66*15^( 1/2))^(1/2)*15^(1/2)+2777947637645821276882712491759014261308302439347 89145405582248649/1492559680981249806771835685227367178860242441331226 211733788508160*(495+66*15^(1/2))^(1/2)+970165972365189633475199730235 4311332350168672670098719512199631/62085465721488962550593165057777038 11225378222344176258892649676800*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495 -66*15^(1/2))^(1/2)-61268438633677359005766723935203033381189423901790 4866736513774537/10554529172653123633600838059822096479083142977985099 6401175044505600*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+41683 417798249786705611926554570672840097491555226126486675979647/639668434 706249917187929579383157362368675331999096947885909360640*(495-66*15^( 1/2))^(1/2)-7009406349232470923845843111584742313487794670152566793436 229067/355371352614583287326627544101754090204819628888387193269949644 800*15^(1/2)*(495-66*15^(1/2))^(1/2)+127141112469296965830509721891596 5186975634520555018469385189147317/20353086558835224637797759344009552 43900330601815308470546075238400*15^(1/2)-3050873466513682484425378607 71652512014043931240894669510340799613/1356872437255681642519850622933 97016260022040121020564703071682560, a[24,23] = 274193/636804, a[22,1] = 1167025/30304848, a[20,16] = 4573876925/1493463420644832*(495+66*15 ^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+7986761337425/7582198904812224*1 5^(1/2)*(495-66*15^(1/2))^(1/2)+37543791335/995642280429888*15^(1/2)+7 8669144834545/995642280429888-6267380675/14360225198508*(495+66*15^(1/ 2))^(1/2)*15^(1/2)+5387414718575/32856195254186304*(495-66*15^(1/2))^( 1/2)-204724622275/8960780523868992*(495+66*15^(1/2))^(1/2)*15^(1/2)*(4 95-66*15^(1/2))^(1/2)-150291531475/62227642526868*(495+66*15^(1/2))^(1 /2), a[17,14] = -86343889643783/1423361572162200*(495+66*15^(1/2))^(1/ 2)*15^(1/2)-4875329406099/718869480890-47503534664513/28754779235600*1 5^(1/2)-46830515603137/189781542954960*(495+66*15^(1/2))^(1/2)-5665955 0856457/355840393040550*15^(1/2)*(495-66*15^(1/2))^(1/2)-2185635443155 7/869832071876900*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-1258 19226953867/189781542954960*(495-66*15^(1/2))^(1/2)-34670051930677/626 2790917513680*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2) , a[19,16] = 23318577/238751570176*(495+66*15^(1/2))^(1/2)*(495-66*15^ (1/2))^(1/2)-19320841301/23636405447424*15^(1/2)*(495-66*15^(1/2))^(1/ 2)-150970273/59687892544*15^(1/2)+5937215723/119375785088+3596875/5595 739926*(495+66*15^(1/2))^(1/2)*15^(1/2)-18335887105/2626267271936*(495 -66*15^(1/2))^(1/2)-29492131/2148764131584*(495+66*15^(1/2))^(1/2)*15^ (1/2)*(495-66*15^(1/2))^(1/2)+493761427/358127355264*(495+66*15^(1/2)) ^(1/2), a[18,15] = 19990678222944016799007968810994718650113/327238104 2960700814201444438358736068377200*(495+66*15^(1/2))^(1/2)*15^(1/2)-88 393911258485420678134471781658464932823/632660334972402157412279258082 6889732195920*(495+66*15^(1/2))^(1/2)+87051869427126857828192491903627 294702861/958576265109700238503453421337407535181200+43969959325147867 53230371514007047435591/45646488814747630404926353397019406437200*15^( 1/2)+452906331221238361922696065114103183779/7083075850564287476626503 113330597550600*(495-66*15^(1/2))^(1/2)+222532788410281602817218939325 6449625076/107839829824841276831638509900458347707885*15^(1/2)*(495-66 *15^(1/2))^(1/2)+110704560801946412846083674130999701608833/4744952512 2930161805920944356201672991469400*(495+66*15^(1/2))^(1/2)*(495-66*15^ (1/2))^(1/2)+10101069885562826853855484798548336576859/118623812807325 40451480236089050418247867350*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66 *15^(1/2))^(1/2), a[16,15] = 418320022266565899999989/7309005634583080 64970225-119156065868215739462012/2192701690374924194910675*15^(1/2)+3 495695795945596671104/146180112691661612994045*(495+66*15^(1/2))^(1/2) -65831485804562645782499/10963508451874620974553375*(495+66*15^(1/2))^ (1/2)*15^(1/2)-4641549371926342688144/32890525355623862923660125*(495+ 66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+761054660389983937 0279/6578105071124772584732025*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2 ))^(1/2)+17714250989146270037792/730900563458308064970225*(495-66*15^( 1/2))^(1/2)-15029054399563919460349/10963508451874620974553375*15^(1/2 )*(495-66*15^(1/2))^(1/2), a[2,1] = 39/88, a[25,14] = -49414375111/155 8819416000*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-16554531058 7/584557281000*15^(1/2)*(495-66*15^(1/2))^(1/2)-2733053737/843517000*1 5^(1/2)-16561593209/3374068000-24743106247/2338229124000*(495+66*15^(1 /2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-1363545178441/155881941600 0*(495-66*15^(1/2))^(1/2)-4721773003/27836061000*(495+66*15^(1/2))^(1/ 2)-3651713009/27836061000*(495+66*15^(1/2))^(1/2)*15^(1/2), a[18,16] = -436840620567893961487535263885/2696403426422943408076694358192*15^(1 /2)-369545114918022904188418990316329/62556559493012287067379309110054 4-18257270145419395739617682201851/625565594930122870673793091100544*( 495+66*15^(1/2))^(1/2)-5983444403437426102960354262185/938348392395184 306010689636650816*(495+66*15^(1/2))^(1/2)*15^(1/2)+107005288406322056 654355448927205/3753393569580737224042758546603264*(495-66*15^(1/2))^( 1/2)+84055437892642261541931143840615/11260180708742211672128275639809 792*15^(1/2)*(495-66*15^(1/2))^(1/2)+1012338871884896232541346539753/3 753393569580737224042758546603264*(495+66*15^(1/2))^(1/2)*15^(1/2)*(49 5-66*15^(1/2))^(1/2)+4578034999918124747163891877361/37533935695807372 24042758546603264*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[1 8,14] = 4230042827581684523751752104701705965843/222105102055991070418 278663608820791814660*(495+66*15^(1/2))^(1/2)*15^(1/2)+875276118514362 37649772046468066688557713/17947887034827561245921508170409760954720+5 125151437791361308235261701221815485791/673045763806033546722056556390 3660358020*15^(1/2)+2558836067301785957586054852230517925601/158646501 46856505029877047400630056558190*(495+66*15^(1/2))^(1/2)+7231113621317 2294095616390215153042369611/84803766239560226886979126105186120511052 0*15^(1/2)*(495-66*15^(1/2))^(1/2)+47734345931480881442983989229731810 2330687/1130716883194136358493055014735814940147360*(495-66*15^(1/2))^ (1/2)+48522121714870311549903684616247887004701/1865682857270324991513 5407743140946512431440*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/ 2))^(1/2)+20583882814114410685607175778197029916327/138198730168172221 5935956129121551593513440*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1 /2), a[17,16] = -3399762661/1690419432009*(495+66*15^(1/2))^(1/2)*(495 -66*15^(1/2))^(1/2)-21792326747/1690419432009*15^(1/2)*(495-66*15^(1/2 ))^(1/2)+15024834410/51224831273*15^(1/2)+843590793/80496163429*(495+6 6*15^(1/2))^(1/2)*15^(1/2)-167470020833/3380838864018*(495-66*15^(1/2) )^(1/2)-656911562/1690419432009*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495- 66*15^(1/2))^(1/2)+23471698999/482976980574*(495+66*15^(1/2))^(1/2)+61 299209505/51224831273, a[25,16] = 571468963/226284800+120370717/226284 800*15^(1/2)-1751767/2036563200*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495- 66*15^(1/2))^(1/2)-590329/169713600*(495+66*15^(1/2))^(1/2)*(495-66*15 ^(1/2))^(1/2)-519210887/22402195200*15^(1/2)*(495-66*15^(1/2))^(1/2)-2 8396679/298695936*(495-66*15^(1/2))^(1/2)+5795203/339427200*(495+66*15 ^(1/2))^(1/2)*15^(1/2)+7675999/113142400*(495+66*15^(1/2))^(1/2), a[16 ,14] = -14939172465658402889149/8976679102595142257770+219312724397669 057743089/359067164103805690310800*15^(1/2)+659570008376205270318919/1 7773824623138381670384600*(495+66*15^(1/2))^(1/2)*15^(1/2)-11242117128 844970311919/157989552205674503736752*(495+66*15^(1/2))^(1/2)-17498238 31034832192631/43088059692456682837296*(495-66*15^(1/2))^(1/2)+4464635 9022841677628361/1615802238467125606398600*15^(1/2)*(495-66*15^(1/2))^ (1/2)-3070220299360933125107/1974869402570931296709400*(495+66*15^(1/2 ))^(1/2)*(495-66*15^(1/2))^(1/2)+18545262695413350953267/1184921641542 5587780256400*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2) , a[25,15] = -3460846507/13082090000+48037163/3270522500*15^(1/2)-1080 37591/64756345500*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^( 1/2)-412156237/46254532500*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^( 1/2)-2890671767/58869405000*15^(1/2)*(495-66*15^(1/2))^(1/2)-461863819 1/19623135000*(495-66*15^(1/2))^(1/2)-848300537/15418177500*(495+66*15 ^(1/2))^(1/2)+429589031/129512691000*(495+66*15^(1/2))^(1/2)*15^(1/2), a[19,15] = 1908413061857/289859328166800-2931986748899/14348036744256 600*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-1254438533/2173944 96125100*15^(1/2)*(495-66*15^(1/2))^(1/2)-857188607021/289859328166800 *15^(1/2)+5379357542731/9565357829504400*(495+66*15^(1/2))^(1/2)*15^(1 /2)+136382862089/144929664083400*(495-66*15^(1/2))^(1/2)-1935518588/85 404980620575*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+ 34225020981/21690153808400*(495+66*15^(1/2))^(1/2), a[17,15] = -486380 4455866/35119141675695*15^(1/2)-62267936631107/9657763960816125*(495+6 6*15^(1/2))^(1/2)*15^(1/2)-5561334169199/58531902792825+2301512129984/ 643850930721075*(495+66*15^(1/2))^(1/2)-331440117351157/96577639608161 25*15^(1/2)*(495-66*15^(1/2))^(1/2)-27173454387092/214616976907025*(49 5-66*15^(1/2))^(1/2)-37986613754722/28973291882448375*(495+66*15^(1/2) )^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-25832795272667/57946583764896 75*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[15,14] = 1159886 19653/292402973068800*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)- 2683538373203/877208919206400*15^(1/2)*(495-66*15^(1/2))^(1/2)-1968290 835/88606961536*15^(1/2)+496725643/7310074326720*(495+66*15^(1/2))^(1/ 2)*15^(1/2)-69268797311/7310074326720*(495-66*15^(1/2))^(1/2)-12449370 1507/877208919206400*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2) )^(1/2)-52034279227/11696118922752*(495+66*15^(1/2))^(1/2)-74896827627 /88606961536, a[21,1] = 1534965908593628117013/2925190767628355078125, a[15,9] = 63389514041678039114265201628308697725/17763406511249121278 3473366421768720384*(495+66*15^(1/2))^(1/2)*15^(1/2)-59944953779325968 21795043959647438677125/3197413172024841830102520595591836966912*(495+ 66*15^(1/2))^(1/2)-586927014740721935830387224699720472375/52757317338 409890196691589827265309954048*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-6 6*15^(1/2))^(1/2)+7925274936394787837453788892272777275575/10551463467 6819780393383179654530619908096*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/ 2))^(1/2)+4365532842363197027475991758966784554875/3197413172024841830 102520595591836966912*(495-66*15^(1/2))^(1/2)-194972669509543955301656 024896547363525/1598706586012420915051260297795918483456*15^(1/2)*(495 -66*15^(1/2))^(1/2)+83754126586110644280691704028211762275/18167120295 595692216491594293135437312*15^(1/2)-298844563511735270450574605564882 9302375/96891308243177025154621836230055665664, a[16,13] = -5008995772 8648885291210634556861014207223642623/82675783318059004473031720491009 0891816255882000*(495+66*15^(1/2))^(1/2)*15^(1/2)+97307976884973653382 06022389185748546702810361177/9204570542743902497997531547999011928887 648819600*(495+66*15^(1/2))^(1/2)-261514512289046720249818307028675676 8542537260957/253125689925457318694932117569972828044410342539000*(495 +66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-63557795213115057 4447470141761417043531594083113/37968853488818597804239817635495924206 661551380850*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-928675811 835143905263926957504331099234953932259/184091410854878049959950630959 9802385777529763920*(495-66*15^(1/2))^(1/2)-37532879399783886931299400 415084904096854083988891/138068558141158537469962973219985178933314732 294000*15^(1/2)*(495-66*15^(1/2))^(1/2)+895188211928147628185622286558 387328719961332/2091947850623614204090348079090684529292647459*15^(1/2 )+1488122595447313901345177744809891103830021016413/697315950207871401 36344935969689484309754915300, a[25,9] = 14983717615076520999898175279 006718901325/8577355584275808249327589714760925401088*(495+66*15^(1/2) )^(1/2)*15^(1/2)+79953215765348899446771774079566331820483/57182370561 83872166218393143173950267392*(495+66*15^(1/2))^(1/2)-2062159779274001 1665293932604020043908997/141526367140550836113905230293555269117952*( 495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-15666467932539 007518488566513824293941973/20966869206007531276134108191637817647104* (495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-3707004449454446972134 7012758337427892153/1906079018727957388739464381057983422464*(495-66*1 5^(1/2))^(1/2)-8918411246208536670588999325959809281/20940809531923359 98370993582705304053*15^(1/2)*(495-66*15^(1/2))^(1/2)+7724360629880208 471040518407144433826297/129959933095088003777690753253953415168*15^(1 /2)+33101027521371172109192250549843371608719819199/984793053016878529 95941463124070766233804800, a[15,10] = 1280481580011067872042906081843 788744919/6705266146214797737008913520090013286400*(495+66*15^(1/2))^( 1/2)*15^(1/2)-830532507857909031976483582994466468573/1341053229242959 547401782704018002657280*(495+66*15^(1/2))^(1/2)-628631066385136591582 4207650794833606289/811337203691990526178078535930891607654400*(495+66 *15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+36367358350110160165 53586602489401592969/162267440738398105235615707186178321530880*(495+6 6*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+4777099095613693839379029295 382810922327/14751585521672555021419609744198029230080*(495-66*15^(1/2 ))^(1/2)-9705196694365591791033907171421616130041/73757927608362775107 098048720990146150400*15^(1/2)*(495-66*15^(1/2))^(1/2)+463619156568945 7788033216342021225893961/1341053229242959547401782704018002657280*15^ (1/2)-12947680617197490104519222676481332902841/1341053229242959547401 782704018002657280, a[25,2] = -1029/400, a[25,6] = -2667/1600, c[4] = \+ 101/152, a[23,6] = -381/544, a[21,11] = -44794043532626317794933177103 220192762017625/167028975954451970498474513047407579017107824+39042282 8672114370457150585026889349975/69595406647688321041031047103086491257 12826*(495+66*15^(1/2))^(1/2)*15^(1/2)+2944144225934645439912079016013 5926728275/4639693776512554736068736473539099417141884*15^(1/2)+107835 852142499187656942652161788723246225/139190813295376642082062094206172 98251425652*(495+66*15^(1/2))^(1/2), c[8] = 17727/277750, a[20,19] = - 110845/348543, a[21,19] = 81770692250/117233567091, a[22,19] = 8287866 88/2261607237, a[23,21] = 253319312163456/659439747650125, a[22,21] = \+ -75739731820032/659439747650125, a[25,19] = 3601857/2273300, a[21,20] \+ = 21260379985/34709636178, a[25,21] = 30334395520464588/16485993691253 125, a[22,20] = 76890112/586342617, a[25,23] = 753491/379050, a[25,20] = 3071471/2273300, a[23,22] = 381/544, a[25,22] = 2667/1600, a[25,24] = 1029/400, a[21,18] = -390422828672114370457150585026889349975/69595 40664768832104103104710308649125712826*(495+66*15^(1/2))^(1/2)*15^(1/2 )+44794043532626317794933177103220192762017625/16702897595445197049847 4513047407579017107824-29441442259346454399120790160135926728275/46396 93776512554736068736473539099417141884*15^(1/2)-1078358521424991876569 42652161788723246225/13919081329537664208206209420617298251425652*(495 +66*15^(1/2))^(1/2), a[25,17] = 112245633/226284800-1120087/28285600*1 5^(1/2)-68125601/7467398400*(495+66*15^(1/2))^(1/2)+993373/4480439040* (495+66*15^(1/2))^(1/2)*15^(1/2), a[25,18] = -621238048935340248204899 2934870163/5513981142404675080846640779279519700*(495+66*15^(1/2))^(1/ 2)*15^(1/2)+56874537829985536110363501647573078867/4411184913923740064 6773126234236157600-477467113712319024289213588759633869/1102796228480 9350161693281558559039400*15^(1/2)-16827638430850958536645296942748214 1/11027962284809350161693281558559039400*(495+66*15^(1/2))^(1/2), a[20 ,17] = 229884359555/8366741852352+46739047805/4183370926176*15^(1/2)-2 0855494525/25100225557056*(495+66*15^(1/2))^(1/2)-34057541825/75300676 671168*(495+66*15^(1/2))^(1/2)*15^(1/2), a[20,18] = -52601571650923977 687781073690186/1325511779651805411892910350460322529*(495+66*15^(1/2) )^(1/2)*15^(1/2)+1461351037003208398982108006046541205/159061413558216 64942714924205523870348-312574630477851429199757289182427/101962444588 600416299454642343101733*15^(1/2)-3126986093611720912749441466631523/1 325511779651805411892910350460322529*(495+66*15^(1/2))^(1/2), a[19,18] = 199389002800788393634510722754176/279672820066436005916238277290942 7897*(495+66*15^(1/2))^(1/2)*15^(1/2)-10254652481568348033538409169201 80809/5593456401328720118324765545818855794+16445717956872940821116448 130678578/2796728200664360059162382772909427897*15^(1/2)+1343763225617 6229690482072355537756/2796728200664360059162382772909427897*(495+66*1 5^(1/2))^(1/2), a[19,17] = 2466730563/34107367168-46877457/4872481024* 15^(1/2)-417081443/102322101504*(495+66*15^(1/2))^(1/2)+221689729/3069 66304512*(495+66*15^(1/2))^(1/2)*15^(1/2), a[18,17] = 6167304375146239 0569674581/351685648446307151385663744-4915309705585387914994781/35168 5648446307151385663744*15^(1/2)-46955489957737999234808063/11605626398 728135995726903552*(495+66*15^(1/2))^(1/2)+6751918613532367335628313/3 4816879196184407987180710656*(495+66*15^(1/2))^(1/2)*15^(1/2), c[6] = \+ 25/152, a[21,5] = -156479396217570600333298656/73273972406867506457421 875, a[21,6] = 11005843237812473721/5193844391648046875, a[23,2] = 112 211/506844, a[22,4] = 390625/3949854672, a[22,5] = 404856925/321658957 2, c[5] = 8225/76912, c[3] = 101/228, c[7] = 2013/3445, a[18,7] = 1899 3974973485227904952341910291889381118708193023824550/12060034665080522 994492944014191159522924996678137100092619*(495+66*15^(1/2))^(1/2)*15^ (1/2)+37364999682821940340524213896266047309233288831638123551875/4288 0123253619637313752689828235233859288877077820800329312+83870865016481 720496920100044907813868655001976949702225/134000385167561366605477155 7132351058102777408681900010291*15^(1/2)+10188221773942597558585056227 9744393868968529126539562125/40200115550268409981643146713970531743083 32226045700030873*(495+66*15^(1/2))^(1/2), a[18,8] = -2754138826372250 84153879395633151790451174339761235000000000/1736570677629765561734121 6750404769836227513968915818501163621*(495+66*15^(1/2))^(1/2)*15^(1/2) -9030706067191500671304838647417536518671082752129344726562500/1929522 975144183957482357416711641092914168218768424277907069-113106637991389 7019748847486012312835922421062622057500000000/19295229751441839574823 57416711641092914168218768424277907069*15^(1/2)-1097562530284460395919 027812165520839910017377896612500000000/578856892543255187244707225013 4923278742504656305272833721207*(495+66*15^(1/2))^(1/2), a[19,6] = -46 012134893475328/76616158064651325, a[19,7] = 5620721356384469075/35456 98345096452809913, a[19,8] = 10436097770007465488281250/34480588269890 588098919643, a[20,6] = 24467886602752/291937806983125, a[20,7] = 1194 4081475573/1322478962029186830, a[20,8] = 1095050362935718750/11254693 571026117461, a[21,4] = 46426400659946403657/610018749211776263125, b[ 20] = 13/56, b[22] = 19/100, b[21] = 21/100, b[23] = 17/100, b[24] = 1 1/100, c[21] = 2013/3445, b[18] = 29/100, c[24] = 39/88, c[23] = 101/2 28, c[22] = 25/152, c[19] = 1/3, c[18] = 51789075/64972747+333240/6497 2747*15^(1/2)+3057458/714700217*(495+66*15^(1/2))^(1/2)+70408/71470021 7*(495+66*15^(1/2))^(1/2)*15^(1/2), b[19] = 27/100, c[20] = 1/5, a[24, 3] = -274193/636804, a[20,10] = 68983297474551316574674372049500652903 8916799/838355228023916751926128515866680316613341143*a[20,11]*(495+66 *15^(1/2))^(1/2)*15^(1/2)-57799442520783622511381375141021879258823374 68020351917269606714904240752881819452757010256477/4553107016368863878 3952708585542704867548001484568201975536636564976676773678297361586556 77607296*(495+66*15^(1/2))^(1/2)*15^(1/2)-9823219876217135627910816350 7503711334331028565/838355228023916751926128515866680316613341143*a[20 ,11]+17811073894998005409723936053026585/60599525887138708165477957969 55504128*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-4358 3346131618725163161650192664655/3029976294356935408273897898477752064* (495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-4827425744409051504275 31405059365444676866008/76214111638537886538738955987880028783031013*a [20,11]*(495+66*15^(1/2))^(1/2)+84512802292818577534401340531783721260 07600853952624530450537286583897155967945634151371752919/3384066025679 5609906991878002768226590745136238530420387223175825320503007463599390 3683867930272*(495+66*15^(1/2))^(1/2)-13954888386903425569378253302344 3195/550904780792170074231617799723227648*(495-66*15^(1/2))^(1/2)-1826 5794041764066413104163434188989441120218996486476982339871214815236770 106518690271433310863867/100168354360115005324695958888193950708605603 266050044346180600442948688902092254195490424907360512*15^(1/2)+643527 6729659950936901445282737804392845587547/83835522802391675192612851586 6680316613341143*a[20,11]*15^(1/2)+1775211130510935382197885405175685/ 42377290830166928787047523055632896*15^(1/2)*(495-66*15^(1/2))^(1/2)+2 1830727047870064010049220470941805413265901080569307165451607195065771 927245993623259957653419948895/162272734063386308626007453398874200147 94107729100107184081257271757687602138945179669448834992402944, a[20,1 3] = -73386988744242103366672422267080822593823256/4904585119217563989 229997497115053593255737*a[20,11]*(495+66*15^(1/2))^(1/2)*15^(1/2)-143 178685300843522978975842175312251707656968/490458511921756398922999749 7115053593255737*a[20,11]*(495+66*15^(1/2))^(1/2)+60498086020737874444 0200065592050036606318250641587984042263733276217820026063822514435783 07823/8790144149917254453109967549476193828032245988639996882015091145 24579430499711382893147040746112*(495+66*15^(1/2))^(1/2)*15^(1/2)+3482 4582594533362615224205558195352878478739779848005861568604145295183451 34629602185475335439/6592608112437940839832475662107145371024184491479 99766151131835893434572874783537169860280559584*(495+66*15^(1/2))^(1/2 )-433948807018695022378898091347998031973552360/4458713744743239990209 08863374095781205067*a[20,11]-1631735910272741447707280640636661004796 82008/445871374474323999020908863374095781205067*a[20,11]*15^(1/2)+145 17312163286399149496689890085/4883724069920184820647992065213824*(495+ 66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-136482432983006310806407969 6735/67829500971113678064555445350192*(495+66*15^(1/2))^(1/2)*15^(1/2) *(495-66*15^(1/2))^(1/2)-1308466558624809904319027296945/3083159135050 621730207065697736*(495-66*15^(1/2))^(1/2)-639048077256390186692861431 47275/147991638482429843049939153491328*15^(1/2)*(495-66*15^(1/2))^(1/ 2)+7332398509447281137091292050092091696984024769184628978781441070267 703401079682492358371244683/544843645656028168581196335711334328183816 9001223138563232494511515988205576723447684795707104*15^(1/2)+11194682 3719478179346673693166399554839082889477918244526395142029036285214217 323045106382778509/799104013628841313919087959043290348002931453512726 98927409919502234493681791943899377003704192, a[14,10] = -307025684909 12439111188538/2861432914057609960190041997*(495+66*15^(1/2))^(1/2)*15 ^(1/2)+6755832884015803347811847721/91565853249843518726081343904*(495 +66*15^(1/2))^(1/2)+100517936407692876551070243/3662634129993740749043 25375616*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-2791 43136095627004211108611/91565853249843518726081343904*(495+66*15^(1/2) )^(1/2)*(495-66*15^(1/2))^(1/2)-1021753700305474025763231075/166483369 54517003404742062528*(495-66*15^(1/2))^(1/2)-2845475421494168148093843 3/33296673909034006809484125056*15^(1/2)*(495-66*15^(1/2))^(1/2)-62483 8597750010736607379013/8324168477258501702371031264*15^(1/2)+208895630 08538679095591532243/16648336954517003404742062528, a[19,13] = -733869 88744242103366672422267080822593823256/4904585119217563989229997497115 053593255737*a[19,11]*(495+66*15^(1/2))^(1/2)*15^(1/2)-143178685300843 522978975842175312251707656968/490458511921756398922999749711505359325 5737*a[19,11]*(495+66*15^(1/2))^(1/2)-70731447915416584889016206096954 40556335504032484072389037082688134818985407096630855499992613/3152931 9904723816917889327645918284739631353967444213321678944227909209153683 10131583799340800*(495+66*15^(1/2))^(1/2)*15^(1/2)+1745797646765587968 2947509913395276345473841607647033713169136005262331955510423871609360 64023/7882329976180954229472331911479571184907838491861053330419736056 97730228842077532895949835200*(495+66*15^(1/2))^(1/2)-4339488070186950 22378898091347998031973552360/4458713744743239990209088633740957812050 67*a[19,11]-163173591027274144770728064063666100479682008/445871374474 323999020908863374095781205067*a[19,11]*15^(1/2)-121535852020857141254 764064119/27568187608417752345536908713600*(495+66*15^(1/2))^(1/2)*(49 5-66*15^(1/2))^(1/2)+7589099400604327350943308035467/55136375216835504 691073817427200*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/ 2)+2638774990174311084272599528753/835399624497507646834451779200*(495 -66*15^(1/2))^(1/2)+133085358767341451000935270561/4515673645932473766 6727123200*15^(1/2)*(495-66*15^(1/2))^(1/2)-12754304205063189647727376 063779546939788405093358475224950898470357996436097948291238109679879/ 2866301809520346992535393422356207703602850360676746665607176747991746 28669846375598527212800*15^(1/2)-2218566465566495266180098207755537403 71882330022732313406166391412803161379475441746353278027/4777169682533 9116542256557039270128393380839344612444426786279133195771444974395933 087868800, a[19,9] = 2223543721376062484038978612772140819847550375/39 3222308554683296648889007567566605040336593*a[19,11]*(495+66*15^(1/2)) ^(1/2)*15^(1/2)+360146264750566047161248627473354527848611259486296323 118830427028862319714561278576203773690955/242673139366161178335581391 979924806897968611341494357022742812351741265536716419531367212541952* (495+66*15^(1/2))^(1/2)*15^(1/2)+1661644147259794908494849034455405930 7420567000/393222308554683296648889007567566605040336593*a[19,11]-3008 572710114978801133809542195/38231588047416797885568790284288*(495+66*1 5^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+8090739019631056999055 30940281485/4243706273263264565298135721555968*(495+66*15^(1/2))^(1/2) *(495-66*15^(1/2))^(1/2)-105076169688424908960027527084898127335349414 8255153503546139281060685129677387916709064679415615/24267313936616117 8335581391979924806897968611341494357022742812351741265536716419531367 212541952*(495+66*15^(1/2))^(1/2)+282198637264740272229946121642305/12 8597159795856501978731385501696*(495-66*15^(1/2))^(1/2)+20096202074345 6330709956275553303985002809939274756780753700734941756207588709359700 0709479739905/80891046455387059445193797326641602299322870447164785674 247604117247088512238806510455737513984*15^(1/2)-162744959789334607207 9604782545865369205399250/39322230855468329664888900756756660504033659 3*a[19,11]*(495+66*15^(1/2))^(1/2)+45216479318404965835919476599765765 471888831375/393222308554683296648889007567566605040336593*a[19,11]*15 ^(1/2)-126778048132170996252726627127707347486608634047123140809595736 075360682518666917644278901874986598534727595/221479331142828538177320 1314116131881179169153780655229805395224586840675677281430075955936995 706670434304-177438983067524553576678905118445/12859715979585650197873 1385501696*15^(1/2)*(495-66*15^(1/2))^(1/2), a[20,12] = 32605442268353 3972382940944248612996897635913850385781247371049011432957465096500/30 1784038385219524590571456153549459998213851893199978104100439448352738 79836333*a[20,11]*(495+66*15^(1/2))^(1/2)*15^(1/2)+1582368239298967156 924245817101592519594222022521414699230759737356169996754285750/301784 0383852195245905714561535494599982138518931999781041004394483527387983 6333*a[20,11]*(495+66*15^(1/2))^(1/2)-10455707291707716935561892422001 2687359300875278507469943621881406025102168848886195831831015895654897 09089380042384255091863485125/4415235756295708192150877661574887091507 0700709356333848394006193482658930674987403673033433158872290654359947 2353061903753206829568*(495+66*15^(1/2))^(1/2)*15^(1/2)-60871410149131 3707425372639189644175066606999731045978784708718206422262913988975289 70215665623589299986562416980507942262939144125/3311426817221781144113 1582461811653186303025532017250386295504645111994198006240552754775074 8691542179907699604264796427814905122176*(495+66*15^(1/2))^(1/2)+37708 7149009509066186452266344630931324668925005610771681436989919670339144 7184250/27434912580474502235506496013959041818019441081199998009463676 31348661261803303*a[20,11]+8578516504152727723124477694208938965011863 98237361908471368108566604429824662000/2743491258047450223550649601395 904181801944108119999800946367631348661261803303*a[20,11]*15^(1/2)+979 103008642060391342055123440766626160244912108003784618486229068074375/ 1766942570220628025432180348242154413343890321922823851014564914877394 3808*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+56842063626884888 066138543864527725958915080138097783372866364251693625/883471285110314 0127160901741210772066719451609614119255072824574386971904*(495+66*15^ (1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+181833489008238739741769 972594521015211628467079492421378267235714464375/133859285622774850411 528814260769273738173509236577564470800372339196544*(495-66*15^(1/2))^ (1/2)+4039542018793757338459161700273433563845654146601981501584928059 4318125/17847904749703313388203841901435903165089801231543675262773382 9785595392*15^(1/2)*(495-66*15^(1/2))^(1/2)-10582989523427632633831597 0395141796554221868053340100963025476410152100714973695185009673638461 882431167217312114667900309739490125/180623280939369880587990449791699 9264707437756291850021070663889733381501709431302877533185901953866435 86017965962616233353584612096*15^(1/2)-2555970554568994878839104326065 4892287062384094609504415324423863578314550620076969269724886950879597 5791897561612752240756472875/55465463208773186116379686716321181167125 3725254675271325246089277869338771512759980817806203578647761664418750 07712646508086784, a[14,11] = -131305632444106618723157224706857241760 74235474873559/1684472252639923552532001908581242221889072724213841856 *(495+66*15^(1/2))^(1/2)*15^(1/2)+216344025468221065029564055578401998 9685985966394801/78959636842496416524937589464745729151050283947523837 *(495+66*15^(1/2))^(1/2)-262033653903945943315331508528420647097689137 2062198311/2834966801192991338911359212142230659439309394851895843648* (495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+1567417592090996841749 224823127656355473861331736569451/566993360238598267782271842428446131 8878618789703791687296*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/ 2))^(1/2)+960434983056913523969908257106104327413288387315836063/85908 084884636101179132097337643353316342708934905934656*(495-66*15^(1/2))^ (1/2)-2107085918307670902159884356898641932607128641690997959/51544850 9307816607074792584025860119898056253609435607936*15^(1/2)*(495-66*15^ (1/2))^(1/2)+63197323865987155320437607552839368481975380037653735/561 490750879974517510667302860414073963024241404613952*15^(1/2)-196251321 827756638854506220121563148477341044141985823/561490750879974517510667 302860414073963024241404613952, a[16,10] = 170328089215324137693260951 01435868425512701950321/2284930083473257768173058734405054028394965310 81600*(495+66*15^(1/2))^(1/2)*15^(1/2)-3149271718657689721534442073886 4818541314974948753/11424650417366288840865293672025270141974826554080 0*(495+66*15^(1/2))^(1/2)+26366057151537512209022950975149759973777226 90725299/3041241941102906089438341175493126911793698828696096000*(495+ 66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-121284813075941151 6381383562111404947777471603683549/12164967764411624357753364701972507 6471747953147843840*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-27 42291030606566024225316251947419989058495068734689/1382382700501320949 7447005343150576871789540130436800*(495-66*15^(1/2))^(1/2)+12060316683 436861169241947489803239449824641591213/691191350250660474872350267157 528843589477006521840*15^(1/2)*(495-66*15^(1/2))^(1/2)+670838308222019 8976386948910077685899871881611055887/50268461836411670899807292156911 18862468923683795200*15^(1/2)-2234453575085500446794327256376705620346 9038581346723/5026846183641167089980729215691118862468923683795200, a[ 25,10] = -289615505775241581098990702550013952492757/29097417663212797 3015045830605969822720000*(495+66*15^(1/2))^(1/2)*15^(1/2)-82637560006 838181681227476570558015839783/145487088316063986507522915302984911360 000*(495+66*15^(1/2))^(1/2)+186351556095480401186869715192832670221231 /3200715942953407703165504136665668049920000*(495+66*15^(1/2))^(1/2)*1 5^(1/2)*(495-66*15^(1/2))^(1/2)+47975016305743765021328635026942973499 8683/3200715942953407703165504136665668049920000*(495+66*15^(1/2))^(1/ 2)*(495-66*15^(1/2))^(1/2)+659613600904755105711427456179352195929/142 984853381881067820661341821115392000*(495-66*15^(1/2))^(1/2)+193491757 45772217526368706395431357179269/1322609893782399877341117411845317376 0000*15^(1/2)*(495-66*15^(1/2))^(1/2)-94118187902607258388116970891081 97825199/413315591806999961669099191201661680000*15^(1/2)-212677882133 7403489098990902156033714011143903/60133781430710592823191244246959199 817216000\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 212 "The values for the remaining unknown linking coef ficients can be obtained from a system of equations consisting of the \+ following three conditions together with the row sum conditions for ro ws 14 to 20 and row 25." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[18,j]*c[j]^5,j=2..17)=Sum( a[11,j]*c[j]^5,j=2..10)" "6#/-%$SumG6$*&&%\"aG6$\"#=%\"jG\"\"\"*$&%\"c G6#F,\"\"&F-/F,;\"\"#\"#<-F%6$*&&F)6$\"#6F,F-*$&F06#F,F2F-/F,;F5\"#5" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(a[19,j]*c[j]^5,j=2..18)=Sum(a [10,j]*c[j]^5,j=2..9)" "6#/-%$SumG6$*&&%\"aG6$\"#>%\"jG\"\"\"*$&%\"cG6 #F,\"\"&F-/F,;\"\"#\"#=-F%6$*&&F)6$\"#5F,F-*$&F06#F,F2F-/F,;F5\"\"*" } {TEXT -1 6 ", " }{XPPEDIT 18 0 "Sum(a[20,j]*c[j]^5,j=2..19)=Sum(a[ 9,j]*c[j]^5,j=2..8)" "6#/-%$SumG6$*&&%\"aG6$\"#?%\"jG\"\"\"*$&%\"cG6#F ,\"\"&F-/F,;\"\"#\"#>-F%6$*&&F)6$\"\"*F,F-*$&F06#F,F2F-/F,;F5\"\")" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "cdns5 := [add(a[18,j]*c[j]^5,j=2..17)=add(a[11, j]*c[j]^5,j=2..10),\n add(a[19,j]*c[j]^5,j=2..18)=add(a[10,j ]*c[j]^5,j=2..9),\n add(a[20,j]*c[j]^5,j=2..19)=add(a[9,j]*c [j]^5,j=2..8),\n seq(add(a[i,j],j=1..i-1)=c[i],i=[$14..20,25 ])]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "eqns5 := simplify(expand(subs(e24,[op(cdns5)]))):\nno ps(%);\nindets(eqns5);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<-&%\"aG6$\"#D\"\"\"&F%6$\"#?\"# 6&F%6$F+F(&F%6$\"#>F,&F%6$F1F(&F%6$\"#=F,&F%6$F6F(&F%6$\"# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "e25 := solve(\{op(eqns5)\},indets(eqns5)):\ne26 := ` union`(map(u_->lhs(u_)=simplify(subs(e25,rhs(u_))),e24),e25):\ninfolev el[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "indets(map(rhs,e26));" }}{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 3 "e2 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86234 "e26 := \{a[23,20] = 0, a[23,19] = 0, c[12] = -4882 7895154/211479617260065*(495+66*15^(1/2))^(1/2)*15^(1/2)-1080276228947 /70493205753355*(495+66*15^(1/2))^(1/2)+2523614583531/6408473250305+33 872310083/1281694650061*15^(1/2), a[23,18] = 0, a[25,5] = 0, a[23,17] \+ = 0, a[25,4] = 0, a[25,8] = 0, a[24,15] = 0, a[24,14] = 0, a[23,4] = 0 , a[23,16] = 0, a[23,14] = 0, a[23,15] = 0, a[23,13] = 0, a[23,12] = 0 , a[23,11] = 0, a[23,10] = 0, a[23,9] = 0, a[24,10] = 0, a[23,8] = 0, \+ a[23,3] = 0, a[23,5] = 0, a[24,12] = 0, a[24,13] = 0, a[24,22] = 0, a[ 24,21] = 0, a[24,20] = 0, a[24,18] = 0, a[24,19] = 0, a[24,17] = 0, a[ 24,16] = 0, a[25,3] = -753491/379050, a[24,11] = 0, b[8] = 0, b[5] = 0 , b[4] = 0, b[16] = 31/175+1/100*15^(1/2), b[25] = 1/42, b[13] = 31/17 5-1/100*15^(1/2), b[1] = 1/42, c[16] = 1/2+1/66*(495-66*15^(1/2))^(1/2 ), c[14] = 1/2-1/66*(495-66*15^(1/2))^(1/2), c[25] = 1, b[12] = 0, b[1 7] = 31/175-1/100*15^(1/2), c[17] = 1/2+1/66*(495+66*15^(1/2))^(1/2), \+ b[14] = 31/175+1/100*15^(1/2), c[15] = 1/2, c[13] = 1/2-1/66*(495+66*1 5^(1/2))^(1/2), b[15] = 128/525, a[11,6] = 0, a[24,5] = 0, a[24,1] = 3 9/88, c[11] = 51789075/64972747+333240/64972747*15^(1/2)+3057458/71470 0217*(495+66*15^(1/2))^(1/2)+70408/714700217*(495+66*15^(1/2))^(1/2)*1 5^(1/2), a[25,13] = -55890504392518564023586021030994240051/5930958253 122427625046584152986240000*(495+66*15^(1/2))^(1/2)*15^(1/2)-891100910 60970255586136770621176187511/1976986084374142541682194717662080000*(4 95+66*15^(1/2))^(1/2)+43168201201852472563143389555061879503/652405407 84346703875512425682848640000*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66 *15^(1/2))^(1/2)+60515870791866688668248776153282301603/21746846928115 567958504141894282880000*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/ 2)+1332897194407130790492306856492982943/17810685444812094970109862321 280000*(495-66*15^(1/2))^(1/2)+745926401497595167053150707146538419/40 346654783145766156779484033920000*15^(1/2)*(495-66*15^(1/2))^(1/2)-161 95323390898774863275935548635966849/5990866922345886489946044598976000 0*15^(1/2)-10162782157776673605214552785788829801/85583813176369806999 22920855680000, a[13,5] = 0, a[12,5] = 0, a[11,5] = 0, a[10,5] = 0, a[ 9,5] = 0, a[12,3] = 0, a[11,3] = 0, a[10,3] = 0, a[9,3] = 0, a[8,3] = \+ 0, a[7,3] = 0, a[6,3] = 0, a[13,4] = 0, a[12,4] = 0, a[11,4] = 0, a[10 ,4] = 0, a[9,4] = 0, a[8,4] = 0, a[10,2] = 0, a[9,2] = 0, a[8,2] = 0, \+ a[7,2] = 0, a[6,2] = 0, a[5,2] = 0, a[4,2] = 0, a[13,3] = 0, a[13,2] = 0, a[12,2] = 0, a[11,2] = 0, a[13,8] = 0, a[13,7] = 0, a[13,6] = 0, a [24,8] = 0, a[24,9] = 0, a[24,4] = 0, a[19,3] = 0, a[18,3] = 0, a[17,3 ] = 0, a[16,3] = 0, a[15,3] = 0, a[14,3] = 0, a[17,2] = 0, a[16,2] = 0 , a[15,2] = 0, a[14,2] = 0, a[20,3] = 0, a[19,2] = 0, a[18,2] = 0, a[1 5,5] = 0, a[14,5] = 0, a[20,2] = 0, a[20,5] = 0, a[19,5] = 0, a[18,5] \+ = 0, a[17,5] = 0, a[16,5] = 0, a[14,4] = 0, a[19,4] = 0, a[18,4] = 0, \+ a[17,4] = 0, a[16,4] = 0, a[15,4] = 0, a[16,8] = 0, a[15,8] = 0, a[14, 8] = 0, a[20,4] = 0, a[17,7] = 0, a[16,7] = 0, a[15,7] = 0, a[14,7] = \+ 0, a[17,8] = 0, a[18,6] = 0, a[17,6] = 0, b[9] = -13/56, a[24,2] = 0, \+ a[16,6] = 0, a[15,6] = 0, a[14,6] = 0, a[22,2] = 0, a[22,13] = 0, a[22 ,12] = 0, a[22,11] = 0, a[22,8] = 0, a[22,6] = 0, a[22,3] = 0, a[22,18 ] = 0, a[22,17] = 0, a[22,16] = 0, a[22,15] = 0, a[22,14] = 0, a[21,2] = 0, a[21,13] = 0, a[21,12] = 0, a[21,8] = 0, a[24,6] = 0, a[24,7] = \+ 0, a[21,7] = 0, a[21,3] = 0, a[21,17] = 0, a[21,16] = 0, a[21,15] = 0, a[21,14] = 0, c[9] = 1/5, a[22,9] = -76890112/586342617, c[10] = 1/3, a[19,12] = -767857350621349612232365076031660012507243302330659012983 146444882552290312974735723275/125301689024387019379551780258306601256 56283890472914056062253468072838490524900225469952*(495+66*15^(1/2))^( 1/2)*(495-66*15^(1/2))^(1/2)-75947023249244695105564726549669900033769 0500161932506542821802549732174619192240958225/44298576927813592709942 5485761690004442393874915709082790079668063181158755930816051968*(495- 66*15^(1/2))^(1/2)-861046685699424690828395217817049098521970527974447 5838787845046207537161193773728998275/15947487694012893375579317487420 840159926179496965526980442868050274521715213509377870848*15^(1/2)*(49 5-66*15^(1/2))^(1/2)-2042408915532495628745192823801256307313363719851 1679290002109762392731355710727689275/10278654177781747683478856974314 21338432742037890356231161356729802850032425870721620582*(495+66*15^(1 /2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+10656277089125517574966660 95819531967211680725773152658584457814629399276136215725600075/4393247 2986261414257794263050746116143047326437921561929594677824447718223728 675972096+393925748804209458132757147978051183254121219736447671744888 341365234497125489205466425/439324729862614142577942630507461161430473 26437921561929594677824447718223728675972096*15^(1/2)+4467656289741282 9009454571411766027566469028397688036590664579101642927080848559936690 75/5315829231337631125193105829140280053308726498988508993480956016758 173905071169792623616*(495+66*15^(1/2))^(1/2)+895895184373623104487614 380409620558402928859356077221719972543748424340659718184686725/265791 4615668815562596552914570140026654363249494254496740478008379086952535 584896311808*(495+66*15^(1/2))^(1/2)*15^(1/2), a[14,13] = 159844475107 9441171273/16159279084983021920856*(495+66*15^(1/2))^(1/2)*15^(1/2)+31 02710944829420080669/37704984531627051148664*(495+66*15^(1/2))^(1/2)-6 561603091532524625551/1244264489543692687905912*(495+66*15^(1/2))^(1/2 )*15^(1/2)*(495-66*15^(1/2))^(1/2)-35515474297055779886885/37327934686 31078063717736*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-3971147 0248879690525337/113114953594881153445992*(495-66*15^(1/2))^(1/2)-1929 879449437757437867/16159279084983021920856*15^(1/2)*(495-66*15^(1/2))^ (1/2)+227699360850867462438055/113114953594881153445992*15^(1/2)+18045 4599750998397262245/37704984531627051148664, a[16,11] = 59876407900611 048420430347160229958910540540624222967777878571466744857688055291/699 0649798122254879981052929913741233975618225036381641093813550305368709 8320928000*(495+66*15^(1/2))^(1/2)*15^(1/2)-44373203401444493569819646 011023230257101197200190943252676868247630645717834437/139812995962445 09759962105859827482467951236450072763282187627100610737419664185600*( 495+66*15^(1/2))^(1/2)+13053177800232951792188311988374109264136807345 1591037842513336026487972156959197/38448573889672401839895791114525576 786865900237700099026015974526679527904076510400*(495+66*15^(1/2))^(1/ 2)*(495-66*15^(1/2))^(1/2)-9684873907603915144383947284992917772741600 71205048044579302980535087052391884449/1153457216690172055196873733435 767303605977007131002970780479235800385837122295312000*(495+66*15^(1/2 ))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-7569237502146003892215225738 60117240432945020810456862043431960429821151852359123/1398129959624450 9759962105859827482467951236450072763282187627100610737419664185600*(4 95-66*15^(1/2))^(1/2)+263629071248380528802217292904751753971759998833 8888836782961749234917030681433653/20971949394366764639943158789741223 7019268546751091449232814406509161061294962784000*15^(1/2)*(495-66*15^ (1/2))^(1/2)-662103215957620151249574093269882686165891564446762498247 90888301449276524367/2353754140781903999993620515122471795951386607756 357454913741936129753774354240*15^(1/2)+144321060930505057382652985608 22011759664831069245950328191010631884195945409799/1059189363351856799 99712923180511230817812397349036085471118387125838919845940800, a[17,1 2] = 20625100239507052150530528014406598317940614967904980544019351949 81336150529071932571468645/3526017312020863375109937984757192708972821 60729607164560748995336005930604885069735873664*(495+66*15^(1/2))^(1/2 )*15^(1/2)+21950024233525577202352169639674214394525570114587522671084 049111799201387642682229845120625/105780519360625901253298139542715781 2691846482188821493682246986008017791814655209207620992*(495+66*15^(1/ 2))^(1/2)-537889723444443631793362349165888861752023905392325609673753 0109733166794995746210119273075/38786190432229497126209317832329119798 70103768025678810168238948696065236653735767094610304*(495+66*15^(1/2) )^(1/2)*(495-66*15^(1/2))^(1/2)-44079450885398543650816266846415094218 24352825576746473116241593509983795684422570864451955/1163585712966884 9137862795349698735939610311304077036430504716846088195709961207301283 830912*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-135893 6683294246493835741312037380639535393360471796374863746079946917868890 2194663998720875/35260173120208633751099379847571927089728216072960716 4560748995336005930604885069735873664*(495-66*15^(1/2))^(1/2)-10831101 5737272776218211980307653603313274604723185997807730585656183505668130 41820876945915/1057805193606259012532981395427157812691846482188821493 682246986008017791814655209207620992*15^(1/2)*(495-66*15^(1/2))^(1/2)+ 5035339331149382138694451118606326994182012611675267644687789729295278 859341198095736492525/320547028365533034100903453159744791724801964299 64287687340817757818720964080460885079424*15^(1/2)+1861846813162796801 5723129136344526950721576481827215389915566861437225213489812831788545 975/320547028365533034100903453159744791724801964299642876873408177578 18720964080460885079424, a[14,12] = -289804150241496084702505677480068 44874496969165876629344425090037471180265975/5560259127312703955256964 35008504195893379880002316479053295626653067982544432*(495+66*15^(1/2) )^(1/2)*15^(1/2)-99765719578606126127875725205455669533869458956454230 864234408168976968137375/370683941820846930350464290005669463928919920 001544319368863751102045321696288*(495+66*15^(1/2))^(1/2)+457748579807 2565700171178273501486293615281976114508992388556678726688329375/27801 2956365635197762848217504252097946689940001158239526647813326533991272 216*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+200841771823310260 5354237877201536969910156428731711178806635228757150383625/55602591273 1270395525696435008504195893379880002316479053295626653067982544432*(4 95+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+392060739572280 7442759729614153985440023284311589623651473153296272407874375/91905109 54235874306209858429892631337080659173592007918236291349637487314784*( 495-66*15^(1/2))^(1/2)+31857312923681668727338841524393098989337502250 0910302606787822374914568125/30635036514119581020699528099642104456935 53057864002639412097116545829104928*15^(1/2)*(495-66*15^(1/2))^(1/2)-4 4156875659383230229388062582129428409372723912177086212422885492186671 35125/2808211680460961593564123409133859575219090303042002419461089023 500343346184*15^(1/2)-750866026797031110163627466855126313471852735918 84428191176525250463168342875/1123284672184384637425649363653543830087 6361212168009677844356094001373384736, a[25,12] = 11686181254300765915 83129397248101071100533837792663395374644503559384618275734363510485/1 0470614044082410420808926371787310920931175788273150558000621095722807 1535080121028608*(495+66*15^(1/2))^(1/2)*15^(1/2)+19116389931953195739 797637098521046530277983807510930912377426235711720030722182461760131/ 4711776319837084689364016867304289914419029104722917751100279493075263 21907860544628736*(495+66*15^(1/2))^(1/2)-4224505671049153233171406799 3698913772716227502760330999553990676479270452454394452453399/15548861 8554623794749012556621041567175827960455856285786309223271483686229593 97972748288*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-2226524527 3464534369439671721020535338092592305627692535776859090192037286750097 575722049/310977237109247589498025113242083134351655920911712571572618 44654296737245918795945496576*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66 *15^(1/2))^(1/2)-58741122756010236527426978127832343714077121380697884 47755816170789214870469497570158013/7852960533061807815606694778840483 1906983818412048629185004658217921053651310090771456*(495-66*15^(1/2)) ^(1/2)-183946178835236506437757633466750538581976035499231517299488278 93542264877001589687787601/9423552639674169378728033734608579828838058 20944583550220055898615052643815721089257472*15^(1/2)*(495-66*15^(1/2) )^(1/2)+86264401130708712113446086677375779164102951704783953810116027 54761902195759989492385519/2855622012022475569311525374123812069344866 1240744956067274421170153110418658214825984*15^(1/2)+16072919085587678 4104483221710078960531681882656801045944436902101349673389944846945982 37/1427811006011237784655762687061906034672433062037247803363721058507 6555209329107412992, a[22,10] = -828786688/2261607237, a[17,13] = -426 3816413983153200192448588760470591/93243311620382237700672335844125100 0*(495+66*15^(1/2))^(1/2)*15^(1/2)-17810567793494246927942974354446874 17/72522575704741740433856261212097300*(495+66*15^(1/2))^(1/2)+4890409 671890851596858786913265540229/3190993331008636579089675493332281200*( 495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+15404935302574800991602 822823564436629/47864899965129548686345132399984218000*(495+66*15^(1/2 ))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+5799267746539103233473792702 417045641/145045151409483480867712522424194600*(495-66*15^(1/2))^(1/2) +60944529169619209342899809856869816113/652703181342675663904706350908 8757000*15^(1/2)*(495-66*15^(1/2))^(1/2)-15476916210631878385776709354 13803321/11302219590349362145536040708378800*15^(1/2)-1655491922818426 2713720331805086482661/26371845710815178339584094986217200, a[16,12] = -23020536576734933592912418285474818874519720530815247465648395780450 09423200985468614339798479742045/1343104356935991658798598905284216680 2015031990877597162493524816916542463738666427206376710067109504*(495+ 66*15^(1/2))^(1/2)*15^(1/2)-792319329291265616381463231057070373311514 879500727725199342531353993768154289597302767998992852575/447701452311 9972195995329684280722267338343996959199054164508272305514154579555475 735458903355703168*(495+66*15^(1/2))^(1/2)+737429452162108802425730963 9856463112109600685004705624040243880676462121191669419885726010677690 75/2110592560899415463826369708303769068888076455709336696963268185515 4566728732190099895734830105457792*(495+66*15^(1/2))^(1/2)*(495-66*15^ (1/2))^(1/2)+268752266421672836321602073827210934185377849215401300985 3928426333018327530288075533725009135737155/44322443778887724740353763 8743791504466496055698960706362286318958245901303375992097810431432214 613632*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+123079 9452661775024174129770408812905025522655147973122356184611390811895390 0485191828337179017106925/13431043569359916587985989052842166802015031 990877597162493524816916542463738666427206376710067109504*(495-66*15^( 1/2))^(1/2)+2377098084426487491189646295102989450079588351864125560781 866201266381651054666773249314021201985405/134310435693599165879859890 5284216680201503199087759716249352481691654246373866642720637671006710 9504*15^(1/2)*(495-66*15^(1/2))^(1/2)-18280472189378444289228602175748 4579671718006905348933974655452886877353149406481449951808917440175/45 2223689204037595555083806493002249226095353228201924663081643667223651 97773287633691504074300032*15^(1/2)-9444353728721997212298626165510967 193953653555372643606126864973867917446569402315152828775670675/122222 6187037939447446172449981087160070527981697843039629950388289793653993 872638748419029035136, a[15,13] = -606603048332993922647492911922851/2 947874218331782144082449116871680*(495+66*15^(1/2))^(1/2)*15^(1/2)+160 868397703326523929810394514891/57801455261407493021224492487680*(495+6 6*15^(1/2))^(1/2)-21222358079398713889334503007015519/1621330820082480 17924534701427942400*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+2 4430979816099254085564817485473/28611720354396709045506123781401600*(4 95+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-866287711641431 2827094002696547313/2947874218331782144082449116871680*(495-66*15^(1/2 ))^(1/2)-378396991847365253345251632120523/260106548676333718595510216 1945600*15^(1/2)*(495-66*15^(1/2))^(1/2)-23786446617715796291634345596 229/17865904353525952388378479496192*15^(1/2)+313704936756581279384432 239924553/5254677751037044820111317498880, a[17,11] = -860894248255463 26314476664211425767352894287495414314989486346860821/2478561787156852 389068153584383284313488094897128832374722468389588000*(495+66*15^(1/2 ))^(1/2)*15^(1/2)+7036530330296001919254839255948123706229511388021308 973656944198447/660949809908493970418174289168875816930158639234355299 92599157055680*(495+66*15^(1/2))^(1/2)+3415501318832112086091899257638 3167411849157891882541686237264128309/12830202192341353543411618554454 648210997197114549249939739836369632000*(495+66*15^(1/2))^(1/2)*15^(1/ 2)*(495-66*15^(1/2))^(1/2)-7576475876322330809383185602169500822747595 273068447080370266969753/855346812822756902894107903630309880733146474 303283329315989091308800*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/ 2)+185398820464774480216804066295869270126872623165675323129497604093/ 893175418795262122186722012390372725581295458424804459359448068320*(49 5-66*15^(1/2))^(1/2)-1385194297222359505923798457484412885998372560041 84505631117900226293/3304749049542469852090871445844379084650793196171 776499629957852784000*15^(1/2)*(495-66*15^(1/2))^(1/2)+103539271044799 9643103121176108421273457784120335352459294571705397/16023025694751368 97983452822227577738012505792083285575578161383168*15^(1/2)-1279134531 01066672615218540841280626102605126515825281677549047401409/4005756423 6878422449586320555689443450312644802082139389454034579200, a[21,9] = \+ -21260379985/34709636178, a[25,11] = 128920787551671343348376648677848 87364459631885161414654166871749310538128360356282255762947040291/2849 7408930222627879101099930305297509498618121353812422052614331119206690 9078761613200232238080000*(495+66*15^(1/2))^(1/2)*15^(1/2)-12480141995 6746912533228504120517562720675622369823626493412499565231034612699574 402577387154565829/664939541705194650512358998373790275221634422831588 956514561001059448156121183777097467208555520000*(495+66*15^(1/2))^(1/ 2)-2693752889365901399339990992030284330802789275903520277745199948003 /825810342708872393444191578712108928802655196252902981631633832192000 0*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+86206037733181423374 73522178116724694615075400164606062275312778891/9909724112506468721330 298944545307145631862355034835779579605986304000*(495+66*15^(1/2))^(1/ 2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+9415833269344714094602052035425875 010781868302770380527360876893541/834151861322093326711304624961726190 70975272348778078952690286080000*(495-66*15^(1/2))^(1/2)-2351260393591 35435266466723342816311162099922849416444217438064061/3128069479957849 9751673923436064732151615727130791779607258857280000*15^(1/2)*(495-66* 15^(1/2))^(1/2)-511975650628347237449832088490580845648606388072013087 013572919932259111943700950216968199185543/816879043863875492029925059 427260780370558259006866039944178133979666039460913731077969543680000* 15^(1/2)-1290393536512988930091830278930902869460971301936267208599441 7376461195912450807078157049600542041/20149683081975595470071484799205 765915807103722169362318623060638165095640035872033256582077440000, a[ 15,12] = -118673264953965898239568178186990539375320133228945328781256 3780081271262993935492821075/29859252522666041600461145767521730374799 58799054639506083291835447600675497430644543488*(495+66*15^(1/2))^(1/2 )*15^(1/2)-62787138053343564726123359273066353779726848092727005318663 855357084118475306201635375/124413552177775173335254774031340543228331 616627276646086803826476983361479059610189312*(495+66*15^(1/2))^(1/2)+ 2069579244553928310941747084237830205961522603833836536321316258542043 85359909929293975/4478887878399906240069171865128259556219938198581959 259124937753171401013246145966815232*(495+66*15^(1/2))^(1/2)*(495-66*1 5^(1/2))^(1/2)+5891979785833502419846398661311140104554853460303611448 42253038902651606518412193051375/2687332727039943744041503119076955733 7319629191491755554749626519028406079476875800891392*(495+66*15^(1/2)) ^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+109931052493305895825340776604 8281629166843436849055712163350435869590310234999724676375/74648131306 6651040011528644188043259369989699763659876520822958861900168874357661 135872*(495-66*15^(1/2))^(1/2)+491265714510885901435656358464046529229 4459913660535043858759590531421311858910677650975/89577757567998124801 38343730256519112439876397163918518249875506342802026492291933630464*1 5^(1/2)*(495-66*15^(1/2))^(1/2)-26100793447835537272907704911698240760 37127743703573983625462105790320139194698784616725/2714477502060549236 40555870613833912498178072641330864189390166858872788681584604049408*1 5^(1/2)-12789425881975715683435041476352781333133614948791446840597433 21705314597196773722377875/6786193755151373091013896765345847812454451 8160332716047347541714718197170396151012352, a[21,10] = -81770692250/1 17233567091, a[18,9] = 58430854255030432070621801602135091580895842077 78603496061810016606405773989744218675/1618793792195454314326103615159 47869757665121017243493406225592082722798679429383530496*(495+66*15^(1 /2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+52386928616016519180924708 766339483104148690883985181316719179227214879609227195990525/436817372 4971860848181549437731926644254455646497046647469579468898869170714761 1428864*15^(1/2)*(495-66*15^(1/2))^(1/2)+12743159635160699596199822397 2483480728072236724532962935998385342950133745598128278175/45865824112 2045389059062690961852297646717842882189897984305844234381262925049920 003072*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-359609019171790 0495311282254569337139360867187537629930565174985393074773162804318739 8376598916909373086010218882431250213048875/77801281619871366493921082 0439733197294280574287264570352168627947071266006287273344593751980649 86812662396372358951510966377792*(495+66*15^(1/2))^(1/2)*15^(1/2)-9669 6702427815531868039302296149065719199151557450905776561953160733596973 52763101626940226811743783389217017725058051700571375/5715429320100743 1767802447782533200903161107385657636022197879004376217888432490236517 4473447676670800091066096300837546125824*15^(1/2)+20670610790243638387 17693546658810214399316195139295817954075823298971474400156388153775/3 0577216074803025937270846064123486509781189525479326532287056282292084 1950033280002048*(495-66*15^(1/2))^(1/2)-16299377019098876971786761477 8601649091999597524261566556548368328211739941770860234258328734706686 82274605611331302469947384125/3577070419304430643398670438803371022042 6693070678830820789362204463046712932748199751436872673557155247078791 88917310849028864*(495+66*15^(1/2))^(1/2)-1611432561961693218541781015 4973528628406360067119665672316357482444559905465750545380831125173659 8122557269402242991954956908575/14792875887319570575195927661361534351 4064042645231528528041569187797269828884092376868687245281020677670628 8719837461884090368, a[19,10] = -2151795225636507451109162310143923559 0229/2888839913580347511855374513290961053081600*(495+66*15^(1/2))^(1/ 2)*(495-66*15^(1/2))^(1/2)-1279049075494197865114395209629986016443/10 504872413019445497655907321058040193024*(495-66*15^(1/2))^(1/2)+226592 14536067896633614417137680110863011/2626218103254861374413976830264510 04825600*15^(1/2)*(495-66*15^(1/2))^(1/2)+2387182673647655859044296843 852497619629/577767982716069502371074902658192210616320*(495+66*15^(1/ 2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-526043757001214561547116994 9063600761613/52524362065097227488279536605290200965120*(495+66*15^(1/ 2))^(1/2)*15^(1/2)+4298989093388165599392306128065847615403/9549884011 83585954332355211005276381184-4990702143348148007996489571831817889093 1/23874710029589648858308880275131909529600*15^(1/2)+67840389553449804 388390138705221657103209/262621810325486137441397683026451004825600*(4 95+66*15^(1/2))^(1/2), a[19,9] = -355868881345850140430908227430848059 927075/2370859701228665121582342957138319965275136*(495+66*15^(1/2))^( 1/2)*15^(1/2)-614359056788987858948697620424946665646525/1303972835675 7658168702886264260759809013248*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/ 2))^(1/2)+178220451175383309209554257744380321128025/52158913427030632 674811545057043039236052992*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*1 5^(1/2))^(1/2)-45514293408354922515465996527236381825/1165041622225388 2661338294629672333981696*15^(1/2)*(495-66*15^(1/2))^(1/2)-39479405651 2199861195188666930860982817975/43106540022339365846951690129787635732 2752*(495-66*15^(1/2))^(1/2)+77928163144234885845658078167344358344119 069697625775/3934183930271671997059445068416867252248888700155904-1441 98268363280897966135647057428003102275/1077663500558484146173792253244 69089330688*15^(1/2)+1307533573911285247145069625249145489665775/11854 29850614332560791171478569159982637568*(495+66*15^(1/2))^(1/2), a[10,7 ] = 5620721356384469075/3545698345096452809913, a[10,1] = -17689789589 /433566079650, c[2] = 39/88, a[10,8] = 10436097770007465488281250/3448 0588269890588098919643, a[22,7] = 75739731820032/659439747650125, a[8, 1] = 5673493348726844397056253/158826937830690185546875000, a[8,5] = 8 3874430427399451044535741888/1824862440224861393218994140625, a[10,6] \+ = -46012134893475328/76616158064651325, a[8,6] = -98689436493374054472 6981696/54826948196505455902099609375, a[9,1] = 2526695731/13381669125 0, a[9,6] = 24467886602752/291937806983125, a[7,4] = 46426400659946403 657/610018749211776263125, a[7,1] = 1534965908593628117013/29251907676 28355078125, a[7,6] = 11005843237812473721/5193844391648046875, a[8,7] = 5672317641427470160608354913689/40381865475711712996627696578125000 , a[23,7] = -253319312163456/659439747650125, a[9,8] = 109505036293571 8750/11254693571026117461, a[10,9] = 110139186925/164279737386, a[7,5] = -156479396217570600333298656/73273972406867506457421875, a[5,1] = 8 719384129925/100440274363616, a[5,4] = -2900926450625/200880548727232, a[16,9] = 50063971577508796989932533520141190241209119641155965/57049 9532233022530103277424880151739690209749047396608*(495+66*15^(1/2))^(1 /2)*15^(1/2)-335814103222680807523386900403987635039205386347311925/57 0499532233022530103277424880151739690209749047396608*(495+66*15^(1/2)) ^(1/2)+25861692976938895204697334600993725104726707177012055/627549485 4563247831136051673681669136592307239521362688*(495+66*15^(1/2))^(1/2) *15^(1/2)*(495-66*15^(1/2))^(1/2)-719742462005738228503762575409098422 07760493224762025/1882648456368974349340815502104500740977692171856408 8064*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+33115680250011191 448970159142023181041838885854451525/570499532233022530103277424880151 739690209749047396608*(495-66*15^(1/2))^(1/2)+450633836129874489449406 02137484447778392516290975315/5704995322330225301032774248801517396902 09749047396608*15^(1/2)*(495-66*15^(1/2))^(1/2)+6831137283647196302884 273272119561096079229960021975/740908483419509779354905746597599661935 3373364251904*15^(1/2)-23404899062582426102055804737678235342037391391 480725/2469694944731699264516352488658665539784457788083968, a[5,3] = \+ 6944404306875/200880548727232, a[9,7] = 11944081475573/132247896202918 6830, a[6,1] = 1167025/30304848, a[6,4] = 390625/3949854672, a[6,5] = \+ 404856925/3216589572, a[3,1] = 28078/126711, a[3,2] = 112211/506844, a [4,1] = 101/608, a[4,3] = 303/608, a[12,9] = -281211230909242277181931 3341069996795037876904500378145129471863425104791510387562227247473304 56172743380029/1777219414844154523916564814426187150039165643225617392 1137707234507471866579929942450646636509884128799622+67637408773077542 9952449408713302591862747979275682175058845619651319936902380422980097 6446366900597141/38492948123113591594467507351660973576763388417275663 680176970401792228430972341222548509067597756397660*(495+66*15^(1/2))^ (1/2)*15^(1/2)+3647611941783523639106839681293674145014752901446257471 407287711249070189350233573351451468961784464641/641549135385226526574 4584558610162262793898069545943946696161733632038071828723537091418177 932959399610*(495+66*15^(1/2))^(1/2)-594585803588270123266011580156929 6816414941113900029074561383838890594834114310825436409542274213823424 7/12830982707704530531489169117220324525587796139091887893392323467264 076143657447074182836355865918799220*15^(1/2), a[11,7] = 1899397497348 5227904952341910291889381118708193023824550/12060034665080522994492944 014191159522924996678137100092619*(495+66*15^(1/2))^(1/2)*15^(1/2)+373 64999682821940340524213896266047309233288831638123551875/4288012325361 9637313752689828235233859288877077820800329312+83870865016481720496920 100044907813868655001976949702225/134000385167561366605477155713235105 8102777408681900010291*15^(1/2)+10188221773942597558585056227974439386 8968529126539562125/40200115550268409981643146713970531743083322260457 00030873*(495+66*15^(1/2))^(1/2), a[12,7] = 12039560526382767444395981 6224372224515900903963473583877146503450993695151583738412984486608167 592412726499223108027/295063494385794976532558803599500929144693875761 874497381130571566907020515112489594526125158556423774229187413128270- 4574290713578312637346253714366174333264683073453621195923343653987480 49518495290294028445214608755002615341437/1278161119280030221063715848 3842362102867397693821723949799894804717653043756226536475032495497354 2895485894482620*(495+66*15^(1/2))^(1/2)*15^(1/2)-19409130444629839503 0544188331511201788876606372749180626268993467082976330239046596838838 518978497037496895389/127816111928003022106371584838423621028673976938 21723949799894804717653043756226536475032495497354289548589448262*(495 +66*15^(1/2))^(1/2)+12780472568351676678614075153447418157156395227824 198601931888244825804433658120101910199505548093911257104546897/127816 1119280030221063715848384236210286739769382172394979989480471765304375 62265364750324954973542895485894482620*15^(1/2), a[12,6] = 25108552108 3237108777248115694608871632336099720342563852044971041903837740861882 751285544579556435153690263824896/109059581580566357404472531727925474 9804567879870625150516917051380618003386804945891914477353935936889365 5280625-17224203165087627219883985321280669596094555813328784551128281 5171786743955334377646964770699567651065020416/69046901918687152519450 7956492089110354268996435976670159491643799061730539287715031284885947 411166121788875*(495+66*15^(1/2))^(1/2)*15^(1/2)-248326505096660202626 5671372002316578111995428215292500398470208821705013352242278156416063 385897318005911552/299203241647644327584286781146571947820183231788923 2237357797123129267499003580098468901172438781719861085125*(495+66*15^ (1/2))^(1/2)+989414013347979104246599002035391609046115745254951412455 31026292768496305967031119826379359226270888023523328/1496016208238221 6379214339057328597391009161589446161186788985615646337495017900492344 505862193908599305425625*15^(1/2), a[12,1] = 4536078586418429346535504 9313006780143265609187196347220787941439290058648676039911927050322004 12298285836937/1172606656536926424103513708860616450309534511569707813 821852440031214272905699213661387392662314110822638750-536443589551657 3669420621604632495515121639476730674940078684840492375553763895312868 3448772183267658/13367151790715393045193549113238448870986337808439152 94531482553071844639268719963591518065572670919625*(495+66*15^(1/2))^( 1/2)*15^(1/2)-35279136548504838526715366339693933451439371283610203456 07398270646119601220687075114545149449903002764/2539758840235924678586 7743315153052854874041836034390596098168508365048146105679308238843245 880747472875*(495+66*15^(1/2))^(1/2)+136142098290337291849630976028295 8026582387432115444780575878975425366396745823033150091119530031197073 96/1269879420117962339293387165757652642743702091801719529804908425418 25240730528396541194216229403737364375*15^(1/2), a[13,1] = 49032552280 925998159/1175735428193602866744+1095947848209481903/17636031422904043 00116*15^(1/2)-7475873553226772611/6466544855064815767092*(495+66*15^( 1/2))^(1/2)+5352672249559690835/116397807391166683807656*(495+66*15^(1 /2))^(1/2)*15^(1/2), a[13,10] = 1036874421023669248333689/196556516582 25505790722624+587392382348212140203319/39311303316451011581445248*15^ (1/2)-259903711964372646898395/432424336480961127395897728*(495+66*15^ (1/2))^(1/2)*15^(1/2)-185211952023363244825707/10810608412024028184897 4432*(495+66*15^(1/2))^(1/2), a[13,9] = -162384252323883015198625/1065 0929594044147603352784*15^(1/2)+122383789982155868115425/4686409021379 42494547522496*(495+66*15^(1/2))^(1/2)*15^(1/2)+2265050157236155663680 125/14201239458725530137803712-120963267116719524969875/26035605674330 138585973472*(495+66*15^(1/2))^(1/2), a[11,9] = 2874397467399404298522 6536878679230542480801150/17539834082278474517747485443485973779433565 53311*(495+66*15^(1/2))^(1/2)*15^(1/2)+2919917156154994497064347556110 25178229310468246875/5612746906329111845679195341915511609418740970595 2+1071091749312998743815087165568469189531652126075/175398340822784745 1774748544348597377943356553311*15^(1/2)+35325978470435165189631258165 5773178339865859125/1753983408227847451774748544348597377943356553311* (495+66*15^(1/2))^(1/2), a[13,11] = 1451483888900070743380148776770707 95676982463/1292134149111656271966122106166449116832367968+28938477630 8176063330366809130393474082987849/15505609789339875263593465273997389 401988415616*15^(1/2)-274115022589237178977231727148612453501574585/51 1685123048215883698584354041913850265617715328*(495+66*15^(1/2))^(1/2) *15^(1/2)-24857236960157204698513148202889704149473711/533005336508558 2121860253687936602606933517868*(495+66*15^(1/2))^(1/2), a[13,12] = 64 43329452303863103711607826787077523918806208419166908252875/4818690817 7657952343922139838789383990650083655460342201310944-17149160115186691 1759544690813927486314725154315627741296625/90350452833108660644854012 19773009498246890685398814162745802*15^(1/2)-7090202015237773474138169 907437581683377756583694650388556125/238525195479406864102414592202007 4507537179140945286938964891728*(495+66*15^(1/2))^(1/2)+11992999518505 1271338440264753881499366865434058651556229025/14456072453297385703176 6419516368151971950250966381026603932832*(495+66*15^(1/2))^(1/2)*15^(1 /2), a[11,1] = 36250222198273693341131051026706723455692594208/4999430 483339723613905130972844241982587217763497*(495+66*15^(1/2))^(1/2)*15^ (1/2)+4861393204748134765917698036236910531265683193125/22219691037065 43828402280432375218658927652339332+1484428141946824012327061188206437 00603232458928/555492275926635957100570108093804664731913084833*15^(1/ 2)+143068868482411022237818049543538495099770530576/166647682777990787 1301710324281413994195739254499*(495+66*15^(1/2))^(1/2), a[12,8] = -34 0659803958984878209931438027146134138660774601249541025577546738764928 298414303784311737346180734782282094165416156250/308280706336293206809 6196297896660310632239903888789451271870099917142127007003940833673042 3089458130745933155208375541+41194832565673882203999777509211923642129 3381998104532443768303827226809809726860648464826292767320320631125000 00/3514251750809858381605960008089851362393260494840337712198477138170 31123765375550406811559375414180212098687404767*(495+66*15^(1/2))^(1/2 )*15^(1/2)+26719277739082859443666346678807717091479495492196588967171 26925052525708719967923978519767792683025169075208750000/6677078326538 7309250513240153707175885471949401966416531771065625235913515421354577 29419628132869424029875060690573*(495+66*15^(1/2))^(1/2)-2086803057652 7122590278134042922470681076456345098996480647230076556364761794880724 541437909825056194904756420000000/667707832653873092505132401537071758 8547194940196641653177106562523591351542135457729419628132869424029875 060690573*15^(1/2), a[11,10] = -37768595712400175213251809663599923911 4085793540/40806683117092499328338457010324585780808416836571*(495+66* 15^(1/2))^(1/2)*15^(1/2)-181781480924529622996033569801045149915635836 5559525/652906929873479989253415312165193372492934669385136-1424811775 0070402721956352075060738058061271231816/40806683117092499328338457010 324585780808416836571*15^(1/2)-484426665991669901588562627983527245301 9208362650/40806683117092499328338457010324585780808416836571*(495+66* 15^(1/2))^(1/2), a[11,8] = -275413882637225084153879395633151790451174 339761235000000000/173657067762976556173412167504047698362275139689158 18501163621*(495+66*15^(1/2))^(1/2)*15^(1/2)-9030706067191500671304838 647417536518671082752129344726562500/192952297514418395748235741671164 1092914168218768424277907069-11310663799138970197488474860123128359224 21062622057500000000/1929522975144183957482357416711641092914168218768 424277907069*15^(1/2)-109756253028446039591902781216552083991001737789 6612500000000/57885689254325518724470722501349232787425046563052728337 21207*(495+66*15^(1/2))^(1/2), a[17,9] = 61217740076995834578342595547 1231078477005/881010437527361508680786914250849809507584*(495+66*15^(1 /2))^(1/2)*15^(1/2)+7168264585653936119575060989007511226465125/881010 437527361508680786914250849809507584*(495+66*15^(1/2))^(1/2)-140764211 411413887441926681309100606007705/235729819770834565836210552731984138 2195968*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-38998 031250916842420972974426336223143470825/872200333152087893593979045108 34131141250816*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-9680426 932551728403108583240026318485469625/881010437527361508680786914250849 809507584*(495-66*15^(1/2))^(1/2)-518469213368660090916794129622290057 8262135/2643031312582084526042360742752549428522752*15^(1/2)*(495-66*1 5^(1/2))^(1/2)+6559804249926315934853561094124426777066675/24027557387 1098593276578249341140857138432*15^(1/2)+15296729363916364958642717710 438043125084525/80091857957032864425526083113713619046144, a[17,10] = \+ -523474846258041930813933069750681477537147/87099209683313173801014888 7671193233701120*(495+66*15^(1/2))^(1/2)*15^(1/2)+45393675438869823180 5806756431700677913713/4354960484165658690050744438355966168505600*(49 5+66*15^(1/2))^(1/2)+127559989212810280117526964548619108824367/323679 4954447449026389066812291596476592000*(495+66*15^(1/2))^(1/2)*15^(1/2) *(495-66*15^(1/2))^(1/2)+1185878186169339080648273599898636788893537/2 3952282662911122795279094410957813926780800*(495+66*15^(1/2))^(1/2)*(4 95-66*15^(1/2))^(1/2)+8798257823408093625269275080577035600962103/4354 960484165658690050744438355966168505600*(495-66*15^(1/2))^(1/2)+395241 1632936359566769644071717315106143261/43549604841656586900507444383559 66168505600*15^(1/2)*(495-66*15^(1/2))^(1/2)-5632305914388827225216573 6895642219352623/4498926120005845754184653345409055959200*15^(1/2)-245 6212915091625339793591322898399091106073/19795274928025721318412474719 7998462204800, a[25,7] = -30334395520464588/16485993691253125, a[12,11 ] = 1/5499, a[14,9] = -261797099571548704054336684175/6747960349884233 978989776419904*(495+66*15^(1/2))^(1/2)*15^(1/2)+593450511455747479579 1690125/210873760933882311843430513122*(495+66*15^(1/2))^(1/2)+1251919 4985069398569031057125/6747960349884233978989776419904*(495+66*15^(1/2 ))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+1109504659418215070669199437 5/20243881049652701936969329259712*(495+66*15^(1/2))^(1/2)*(495-66*15^ (1/2))^(1/2)+8033012316259971085532118125/1533627352246416813406767368 16*(495-66*15^(1/2))^(1/2)+1425332275365304763708338125/34080607827698 151409039274848*15^(1/2)*(495-66*15^(1/2))^(1/2)-147777874438893513831 99282875/18589422451471718950385059008*15^(1/2)-4678476059868941353268 7625/1161838903216982434399066188, b[2] = -11/100, b[11] = -29/100, b[ 10] = -27/100, b[7] = -21/100, b[6] = -19/100, b[3] = -17/100, a[20,15 ] = 13/1036, a[20,14] = -50/501, a[19,14] = 21/32, a[12,10] = -11/342, a[23,1] = 28078/126711, a[15,11] = -109730074138310288883902771590462 5360709712131221694471788967065599/22388395214718747101577535278410507 682903636619968393176006827622400*(495+66*15^(1/2))^(1/2)*15^(1/2)+277 794763764582127688271249175901426130830243934789145405582248649/149255 9680981249806771835685227367178860242441331226211733788508160*(495+66* 15^(1/2))^(1/2)+970165972365189633475199730235431133235016867267009871 9512199631/62085465721488962550593165057777038112253782223441762588926 49676800*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-6126 84386336773590057667239352030333811894239017904866736513774537/1055452 91726531236336008380598220964790831429779850996401175044505600*(495+66 *15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+41683417798249786705611926554 570672840097491555226126486675979647/639668434706249917187929579383157 362368675331999096947885909360640*(495-66*15^(1/2))^(1/2)-700940634923 2470923845843111584742313487794670152566793436229067/35537135261458328 7326627544101754090204819628888387193269949644800*15^(1/2)*(495-66*15^ (1/2))^(1/2)+127141112469296965830509721891596518697563452055501846938 5189147317/20353086558835224637797759344009552439003306018153084705460 75238400*15^(1/2)-3050873466513682484425378607716525120140439312408946 69510340799613/1356872437255681642519850622933970162600220401210205647 03071682560, a[24,23] = 274193/636804, a[22,1] = 1167025/30304848, a[2 0,16] = 4573876925/1493463420644832*(495+66*15^(1/2))^(1/2)*(495-66*15 ^(1/2))^(1/2)+7986761337425/7582198904812224*15^(1/2)*(495-66*15^(1/2) )^(1/2)+37543791335/995642280429888*15^(1/2)+78669144834545/9956422804 29888-6267380675/14360225198508*(495+66*15^(1/2))^(1/2)*15^(1/2)+53874 14718575/32856195254186304*(495-66*15^(1/2))^(1/2)-204724622275/896078 0523868992*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-15 0291531475/62227642526868*(495+66*15^(1/2))^(1/2), a[17,14] = -8634388 9643783/1423361572162200*(495+66*15^(1/2))^(1/2)*15^(1/2)-487532940609 9/718869480890-47503534664513/28754779235600*15^(1/2)-46830515603137/1 89781542954960*(495+66*15^(1/2))^(1/2)-56659550856457/355840393040550* 15^(1/2)*(495-66*15^(1/2))^(1/2)-21856354431557/869832071876900*(495+6 6*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-125819226953867/189781542954 960*(495-66*15^(1/2))^(1/2)-34670051930677/6262790917513680*(495+66*15 ^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[19,16] = 23318577/23 8751570176*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-19320841301 /23636405447424*15^(1/2)*(495-66*15^(1/2))^(1/2)-150970273/59687892544 *15^(1/2)+5937215723/119375785088+3596875/5595739926*(495+66*15^(1/2)) ^(1/2)*15^(1/2)-18335887105/2626267271936*(495-66*15^(1/2))^(1/2)-2949 2131/2148764131584*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^ (1/2)+493761427/358127355264*(495+66*15^(1/2))^(1/2), a[18,15] = 19990 678222944016799007968810994718650113/327238104296070081420144443835873 6068377200*(495+66*15^(1/2))^(1/2)*15^(1/2)-88393911258485420678134471 781658464932823/6326603349724021574122792580826889732195920*(495+66*15 ^(1/2))^(1/2)+87051869427126857828192491903627294702861/95857626510970 0238503453421337407535181200+4396995932514786753230371514007047435591/ 45646488814747630404926353397019406437200*15^(1/2)+4529063312212383619 22696065114103183779/7083075850564287476626503113330597550600*(495-66* 15^(1/2))^(1/2)+2225327884102816028172189393256449625076/1078398298248 41276831638509900458347707885*15^(1/2)*(495-66*15^(1/2))^(1/2)+1107045 60801946412846083674130999701608833/4744952512293016180592094435620167 2991469400*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+10101069885 562826853855484798548336576859/118623812807325404514802360890504182478 67350*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[16,1 5] = 418320022266565899999989/730900563458308064970225-119156065868215 739462012/2192701690374924194910675*15^(1/2)+3495695795945596671104/14 6180112691661612994045*(495+66*15^(1/2))^(1/2)-65831485804562645782499 /10963508451874620974553375*(495+66*15^(1/2))^(1/2)*15^(1/2)-464154937 1926342688144/32890525355623862923660125*(495+66*15^(1/2))^(1/2)*15^(1 /2)*(495-66*15^(1/2))^(1/2)+7610546603899839370279/6578105071124772584 732025*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+177142509891462 70037792/730900563458308064970225*(495-66*15^(1/2))^(1/2)-150290543995 63919460349/10963508451874620974553375*15^(1/2)*(495-66*15^(1/2))^(1/2 ), a[2,1] = 39/88, a[25,14] = -49414375111/1558819416000*(495+66*15^(1 /2))^(1/2)*(495-66*15^(1/2))^(1/2)-165545310587/584557281000*15^(1/2)* (495-66*15^(1/2))^(1/2)-2733053737/843517000*15^(1/2)-16561593209/3374 068000-24743106247/2338229124000*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495 -66*15^(1/2))^(1/2)-1363545178441/1558819416000*(495-66*15^(1/2))^(1/2 )-4721773003/27836061000*(495+66*15^(1/2))^(1/2)-3651713009/2783606100 0*(495+66*15^(1/2))^(1/2)*15^(1/2), a[18,16] = -4368406205678939614875 35263885/2696403426422943408076694358192*15^(1/2)-36954511491802290418 8418990316329/625565594930122870673793091100544-1825727014541939573961 7682201851/625565594930122870673793091100544*(495+66*15^(1/2))^(1/2)-5 983444403437426102960354262185/938348392395184306010689636650816*(495+ 66*15^(1/2))^(1/2)*15^(1/2)+107005288406322056654355448927205/37533935 69580737224042758546603264*(495-66*15^(1/2))^(1/2)+8405543789264226154 1931143840615/11260180708742211672128275639809792*15^(1/2)*(495-66*15^ (1/2))^(1/2)+1012338871884896232541346539753/3753393569580737224042758 546603264*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+457 8034999918124747163891877361/3753393569580737224042758546603264*(495+6 6*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[18,14] = 4230042827581684 523751752104701705965843/222105102055991070418278663608820791814660*(4 95+66*15^(1/2))^(1/2)*15^(1/2)+875276118514362376497720464680666885577 13/17947887034827561245921508170409760954720+5125151437791361308235261 701221815485791/6730457638060335467220565563903660358020*15^(1/2)+2558 836067301785957586054852230517925601/158646501468565050298770474006300 56558190*(495+66*15^(1/2))^(1/2)+7231113621317229409561639021515304236 9611/848037662395602268869791261051861205110520*15^(1/2)*(495-66*15^(1 /2))^(1/2)+477343459314808814429839892297318102330687/1130716883194136 358493055014735814940147360*(495-66*15^(1/2))^(1/2)+485221217148703115 49903684616247887004701/18656828572703249915135407743140946512431440*( 495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+20583882814114 410685607175778197029916327/138198730168172221593595612912155159351344 0*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[17,16] = -3399762 661/1690419432009*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-2179 2326747/1690419432009*15^(1/2)*(495-66*15^(1/2))^(1/2)+15024834410/512 24831273*15^(1/2)+843590793/80496163429*(495+66*15^(1/2))^(1/2)*15^(1/ 2)-167470020833/3380838864018*(495-66*15^(1/2))^(1/2)-656911562/169041 9432009*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+23471 698999/482976980574*(495+66*15^(1/2))^(1/2)+61299209505/51224831273, a [25,16] = 571468963/226284800+120370717/226284800*15^(1/2)-1751767/203 6563200*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-59032 9/169713600*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-519210887/ 22402195200*15^(1/2)*(495-66*15^(1/2))^(1/2)-28396679/298695936*(495-6 6*15^(1/2))^(1/2)+5795203/339427200*(495+66*15^(1/2))^(1/2)*15^(1/2)+7 675999/113142400*(495+66*15^(1/2))^(1/2), a[16,14] = -1493917246565840 2889149/8976679102595142257770+219312724397669057743089/35906716410380 5690310800*15^(1/2)+659570008376205270318919/1777382462313838167038460 0*(495+66*15^(1/2))^(1/2)*15^(1/2)-11242117128844970311919/15798955220 5674503736752*(495+66*15^(1/2))^(1/2)-1749823831034832192631/430880596 92456682837296*(495-66*15^(1/2))^(1/2)+44646359022841677628361/1615802 238467125606398600*15^(1/2)*(495-66*15^(1/2))^(1/2)-307022029936093312 5107/1974869402570931296709400*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2 ))^(1/2)+18545262695413350953267/11849216415425587780256400*(495+66*15 ^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[25,15] = -3460846507 /13082090000+48037163/3270522500*15^(1/2)-108037591/64756345500*(495+6 6*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-412156237/462545325 00*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-2890671767/58869405 000*15^(1/2)*(495-66*15^(1/2))^(1/2)-4618638191/19623135000*(495-66*15 ^(1/2))^(1/2)-848300537/15418177500*(495+66*15^(1/2))^(1/2)+429589031/ 129512691000*(495+66*15^(1/2))^(1/2)*15^(1/2), a[19,15] = 190841306185 7/289859328166800-2931986748899/14348036744256600*(495+66*15^(1/2))^(1 /2)*(495-66*15^(1/2))^(1/2)-1254438533/217394496125100*15^(1/2)*(495-6 6*15^(1/2))^(1/2)-857188607021/289859328166800*15^(1/2)+5379357542731/ 9565357829504400*(495+66*15^(1/2))^(1/2)*15^(1/2)+136382862089/1449296 64083400*(495-66*15^(1/2))^(1/2)-1935518588/85404980620575*(495+66*15^ (1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+34225020981/216901538084 00*(495+66*15^(1/2))^(1/2), a[17,15] = -4863804455866/35119141675695*1 5^(1/2)-62267936631107/9657763960816125*(495+66*15^(1/2))^(1/2)*15^(1/ 2)-5561334169199/58531902792825+2301512129984/643850930721075*(495+66* 15^(1/2))^(1/2)-331440117351157/9657763960816125*15^(1/2)*(495-66*15^( 1/2))^(1/2)-27173454387092/214616976907025*(495-66*15^(1/2))^(1/2)-379 86613754722/28973291882448375*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66 *15^(1/2))^(1/2)-25832795272667/5794658376489675*(495+66*15^(1/2))^(1/ 2)*(495-66*15^(1/2))^(1/2), a[15,14] = 115988619653/292402973068800*(4 95+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-2683538373203/8772089192 06400*15^(1/2)*(495-66*15^(1/2))^(1/2)-1968290835/88606961536*15^(1/2) +496725643/7310074326720*(495+66*15^(1/2))^(1/2)*15^(1/2)-69268797311/ 7310074326720*(495-66*15^(1/2))^(1/2)-124493701507/877208919206400*(49 5+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-52034279227/1169 6118922752*(495+66*15^(1/2))^(1/2)-74896827627/88606961536, a[21,1] = \+ 1534965908593628117013/2925190767628355078125, a[15,9] = 6338951404167 8039114265201628308697725/177634065112491212783473366421768720384*(495 +66*15^(1/2))^(1/2)*15^(1/2)-5994495377932596821795043959647438677125/ 3197413172024841830102520595591836966912*(495+66*15^(1/2))^(1/2)-58692 7014740721935830387224699720472375/52757317338409890196691589827265309 954048*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+792527 4936394787837453788892272777275575/10551463467681978039338317965453061 9908096*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+43655328423631 97027475991758966784554875/3197413172024841830102520595591836966912*(4 95-66*15^(1/2))^(1/2)-194972669509543955301656024896547363525/15987065 86012420915051260297795918483456*15^(1/2)*(495-66*15^(1/2))^(1/2)+8375 4126586110644280691704028211762275/18167120295595692216491594293135437 312*15^(1/2)-2988445635117352704505746055648829302375/9689130824317702 5154621836230055665664, a[18,13] = -3056004933157205556790148421552416 84367020222119264148349548133336308182412832197537/3256042879322643548 93689648529460589911190126355463998435653228466053076361523641600*(495 +66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-19752132416622076862594456 7239798771583989327433318346783601705495197399290502949/80087634772792 29508404408907159105418909635142548799646685685469944241350883600*(495 -66*15^(1/2))^(1/2)-10187485126941781093913191486652146182845306809721 675743001592205156693166115699/179038225440174392203851057405552855672 2313644641647825207179186893722617363200*15^(1/2)*(495-66*15^(1/2))^(1 /2)-241295471832628828149971850650908505325730192791616990377777272852 79340788139240051/1221016079745991330851336181985477212166962973832989 99413369960674769903635571365600*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495 -66*15^(1/2))^(1/2)+25653884501146099146676483794880477495946548598470 5648741526280186929564500121638337/88801169436072096789188085962580160 884870034460581090482450880490741748098597356800*(495+66*15^(1/2))^(1/ 2)*15^(1/2)+3421753718171991841875105232051230717045834060382685988879 99231125105391796795948789/8969815094552737049412937976018198069178791 35965465560428796772633755031298963200+1704355823065934493267387393245 95543646943918889343464899535898127796089046758287/2002190869319807377 101102226789776354727408785637199911671421367486060337720900*15^(1/2)+ 4372357793745988000497129953214099097313499400980253824493391287428347 3618621104849/29600389812024032263062695320860053628290011486860363494 15029349691391603286578560*(495+66*15^(1/2))^(1/2), a[20,13] = -783823 9033617127272918554018074439195/55317291376776610772170386390401622604 8*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-19331104079432470130 6535416619327731075/553172913767766107721703863904016226048*(495-66*15 ^(1/2))^(1/2)-69718449299923616075612199046796736715/16595187413032983 23165111591712048678144*15^(1/2)*(495-66*15^(1/2))^(1/2)-1820126529954 499188102801413491656415/1659518741303298323165111591712048678144*(495 +66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+21675022039698373 6147998000913185820269/29534484573459373716743207740907071232+24599342 82985284946018338927580704993/15903184001093508924400188783565346048*1 5^(1/2)+4892899586426696310153662786810530160425/204673978094073459857 03042964448600363776*(495+66*15^(1/2))^(1/2)+7865302385014539228109068 99584913106669/61402193428222037957109128893345801091328*(495+66*15^(1 /2))^(1/2)*15^(1/2), a[14,1] = 8556079509657666761723/5578979874957880 269648+5812535369870579924495/552319007620830146695152*(495+66*15^(1/2 ))^(1/2)*15^(1/2)+31773543633165214073989/552319007620830146695152*(49 5+66*15^(1/2))^(1/2)-99315299134019944838123/2816826938866233748145275 2*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-29356627452550229437 31/31614219291427988194672*(495-66*15^(1/2))^(1/2)+5893191651919135641 27/1859659958319293423216*15^(1/2)-54905516106695663805167/25607517626 05667043768432*15^(1/2)*(495-66*15^(1/2))^(1/2)-2097848562299629271968 7/28168269388662337481452752*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66* 15^(1/2))^(1/2), a[15,1] = 2321094699412275858129891984708701/22245140 483572994400275713991669760*(495+66*15^(1/2))^(1/2)*15^(1/2)+167988835 36211824920144287856245/494336455190510986672793644259328*(495+66*15^( 1/2))^(1/2)-3894420709139689924988362031238691/52434973996993486800649 8972660787200*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-88608004 9830721400360956597191109/3177877211938999200039387713095680*(495-66*1 5^(1/2))^(1/2)+3289096597943947246887140021725723/14444896417904541818 36085324134400*15^(1/2)-2011985254838826786359655627979213/15889386059 694996000196938565478400*15^(1/2)*(495-66*15^(1/2))^(1/2)-279293761233 5351455284866817848047/524349739969934868006498972660787200*(495+66*15 ^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+10393852690622568797814 8938712023/32099769817565648485246340536320, a[25,1] = -36140638630697 902034896926287498832577/14918659998274712319184796318242560000*(495+6 6*15^(1/2))^(1/2)*15^(1/2)-944082780731031064844598617375201471/110508 592579812683845813306061056000*(495+66*15^(1/2))^(1/2)+510862554049017 798054791876117311841/887055459356874786546123024327936000*(495+66*15^ (1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+263426166862833108899557840948931 74507/1657628888697190257687199590915840000*(495-66*15^(1/2))^(1/2)-11 75672536943812772325376233621682909/1808322424033298462931490462817280 0*15^(1/2)+1389359394679032887528595210889829651/331525777739438051537 439918183168000*15^(1/2)*(495-66*15^(1/2))^(1/2)-360798261146212361008 53644879276413199/150693535336108205244290871901440000+151659281119471 06907569089249137166679/98463155988613101306619655700400896000*(495+66 *15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[16,13] = -5008995 7728648885291210634556861014207223642623/82675783318059004473031720491 0090891816255882000*(495+66*15^(1/2))^(1/2)*15^(1/2)+97307976884973653 38206022389185748546702810361177/9204570542743902497997531547999011928 887648819600*(495+66*15^(1/2))^(1/2)-261514512289046720249818307028675 6768542537260957/253125689925457318694932117569972828044410342539000*( 495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-63557795213115 0574447470141761417043531594083113/37968853488818597804239817635495924 206661551380850*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-928675 811835143905263926957504331099234953932259/184091410854878049959950630 9599802385777529763920*(495-66*15^(1/2))^(1/2)-37532879399783886931299 400415084904096854083988891/138068558141158537469962973219985178933314 732294000*15^(1/2)*(495-66*15^(1/2))^(1/2)+895188211928147628185622286 558387328719961332/2091947850623614204090348079090684529292647459*15^( 1/2)+1488122595447313901345177744809891103830021016413/697315950207871 40136344935969689484309754915300, a[17,1] = -1260021157724277258078211 24630009847749/98508096938135225375048005763057748000*(495+66*15^(1/2) )^(1/2)*15^(1/2)-4738514655370271016563082388009065077/109453441042372 4726389422286256197200*(495+66*15^(1/2))^(1/2)+10558030863097979669781 387952703084117/36119635543982915970850935446454507600*(495+66*15^(1/2 ))^(1/2)*(495-66*15^(1/2))^(1/2)+1339201478984899784788527037931881671 7/1641801615635587089584133429384295800*(495-66*15^(1/2))^(1/2)-226069 7230550197762984719280247515723/66335418813559074326631653712496800*15 ^(1/2)+24127968526290358652281247254688148439/109453441042372472638942 22862561972000*15^(1/2)*(495-66*15^(1/2))^(1/2)-1352056168662838453766 690920686222721/11055903135593179054438608952082800+887032834579694469 49814908429088187421/1083589066319487479125528063393635228000*(495+66* 15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[16,1] = 6550741642 979927236940687185795693431683035981009/173647982960895208298030407890 234395098095063618000*(495+66*15^(1/2))^(1/2)*15^(1/2)+239701909410461 4830947972053269321843527994389417/69459193184358083319212163156093758 039238025447200*(495+66*15^(1/2))^(1/2)-571435644966971560672507488405 55825002924895429/7717688131595342591023573684010417559915336160800*(4 95+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-129144187875587004533197 79687341915438019588819797/6945919318435808331921216315609375803923802 5447200*(495-66*15^(1/2))^(1/2)+58206235202453456849903907222972776909 946608677/70160801196321296282032488036458341453775783280*15^(1/2)-210 52267454309616203231069587123319808049773565493/5209439488826856248940 91223670703185294285190854000*15^(1/2)*(495-66*15^(1/2))^(1/2)+3712823 721091037536407712093468834025705893906799/210482403588963888846097464 1093750243613273498400-11326333231712796658708412959260883935750157480 3/86823991480447604149015203945117197549047531809000*(495+66*15^(1/2)) ^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[19,11] = 774911882680968255 4272078575097904489174294395032914704063121197822692410314582593231229 2555372965009/19370857330844516693022424473174024145442419010685860111 806796435328956015605233987854468968693555200-588620600657265756588651 9520908644896743316427000383179234148439070422999135997840650487090907 4241991/58112571992533550079067273419522072436327257032057580335420389 305986868046815701963563406906080665600*15^(1/2)+491806672298365068495 8528662599164237411830488685030827213733869591163344320778763442231111 531742907/710264768797632278744155564016380885332888697058481537432915 86929539505390552524622133052885209702400*(495+66*15^(1/2))^(1/2)*15^( 1/2)-58135670261497403154583045492582057564204697218057371965841211275 622173835957788919775557819996783433/213079430639289683623246669204914 265599866609117544461229874760788618516171657573866399158655629107200* (495+66*15^(1/2))^(1/2)-1147728383669431268275744411313526514112982076 3187398956357029493783091/66403359383352123656214770757864334218294880 967299716834513016182988800*(495-66*15^(1/2))^(1/2)+317661693355437506 6914616825066095289306736026770132799270513996309323/66403359383352123 656214770757864334218294880967299716834513016182988800*15^(1/2)*(495-6 6*15^(1/2))^(1/2)-7086974820056630734123506856050282659018243270346674 8607371778574983/22134453127784041218738256919288111406098293655766572 278171005394329600*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^ (1/2)+2412848582403909103906228320945372000450472090453967539624031443 700591/199210078150056370968644312273593002654884642901899150503539048 548966400*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2), a[19,1] = - 459600300034341496625739380769016237/549820581859872275778584851877237 3760*(495+66*15^(1/2))^(1/2)*15^(1/2)+62925390199897196083734053046296 02697/544322376041273553020799003358465002240*(495+66*15^(1/2))^(1/2)* (495-66*15^(1/2))^(1/2)+2444752418755114497905831216657566931/20618271 819745210341696931945396401600*15^(1/2)*(495-66*15^(1/2))^(1/2)-810064 364408893226526141218762152619/405273156162068016544411438730150400*15 ^(1/2)-1075597530118626492744265148327032999182754993/2189013353423936 89861381713180247706322105600+58067346194668580596869363743519876791/1 64946174557961682733575455563171212800*(495-66*15^(1/2))^(1/2)+5076767 164466705468418646651216427663/108864475208254710604159800671693000448 0*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-14999908820 34654599930539470245989661/10996411637197445515571697037544747520*(495 +66*15^(1/2))^(1/2), a[20,11] = -3566641542342810891315060420317971636 8932641780035054879318761283843042000810748101187443714473307443/31276 7581286010075602655656719426756624819078609770033231632029872935115425 73211380403728119368869888+3206977625448427051410532017836271647665481 5900344702023325549732064112935925224933529700126403209331/12399000543 8382565685338492485201321376267563306015977459682697556770706472343802 258029065044640877056*15^(1/2)+921427622567902389643414170678708514583 737480648270300616939431095750085530154874376456796809501999055/127296 4055834061007702808522848066899463013649941764035252742361582845919782 7297031824317344583130044416*(495+66*15^(1/2))^(1/2)-66398376215377418 5033171768387452850724418088364103730339768393238036275071547914015402 265452481144181/381889216750218302310842556854420069838904094982529210 57582270847485377593481891095472952033749390133248*(495+66*15^(1/2))^( 1/2)*15^(1/2)+21727011714526164313126367711800902518880207216184789325 716783947425/428695423094174102152233846069848704368646806420036089069 0765181754368*(495-66*15^(1/2))^(1/2)-40950495276592201170692548714279 82642469480207239662098478593602855/1428984743647247007174112820232829 014562156021400120296896921727251456*15^(1/2)*(495-66*15^(1/2))^(1/2)- 6785384615431114736750412060998612811890200669417591820903109548665/12 860862692825223064567015382095461131059404192601082672072295545263104* (495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+4806853622691383439067 725118109319422719971691792563230487001209739/257217253856504461291340 30764190922262118808385202165344144591090526208*(495+66*15^(1/2))^(1/2 )*15^(1/2)*(495-66*15^(1/2))^(1/2), a[20,1] = 263262388393429524194190 876292295802929/22064332555889684446074348186760858200576*(495+66*15^( 1/2))^(1/2)*15^(1/2)-2375016395154214045953940632582376355/15137691774 31101011892834698883801082368*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2) )^(1/2)-9916197885379779218782851350763280715/596333312321342822866874 275317861032448*15^(1/2)*(495-66*15^(1/2))^(1/2)+400003500569460473929 1729270561983861289/14040938899202526465683676118847818854912*15^(1/2) +229485729717192925526434118363214850725041748257/35146127675546190708 5362178705734545724595520000-28435572471737976361393553929007702695/59 6333312321342822866874275317861032448*(495-66*15^(1/2))^(1/2)-14236949 00654645062374478706714962943/2186555478511590350511872342832157118976 *(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+138267605315 337601836997531337210725843/7354777518629894815358116062253619400192*( 495+66*15^(1/2))^(1/2), a[18,1] = 106815588779195679046489805465253523 4249817931699934761175203870354825716259613369357959223340026814898681 74021593660152067011/1358937513831178186072931696666480126053201427660 4544322968809888111146750102863088986818658980392795005953318906400338 6489600*(495+66*15^(1/2))^(1/2)*15^(1/2)-21993188566425526967222364073 9602218876634985914075105506213544726970497809445801949/12285315774439 4674761522531889034843247278810025364059367552704119020164908404882560 0*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)-83922678257503559816 82092107217453220314538752351115312149683842459927476902123751/6204704 9365855896344203298933855981438019601022911141094723587938899073186063 07200*15^(1/2)*(495-66*15^(1/2))^(1/2)+2125063537026051779925850044101 6652346456200702204721764150826482714854775243287445996080422948576220 9627746621460160642034873/10066203806156875452392086641973926859653343 9085959587577546739911934420371132319177680138214669576259303357917825 18769369600*15^(1/2)+6789915451964158745831230683322046161698361857729 9562598644132838144184541556244739885925680960291215835059789605937882 31123/2448536060957077812744021075074738965861624193982800778913299078 938945360378894250267895253870341044145216814217369430387200*(495+66*1 5^(1/2))^(1/2)+3518523909126723296462742349232352685813161470160091135 8774009303961204980455733666421047828170384292116067841556278805382540 07/4529791712770593953576438988888267086844004758868181440989603296037 0489167009543629956062196601309316686511063021334462163200-37227920911 9102084785638981916764407719360460438217301760961596280020153233088299 703/744564592390270756130439587206271777256235212274933693136683055266 78887823275686400*(495-66*15^(1/2))^(1/2)-3706501230285034842733646758 70662476598039236095161246958845614424717013201196729889/7371189464663 6804856913519133420905948367286015218435620531622471412098945042929536 00*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2), a[18,11] \+ = -8004897176315216703336668122840105968047929762501300039608078033546 71278668512942662360613098309942450795621281881/2724943474059642452768 9758288106773860395522283496261076255438220815343817411812829260252700 69600306954880250470400*15^(1/2)+3910169363743972318677602215045354210 3713191296257652129693244099039800992898216180993855483593666033833221 58906161/2724943474059642452768975828810677386039552228349626107625543 822081534381741181282926025270069600306954880250470400-102492808343055 1273658236798849184557611892303847349993776260312199894419303184454459 7316855538300342768366146983/27003944337527988270683544249475181303094 6617223836821476405243629701605397774721731407909646536967355889033830 400*(495+66*15^(1/2))^(1/2)+138992879548267548900244910151298406649320 0653021844378954223348291328121166020604789842756916404248759482001883 663/899231346439682009413762023507523537393052235355376615516429461286 90634597458982336558833912296810129511048265523200*(495+66*15^(1/2))^( 1/2)*15^(1/2)-74695217695538247088258269657112623740462036699232805350 987271127489215828282196722035905075955225999274830614053/197830896216 7300420710276451716551782264714917781828554136144814831193961144097611 40429434607052982284924306184151040*(495+66*15^(1/2))^(1/2)*(495-66*15 ^(1/2))^(1/2)-67212055644078389955942040389500846357587015235477396037 4745945636955651980074479279478527131010926797688112051041/29674634432 5095063106541467757482767339707237667274283120421722224679094171614641 7106441519105794734273864592762265600*(495+66*15^(1/2))^(1/2)*15^(1/2) *(495-66*15^(1/2))^(1/2)-921253456254108209275601415179072307148007906 84738253561430738509403183732424910556198591493337721307996179285867/2 7249434740596424527689758288106773860395522283496261076255438220815343 81741181282926025270069600306954880250470400*(495-66*15^(1/2))^(1/2)+1 0942089800203365298424918818319500980392685933578109620771861125028042 179388045253184481116650201929600954882927/272494347405964245276897582 8810677386039552228349626107625543822081534381741181282926025270069600 306954880250470400*15^(1/2)*(495-66*15^(1/2))^(1/2), a[25,9] = 1498371 7615076520999898175279006718901325/85773555842758082493275897147609254 01088*(495+66*15^(1/2))^(1/2)*15^(1/2)+7995321576534889944677177407956 6331820483/5718237056183872166218393143173950267392*(495+66*15^(1/2))^ (1/2)-20621597792740011665293932604020043908997/1415263671405508361139 05230293555269117952*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2) )^(1/2)-15666467932539007518488566513824293941973/20966869206007531276 134108191637817647104*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)- 37070044494544469721347012758337427892153/1906079018727957388739464381 057983422464*(495-66*15^(1/2))^(1/2)-891841124620853667058899932595980 9281/2094080953192335998370993582705304053*15^(1/2)*(495-66*15^(1/2))^ (1/2)+7724360629880208471040518407144433826297/12995993309508800377769 0753253953415168*15^(1/2)+33101027521371172109192250549843371608719819 199/98479305301687852995941463124070766233804800, a[15,10] = 128048158 0011067872042906081843788744919/67052661462147977370089135200900132864 00*(495+66*15^(1/2))^(1/2)*15^(1/2)-8305325078579090319764835829944664 68573/1341053229242959547401782704018002657280*(495+66*15^(1/2))^(1/2) -6286310663851365915824207650794833606289/8113372036919905261780785359 30891607654400*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2 )+3636735835011016016553586602489401592969/162267440738398105235615707 186178321530880*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+477709 9095613693839379029295382810922327/14751585521672555021419609744198029 230080*(495-66*15^(1/2))^(1/2)-970519669436559179103390717142161613004 1/73757927608362775107098048720990146150400*15^(1/2)*(495-66*15^(1/2)) ^(1/2)+4636191565689457788033216342021225893961/1341053229242959547401 782704018002657280*15^(1/2)-12947680617197490104519222676481332902841/ 1341053229242959547401782704018002657280, a[25,2] = -1029/400, a[25,6] = -2667/1600, c[4] = 101/152, a[23,6] = -381/544, a[21,11] = -4479404 3532626317794933177103220192762017625/16702897595445197049847451304740 7579017107824+390422828672114370457150585026889349975/6959540664768832 104103104710308649125712826*(495+66*15^(1/2))^(1/2)*15^(1/2)+294414422 59346454399120790160135926728275/4639693776512554736068736473539099417 141884*15^(1/2)+107835852142499187656942652161788723246225/13919081329 537664208206209420617298251425652*(495+66*15^(1/2))^(1/2), a[19,13] = \+ -4882955120799037460291588304791519359/6557484890926201927469260361633 4931200*(495+66*15^(1/2))^(1/2)*15^(1/2)+67234637158667343009509554792 3571287/7286094323251335474965844846259436800*(495+66*15^(1/2))^(1/2)* (495-66*15^(1/2))^(1/2)+59545107117705549379062475922066467/5464570742 438501606224383634694577600*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*1 5^(1/2))^(1/2)+3829111840300174831719071777740427111/13114969781852403 854938520723266986240*15^(1/2)*(495-66*15^(1/2))^(1/2)+704627301948185 6090799541819933939007/2732285371219250803112191817347288800*(495-66*1 5^(1/2))^(1/2)-31545024863374033548084172963322669667/6623722112046668 61360531349659948800-83367761677852322097620144069511383/3311861056023 3343068026567482997440*15^(1/2)-19631029211451835356987964416945100783 /10929141484877003212448767269389155200*(495+66*15^(1/2))^(1/2), a[18, 10] = -303017357319290152400576138779269016028191165710055867688330876 59742283387306993680443359/1086247047626574733667191532305392205062800 870914223620455055201609560205641888718334566400*(495+66*15^(1/2))^(1/ 2)*(495-66*15^(1/2))^(1/2)-2302770298726189785416396406228642529842122 8130032644352877836779982605006167566388852181/19749946320483176975767 118769188949182960015834804065826455549120173821920761613060628480*(49 5-66*15^(1/2))^(1/2)-1512040936209135379532460652121334386322097593355 492957670049462272663472960326825453151/266891166493015905077934037421 4722862562164302000549436007506637861327286589407170355200*15^(1/2)*(4 95-66*15^(1/2))^(1/2)-262626982606352506810327995419446515953554018815 02003759930532274663423400181099254988417/1086247047626574733667191532 305392205062800870914223620455055201609560205641888718334566400*(495+6 6*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+2485458649408992108 5967802667179779788373490560995482897978694888783083040221706989739030 4725764404913166833468610666607088465438713/36027071341627045846426483 9514993074358881949403280161189813654153930018739138584406788289391951 64940887427359714563885763263242240+4934455681996063743473558807093611 9713832369231288953649324257361029811255774556431497903027675524021699 592792181718436920178992679/720541426832540916928529679029986148717763 8988065603223796273083078600374782771688135765787839032988177485471942 912777152652648448*15^(1/2)-811685935822713388812235735090616295280469 4244604167313183677300806602622689740196024168928834591968743354480035 6903812984350215729/19814889237894875215534566173324619089738507217180 4088654397509784661510306526221423733559165573407174880850478430101371 6979478323200*(495+66*15^(1/2))^(1/2)+65815063973057057698878486568723 8831934362307589363002894590901905124773282923196855334623086895861040 404376716013821784346748798549/198148892378948752155345661733246190897 3850721718040886543975097846615103065262214237335591655734071748808504 784301013716979478323200*(495+66*15^(1/2))^(1/2)*15^(1/2), c[8] = 1772 7/277750, a[20,19] = -110845/348543, a[21,19] = 81770692250/1172335670 91, a[22,19] = 828786688/2261607237, a[23,21] = 253319312163456/659439 747650125, a[22,21] = -75739731820032/659439747650125, a[25,19] = 3601 857/2273300, a[21,20] = 21260379985/34709636178, a[25,21] = 3033439552 0464588/16485993691253125, a[22,20] = 76890112/586342617, a[25,23] = 7 53491/379050, a[25,20] = 3071471/2273300, a[18,12] = 21014688996466416 2297907987828806282951127540176737051451419897493339496110026863561839 76308891612384929630135716562585047381800180720575/2473553805833915491 9154232226211212476905678998965417683451268303223190966242791706632174 078646863063894639248547636411964481184007256192*(495+66*15^(1/2))^(1/ 2)*(495-66*15^(1/2))^(1/2)+5893587288441764228035680974219260429076484 5067106137708724382713649238026769937446204407008225143101061195526408 56193517123968122469475/2498539197812035850419619416789011361303603939 2894361296415422528508273703275547178416337453178649559489534594492562 0322873547313204608*(495-66*15^(1/2))^(1/2)+37733236001994618038051413 7614965702161130581762872520759226006826474916922635380046171661834833 62525193388445947600803394566000724784925/5996494074748886041007086600 2936272671286494542946467111397014068419856887861313228199209887628758 94277488302678214887748965135516910592*15^(1/2)*(495-66*15^(1/2))^(1/2 )+57507371107922242852436696989437980762854317996170121337930959620536 46274378034007012190905040586084856252151706668737140102916424759275/2 4735538058339154919154232226211212476905678998965417683451268303223190 966242791706632174078646863063894639248547636411964481184007256192*(49 5+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-2171436379244693 7492131702372553449156260984047991715976308583476474712312945036288169 192358644411188707880367670403041934772097466015425/605706472196857175 8593016767973360875887524701307723950646163037214126958369829619010021 2007705817113914023259375907957060253894110208-52346382003618596104715 7192682527652689803768477223404773488830189696868614835124465567897428 25403057336675945458415561814433493136086225/5451358249771714582733715 0911760247882987722311769515555815467334927142625328466571090190806935 2354025226209334383171613542285046991872*15^(1/2)-70073590104070299212 6025462759147869059718309106219282804859734414835613305079035782367743 1188412884777531586870904542267521330590123375/54513582497717145827337 1509117602478829877223117695155558154673349271426253284665710901908069 352354025226209334383171613542285046991872*(495+66*15^(1/2))^(1/2)-242 9204035067344619100142087522767195715210927591649237632390596060555415 80984976145534104248610578541643675574323022677887729203981275/6814197 8122146432284171438639700309853734652889711894444769334168658928281660 583213862738508669044253153276166797896451692785630873984*(495+66*15^( 1/2))^(1/2)*15^(1/2), a[23,22] = 381/544, a[25,22] = 2667/1600, a[25,2 4] = 1029/400, a[21,18] = -390422828672114370457150585026889349975/695 9540664768832104103104710308649125712826*(495+66*15^(1/2))^(1/2)*15^(1 /2)+44794043532626317794933177103220192762017625/167028975954451970498 474513047407579017107824-29441442259346454399120790160135926728275/463 9693776512554736068736473539099417141884*15^(1/2)-10783585214249918765 6942652161788723246225/13919081329537664208206209420617298251425652*(4 95+66*15^(1/2))^(1/2), a[25,17] = 112245633/226284800-1120087/28285600 *15^(1/2)-68125601/7467398400*(495+66*15^(1/2))^(1/2)+993373/448043904 0*(495+66*15^(1/2))^(1/2)*15^(1/2), a[25,18] = -6212380489353402482048 992934870163/5513981142404675080846640779279519700*(495+66*15^(1/2))^( 1/2)*15^(1/2)+56874537829985536110363501647573078867/44111849139237400 646773126234236157600-477467113712319024289213588759633869/11027962284 809350161693281558559039400*15^(1/2)-168276384308509585366452969427482 141/11027962284809350161693281558559039400*(495+66*15^(1/2))^(1/2), a[ 20,17] = 229884359555/8366741852352+46739047805/4183370926176*15^(1/2) -20855494525/25100225557056*(495+66*15^(1/2))^(1/2)-34057541825/753006 76671168*(495+66*15^(1/2))^(1/2)*15^(1/2), a[20,18] = -526015716509239 77687781073690186/1325511779651805411892910350460322529*(495+66*15^(1/ 2))^(1/2)*15^(1/2)+1461351037003208398982108006046541205/1590614135582 1664942714924205523870348-312574630477851429199757289182427/1019624445 88600416299454642343101733*15^(1/2)-3126986093611720912749441466631523 /1325511779651805411892910350460322529*(495+66*15^(1/2))^(1/2), a[19,1 8] = 199389002800788393634510722754176/2796728200664360059162382772909 427897*(495+66*15^(1/2))^(1/2)*15^(1/2)-102546524815683480335384091692 0180809/5593456401328720118324765545818855794+164457179568729408211164 48130678578/2796728200664360059162382772909427897*15^(1/2)+13437632256 176229690482072355537756/2796728200664360059162382772909427897*(495+66 *15^(1/2))^(1/2), a[19,17] = 2466730563/34107367168-46877457/487248102 4*15^(1/2)-417081443/102322101504*(495+66*15^(1/2))^(1/2)+221689729/30 6966304512*(495+66*15^(1/2))^(1/2)*15^(1/2), a[18,17] = 61673043751462 390569674581/351685648446307151385663744-4915309705585387914994781/351 685648446307151385663744*15^(1/2)-46955489957737999234808063/116056263 98728135995726903552*(495+66*15^(1/2))^(1/2)+6751918613532367335628313 /34816879196184407987180710656*(495+66*15^(1/2))^(1/2)*15^(1/2), a[20, 9] = 7298871622239329623715815186148827947671875/108791075311428927618 8341700307003841728043008*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1 /2)+5288207714188347577837606590854470502609375/3896099115836297683319 1300561958815268497408*(495-66*15^(1/2))^(1/2)-69216254730143769881673 30348092464371875/3246749263196914736099275046829901272374784*15^(1/2) *(495-66*15^(1/2))^(1/2)-1945633320665070109801589422288338107020625/3 535709947621440147612110525997762485616139776*(495+66*15^(1/2))^(1/2)* 15^(1/2)*(495-66*15^(1/2))^(1/2)+8212154146476729949046334796100054243 8436875/3264701876769885911698882802971196020586738688*(495+66*15^(1/2 ))^(1/2)*15^(1/2)-9912085223730232602283883117721485996822256875/33636 32236672003666598848948515777718180276224+6110920189917429274331180095 72638755855894375/2522724177504002749949136711386833288635207168*15^(1 /2)-1070798742125075822882594167231186215036359375/6529403753539771823 397765605942392041173477376*(495+66*15^(1/2))^(1/2), a[20,10] = 751771 6654767151923866093586493111124114805/50749538116145020829903394110409 24362443286528*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+7931940 338055589383939670850354289468599445/461359437419500189362758128276447 669313026048*(495-66*15^(1/2))^(1/2)-330276171259826443543763399494010 752592415/28834964838718761835172383017277979332064128*15^(1/2)*(495-6 6*15^(1/2))^(1/2)-3481832483544859671058925604559425189874465/50749538 11614502082990339411040924362443286528*(495+66*15^(1/2))^(1/2)*15^(1/2 )*(495-66*15^(1/2))^(1/2)-91544354870937946654347385587851976580816655 01275/20803263247100278379508254661687910140554865036288+2564903944839 18801658407471165705550946767215/7759226902055230457464568521012983529 35543808*15^(1/2)-367266419049193918159970019933091678466272311/853514 9592260753503211025373114281882290981888*(495+66*15^(1/2))^(1/2)+25614 2177443986017417526128325881170757327379/17070299184521507006422050746 228563764581963776*(495+66*15^(1/2))^(1/2)*15^(1/2), c[6] = 25/152, a[ 20,12] = 1673845687571065427562129743768741570967260037618468257096611 80734865508251052756116559484375/7003915575067319386021040632087784155 22564150076562836476404783375905089062051656311049021696*(495-66*15^(1 /2))^(1/2)+62226784344209778737992091950022645827412749893461431840085 4168467533588307554276250085428125/84046986900807832632252487585053409 86270769800918754037716857400510861068744619875732588260352*15^(1/2)*( 495-66*15^(1/2))^(1/2)+41462887729552019805371870049861024230202788946 072344199665488911308217858192580518166859375/513620475504936754974876 3130197708380498803767228127467493635078089970653121712146281026159104 *(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+782203071139311366473 9083383593812109628284850359406808185280228405150834282952015936713456 25/2773550567726658476864332090306762525469354034303188832446562942168 58415268572455899175412591616*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66 *15^(1/2))^(1/2)-13873040092813279500757243715913642473146027727517184 78676849402148309628012534216585473930625/4283386384751914335252537246 07021510319584686823683057018627718758817988351998533611715930624-1095 8231425068564239742723275246782367958945478261641070545379457766371801 56790511899001900625/8566772769503828670505074492140430206391693736473 66114037255437517635976703997067223431861248*15^(1/2)-2031884074659016 2862540612332712031404104267457288421844253603261305269322570101938622 04959375/1727632508516605448551856689248320091622324903522188329975131 7989938992196863940855672542535168*(495+66*15^(1/2))^(1/2)-16096472674 6024213488465067353505933314432826249126659948720148860831907152392883 2895427058125/34552650170332108971037133784966401832446498070443766599 502635979877984393727881711345085070336*(495+66*15^(1/2))^(1/2)*15^(1/ 2), a[21,5] = -156479396217570600333298656/73273972406867506457421875, a[21,6] = 11005843237812473721/5193844391648046875, a[23,2] = 112211/ 506844, a[22,4] = 390625/3949854672, a[22,5] = 404856925/3216589572, c [5] = 8225/76912, c[3] = 101/228, c[7] = 2013/3445, a[18,7] = 18993974 973485227904952341910291889381118708193023824550/120600346650805229944 92944014191159522924996678137100092619*(495+66*15^(1/2))^(1/2)*15^(1/2 )+37364999682821940340524213896266047309233288831638123551875/42880123 253619637313752689828235233859288877077820800329312+838708650164817204 96920100044907813868655001976949702225/1340003851675613666054771557132 351058102777408681900010291*15^(1/2)+101882217739425975585850562279744 393868968529126539562125/402001155502684099816431467139705317430833222 6045700030873*(495+66*15^(1/2))^(1/2), a[18,8] = -27541388263722508415 3879395633151790451174339761235000000000/17365706776297655617341216750 404769836227513968915818501163621*(495+66*15^(1/2))^(1/2)*15^(1/2)-903 0706067191500671304838647417536518671082752129344726562500/19295229751 44183957482357416711641092914168218768424277907069-1131066379913897019 748847486012312835922421062622057500000000/192952297514418395748235741 6711641092914168218768424277907069*15^(1/2)-10975625302844603959190278 12165520839910017377896612500000000/5788568925432551872447072250134923 278742504656305272833721207*(495+66*15^(1/2))^(1/2), a[19,6] = -460121 34893475328/76616158064651325, a[19,7] = 5620721356384469075/354569834 5096452809913, a[19,8] = 10436097770007465488281250/344805882698905880 98919643, a[20,6] = 24467886602752/291937806983125, a[20,7] = 11944081 475573/1322478962029186830, a[20,8] = 1095050362935718750/112546935710 26117461, a[21,4] = 46426400659946403657/610018749211776263125, b[20] \+ = 13/56, b[22] = 19/100, b[21] = 21/100, b[23] = 17/100, b[24] = 11/10 0, c[21] = 2013/3445, b[18] = 29/100, c[24] = 39/88, c[23] = 101/228, \+ c[22] = 25/152, c[19] = 1/3, c[18] = 51789075/64972747+333240/64972747 *15^(1/2)+3057458/714700217*(495+66*15^(1/2))^(1/2)+70408/714700217*(4 95+66*15^(1/2))^(1/2)*15^(1/2), b[19] = 27/100, c[20] = 1/5, a[24,3] = -274193/636804, a[14,10] = -30702568490912439111188538/28614329140576 09960190041997*(495+66*15^(1/2))^(1/2)*15^(1/2)+6755832884015803347811 847721/91565853249843518726081343904*(495+66*15^(1/2))^(1/2)+100517936 407692876551070243/366263412999374074904325375616*(495+66*15^(1/2))^(1 /2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-279143136095627004211108611/91565 853249843518726081343904*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/ 2)-1021753700305474025763231075/16648336954517003404742062528*(495-66* 15^(1/2))^(1/2)-28454754214941681480938433/332966739090340068094841250 56*15^(1/2)*(495-66*15^(1/2))^(1/2)-624838597750010736607379013/832416 8477258501702371031264*15^(1/2)+20889563008538679095591532243/16648336 954517003404742062528, a[14,11] = -13130563244410661872315722470685724 176074235474873559/168447225263992355253200190858124222188907272421384 1856*(495+66*15^(1/2))^(1/2)*15^(1/2)+21634402546822106502956405557840 19989685985966394801/7895963684249641652493758946474572915105028394752 3837*(495+66*15^(1/2))^(1/2)-26203365390394594331533150852842064709768 91372062198311/2834966801192991338911359212142230659439309394851895843 648*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2)+156741759209099684 1749224823127656355473861331736569451/56699336023859826778227184242844 61318878618789703791687296*(495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15 ^(1/2))^(1/2)+960434983056913523969908257106104327413288387315836063/8 5908084884636101179132097337643353316342708934905934656*(495-66*15^(1/ 2))^(1/2)-2107085918307670902159884356898641932607128641690997959/5154 48509307816607074792584025860119898056253609435607936*15^(1/2)*(495-66 *15^(1/2))^(1/2)+63197323865987155320437607552839368481975380037653735 /561490750879974517510667302860414073963024241404613952*15^(1/2)-19625 1321827756638854506220121563148477341044141985823/56149075087997451751 0667302860414073963024241404613952, a[16,10] = 17032808921532413769326 095101435868425512701950321/228493008347325776817305873440505402839496 531081600*(495+66*15^(1/2))^(1/2)*15^(1/2)-314927171865768972153444207 38864818541314974948753/1142465041736628884086529367202527014197482655 40800*(495+66*15^(1/2))^(1/2)+2636605715153751220902295097514975997377 722690725299/3041241941102906089438341175493126911793698828696096000*( 495+66*15^(1/2))^(1/2)*15^(1/2)*(495-66*15^(1/2))^(1/2)-12128481307594 11516381383562111404947777471603683549/1216496776441162435775336470197 25076471747953147843840*(495+66*15^(1/2))^(1/2)*(495-66*15^(1/2))^(1/2 )-2742291030606566024225316251947419989058495068734689/138238270050132 09497447005343150576871789540130436800*(495-66*15^(1/2))^(1/2)+1206031 6683436861169241947489803239449824641591213/69119135025066047487235026 7157528843589477006521840*15^(1/2)*(495-66*15^(1/2))^(1/2)+67083830822 20198976386948910077685899871881611055887/5026846183641167089980729215 691118862468923683795200*15^(1/2)-223445357508550044679432725637670562 03469038581346723/5026846183641167089980729215691118862468923683795200 , a[25,10] = -289615505775241581098990702550013952492757/2909741766321 27973015045830605969822720000*(495+66*15^(1/2))^(1/2)*15^(1/2)-8263756 0006838181681227476570558015839783/14548708831606398650752291530298491 1360000*(495+66*15^(1/2))^(1/2)+18635155609548040118686971519283267022 1231/3200715942953407703165504136665668049920000*(495+66*15^(1/2))^(1/ 2)*15^(1/2)*(495-66*15^(1/2))^(1/2)+4797501630574376502132863502694297 34998683/3200715942953407703165504136665668049920000*(495+66*15^(1/2)) ^(1/2)*(495-66*15^(1/2))^(1/2)+659613600904755105711427456179352195929 /142984853381881067820661341821115392000*(495-66*15^(1/2))^(1/2)+19349 175745772217526368706395431357179269/132260989378239987734111741184531 73760000*15^(1/2)*(495-66*15^(1/2))^(1/2)-9411818790260725838811697089 108197825199/413315591806999961669099191201661680000*15^(1/2)-21267788 21337403489098990902156033714011143903/6013378143071059282319124424695 9199817216000\}:" }}}{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 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 429 "for ii from 2 to 25 do\n print(` `);print(`__________________________________________________`);print(` `);\n print(c[ii]=subs(e26,c[ii]));print(``); \n for jj to ii-1 do \n print(a[ii,jj]=subs(e26,a[ii,jj]));\n end do:\nend do:\nprin t(`__________________________________________________`);print(``);\nfo r ii from 1 to 25 do\n print(b[ii]=subs(e26,b[ii])); \nend do:\nprin t(`__________________________________________________`);" }}{PARA 11 " " 1 "" {XPPMATH 20 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7368421052631578947368421052631578947368421052631578947368421052631578 9473684210526316, c[8] = .63823582358235823582358235823582358235823582 3582358235823582358235823582358235823582358235823582358235823582358235 8235823582358235823582358235823582358235823582358235823582358235823582 3582358235823582358235823582358235823582358235823582358235823582358235 8235823582358235823582358235823582358235823582358235823582358235823582 3582358235823582358235823582358235823582358235823582358235823582358235 8235823582358235823582358235823582358235823582358235823582358235823582 3582358235823582358235823582358235823582358235823582358235823582358235 8235823582358235823582358235823582358235823582358235823582358235823582 3582358235823582358235823582358235823582358235823582358235823582358235 8235823582358235823582358235823582358235823582358235823582358235823582 358235823582358235823582358235823582358235823582358235823582358236e-1, a[22,12] = 0., a[25,7] = -1.84001013760904971548055102860399278075185 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7876542811000502327087343293056134026371095557702928218502059903975956 1668071704087222000935343665013109260514864939122276306284364970892286 1102510826075471415400807514629402372932639409596885532234307989396682 7859074257300401342988024231459692053473297911293996217457371603246900 65034008339261803496326119352472643159594955304357123662811271372318, \+ a[22,10] = -.366459159858091663862145662208985936314458300435656060822 9075984337239720284817960192970500297351144353452561931291697577814215 3159372827033450105642724382562629728620734865467712508916065163793955 4400178990937673586866046980198976078886680711483768567371276058575859 6066961559691896228222053571364655126455098091817788076878178118422778 9946712131077249466725154452625232733989522514072146117739010400946996 9696599445396981633376335008606094233151766307334291573103946518720836 5835274341227269427949747934061815172728862292723553041937847318623521 0123710795324095436647207722036485506700737533941663806234097224902008 9247264820279667331114045245690907735629959871763533802310697151346266 2305762687122140651338922117182807723744474381517023771356104835456891 49207475762954502784870598643207295325788701479999730, a[12,11] = .181 8512456810329150754682669576286597563193307874158937988725222767775959 2653209674486270230951082014911802145844699036188397890525550100018185 1245681032915075468266957628659756319330787415893798872522276777595926 5320967448627023095108201491180214584469903618839789052555010001818512 4568103291507546826695762865975631933078741589379887252227677759592653 2096744862702309510820149118021458446990361883978905255501000181851245 6810329150754682669576286597563193307874158937988725222767775959265320 9674486270230951082014911802145844699036188397890525550100018185124568 1032915075468266957628659756319330787415893798872522276777595926532096 7448627023095108201491180214584469903618839789052555010001818512456810 3291507546826695762865975631933078741589379887252227677759592653209674 4862702309510820149118021458446990362e-3, a[22,9] = -.1311351243636448 8239134765126581273214871911655706922630186371051381380316757702092802 1679174652249437294441110017421776455999956762481073416500441754517734 4665022020734337992013976360855243786586298911307004655266257066216286 9870330438559952056154226292577331113559497586374486574289038928923701 2768594304650381570337057727461758079917973965040988995688164348456356 5332656009208349936467265179191298660114279225246900311870047815405510 5975692706641516388361039088516398936767033599401491227440491503622019 6834165987289987485252159318994205055369529791487082031426004976881972 0637839974712259402423753892001338186884682816770250217033090057651395 3103975043314990695960276754026221498411056141941666164102139619846189 6894661504708602820183544666343091346539458515941371527493796344671975 293243949893548331316330, a[23,2] = .221391591890206848655602118205996 3223398126445217857960240231708375752697082336971533647433924442234691 5421707665474978494369076086527610073316444507580241652263812928632873 2312111813496855048101585497707381363891059181917907679680532866128433 9954700065503389603112594802345494866270489539187600129428384276029705 3925862790128718106557441737497139159189020684865560211820599632233981 2644521785796024023170837575269708233697153364743392444223469154217076 6547497849436907608652761007331644450758024165226381292863287323121118 1349685504810158549770738136389105918191790767968053286612843399547000 6550338960311259480234549486627048953918760012942838427602970539258627 9012871810655744173749713915918902068486556021182059963223398126445217 8579602402317083757526970823369715336474339244422346915421707665474978 4943691, a[21,5] = -2.135538596825204678401302463372339475781988628449 6838914860810183532951311523669999630955096610105172568189229693741472 0073668269856716050050698565103527679675460727842257601516773932261488 3863010688757468550147396673563841995250779674178382082797372429634126 5548157140207604743205794740179874491026750037853970927941815122023460 2559090707975927607358325769758250374373335142682283343796379900834415 8211917896768686435756027519593545002651406156996380604290866022461204 3500618312729927559488817257536666016423241841839021650075291822332254 2509698632552171055191797077709599964741184853233017884007425056649785 9472908605644742965371135354457758654467659391959399901399212027561820 5221340551071780246774103622090650877054094890274145923935118878383355 3282554062987454828458174211942204306358859741165120824357627, a[21,6] = 2.11901674518982552652433947086665273073363282364418699259408723158 5247087949150559861604221245369796435075656697867958957181212561893723 5485139948183464156395062881423485495074974546596660803651229773772179 2202996291681440343625276333353806383060467685871917948288840431560593 0060205198645316029676782958817865273908786366463244308655085720571511 1009236450990583060346341822722238881203911120415449728215975560579715 8186154718733771507452556527242968672096926861189185529163226240759077 4256906462345337007211427482609534721350448363601127794765968641459492 0168747467504512830334338064673918532110778287901352525658999510729340 2259046081798161296937035476689930560448890267173600824411035556425665 1007826586927951030290969971546632454591835291090872756779486767058674 19517454951148419823915165284343176401561338, a[22,5] = .1258652731216 4027621239829101827356169766255773958593185416196455915140919943267166 6946509643164384417770536750344224519546505574507284387851021734270548 0859526954904895152722331837491886266675989833122545483399956753947966 8499030997915577399627284497084727849139467396121994267287265830879837 2240696924077449567756044444454227062326620015542349709514011941838130 1647768943273810998974413139706591077638424924919205700888220127575542 6095126319709401831014827402418750364586458343464405162848050170822353 2100663043497549459816504062209898876088254631697848456495586748734258 4719415983979941846307770098062109864975959699467682039727759211923478 8093132573271925038747218819871247160780138225232050214456145106224326 2162761248919450267993345344340375172987721170166126497658122731736568 596946256592539870361800700, a[22,16] = 0., a[20,15] = .12548262548262 5482625482625482625482625482625482625482625482625482625482625482625482 6254826254826254826254826254826254826254826254826254826254826254826254 8262548262548262548262548262548262548262548262548262548262548262548262 5482625482625482625482625482625482625482625482625482625482625482625482 6254826254826254826254826254826254826254826254826254826254826254826254 8262548262548262548262548262548262548262548262548262548262548262548262 5482625482625482625482625482625482625482625482625482625482625482625482 6254826254826254826254826254826254826254826254826254826254826254826254 8262548262548262548262548262548262548262548262548262548262548262548262 5482625482625482625482625482625482625482625482625482625482625482625482 6254826254826254826254826254826254826254826254826254826254826254826254 82625482625482625482625483e-1, a[20,14] = -.99800399201596806387225548 9021956087824351297405189620758483033932135728542914171656686626746506 9860279441117764471057884231536926147704590818363273453093812375249500 9980039920159680638722554890219560878243512974051896207584830339321357 2854291417165668662674650698602794411177644710578842315369261477045908 1836327345309381237524950099800399201596806387225548902195608782435129 7405189620758483033932135728542914171656686626746506986027944111776447 1057884231536926147704590818363273453093812375249500998003992015968063 8722554890219560878243512974051896207584830339321357285429141716566866 2674650698602794411177644710578842315369261477045908183632734530938123 7524950099800399201596806387225548902195608782435129740518962075848303 3932135728542914171656686626746506986027944111776447105788423153692614 77045908183633e-1, a[19,14] = .656250000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0, b[25] = .2380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095238095238095238095 2380952380952380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095238095238095238095 2380952380952380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095238095238095238095 2380952380952380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095e-1, b[17] = .1384 1302368078297400535020314503314674881364008994123459126711948172231193 7773066807664269088556121963694120788533360672725247452751220646700343 9396816451077464553092327222070443036618305682499314843929725765177652 1112301625849467258821085734398345451668041645205639413835136475082312 8333459323605259853589841842749659312761869952712818249387006911809025 6862367943012831670413138470095538684098372764758307710658990566657312 3037771812950469114150177758484750034828683501059624660708474025547251 9505366522427534985270795675213646004990868608590654582031912821507960 5813539103943362021352217235596339306766427796104258444233104684065269 4418351160628470679257209344607861590155652499403410260009756050703783 7374048371533143402930714158521510985207144468282803206034018166555628 734464367559769568517340527182568827, b[15] = .24380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095238095238095238095 2380952380952380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095238095238095238095 2380952380952380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095238095238095238095 2380952380952380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095238095238095238095 23809523809523810, b[13] = .138413023680782974005350203145033146748813 6400899412345912671194817223119377730668076642690885561219636941207885 3336067272524745275122064670034393968164510774645530923272220704430366 1830568249931484392972576517765211123016258494672588210857343983454516 6804164520563941383513647508231283334593236052598535898418427496593127 6186995271281824938700691180902568623679430128316704131384700955386840 9837276475830771065899056665731230377718129504691141501777584847500348 2868350105962466070847402554725195053665224275349852707956752136460049 9086860859065458203191282150796058135391039433620213522172355963393067 6642779610425844423310468406526944183511606284706792572093446078615901 5565249940341026000975605070378373740483715331434029307141585215109852 07144468282803206034018166555628734464367559769568517340527182568827, \+ b[14] = .2158726906049313117089355111406811389654720741957730511230185 9480399197377651264747805001662572959232202016492575235361298903826153 4493639013941774604069177967830405052992078669982052455146035782801321 3131377679490745912694557910416975034283703022597690338692622293201473 6292096346258595225496210902586069587244296462640152384433300146746489 8707373905260028048919984431118672971867276160417304448437809883514648 3866576199830553365675847810231442125081294382139374274507036894676786 2954545881319478034776328675072901491861050063970929484567712363113225 3801464206325132931803891378083579063990726080355037642934675288441290 9752458791873415307741080010074931421922682070983841577607202516116856 1672520724787691166591418114231135500012719277472907856981743148250825 1696119158656979821346725944717196945187103145459, b[12] = 0., b[8] = \+ 0., b[4] = 0., b[5] = 0., b[1] = .238095238095238095238095238095238095 2380952380952380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095238095238095238095 2380952380952380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095238095238095238095 2380952380952380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095238095238095238095238095238095238095238095238095238095238095238095 2380952380952380952380952380952380952380952380952380952380952380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095e-1, a[25,16] = 2.509716434086569985363077956674238286724267256774 0327574803612640017713077253057535192008459943567793402995820886428322 8107283543157519670843072225522611862017379182427927367746300350734508 8070305474587440547315610105546317241237948332537171865150267877071059 4876539668949625649685981886300865518689182359484898822211343324053994 5712063876286541398103004695809684222701437637085513273190466604000401 5386444487379422578702683299049161131099120556847741029047223646418445 6106973143367171550472927175401551809720083137503119876612879951637266 4412576210990780924059111695290120408014873114946750324022475284410905 2461717283105594168498131508399778129204080149150757111037065035690023 9407776688904280007522596050964971013956109280970128482292749215888379 9663363570562389939763882595325389843339709817835260638503141, a[25,8] = 0., a[25,4] = 0., a[25,5] = 0., a[24,22] = 0., a[24,21] = 0., a[24, 19] = 0., a[24,20] = 0., a[24,18] = 0., a[24,17] = 0., a[24,15] = 0., \+ a[24,16] = 0., a[24,14] = 0., a[24,12] = 0., a[24,13] = 0., a[24,11] = 0., a[24,10] = 0., a[24,8] = 0., a[24,9] = 0., a[24,7] = 0., a[24,5] \+ = 0., a[24,6] = 0., a[24,4] = 0., a[24,2] = 0., a[23,19] = 0., a[23,20 ] = 0., a[23,18] = 0., a[23,16] = 0., a[23,17] = 0., a[23,15] = 0., a[ 23,14] = 0., a[23,12] = 0., a[23,13] = 0., a[23,11] = 0., a[23,9] = 0. , a[23,10] = 0., a[23,8] = 0., c[21] = .584325108853410740203193033381 7126269956458635703918722786647314949201741654571843251088534107402031 9303338171262699564586357039187227866473149492017416545718432510885341 0740203193033381712626995645863570391872278664731494920174165457184325 1088534107402031930333817126269956458635703918722786647314949201741654 5718432510885341074020319303338171262699564586357039187227866473149492 0174165457184325108853410740203193033381712626995645863570391872278664 7314949201741654571843251088534107402031930333817126269956458635703918 7227866473149492017416545718432510885341074020319303338171262699564586 3570391872278664731494920174165457184325108853410740203193033381712626 9956458635703918722786647314949201741654571843251088534107402031930333 8171262699564586357039187227866473149492017416545718432510885341074020 3193033382, a[13,1] = .17316358058937036844488290897349420704455623821 1674542011409432672270520317477514033191991486928795350797581247852330 8559919696642501592249906993725058518746651169159757443425173874712716 4402112416672944597652475668288317221880996470585582395534925143091237 6190232246768551841745602822724701228027954616754722658009066123052573 8917632542759888817850135757961224138931547602775835118934063625541596 5619197880580510768219362439174917181119220612949970782100682319472346 3531874377158422554348468485059395043628789090926624884190754212563496 2140253043857935081071321315798307415681361292302975871637930209439791 0349141967664724090598649578607586382430872662376603154349452077458074 7635426815461547108376658489841042059286773575332692078650826215379249 243939607172145532352297360307687093652756549135526784888973053e-1, a[ 13,2] = 0., a[23,4] = 0., a[23,5] = 0., a[23,3] = 0., a[22,18] = 0., a [22,17] = 0., a[22,14] = 0., a[22,15] = 0., a[22,13] = 0., a[22,11] = \+ 0., a[22,8] = 0., a[22,6] = 0., a[22,2] = 0., a[22,3] = 0., a[21,17] = 0., a[21,15] = 0., a[21,16] = 0., a[21,14] = 0., a[21,12] = 0., a[21, 13] = 0., a[21,7] = 0., a[21,8] = 0., a[21,3] = 0., a[21,2] = 0., c[23 ] = .44298245614035087719298245614035087719298245614035087719298245614 0350877192982456140350877192982456140350877192982456140350877192982456 1403508771929824561403508771929824561403508771929824561403508771929824 5614035087719298245614035087719298245614035087719298245614035087719298 2456140350877192982456140350877192982456140350877192982456140350877192 9824561403508771929824561403508771929824561403508771929824561403508771 9298245614035087719298245614035087719298245614035087719298245614035087 7192982456140350877192982456140350877192982456140350877192982456140350 8771929824561403508771929824561403508771929824561403508771929824561403 5087719298245614035087719298245614035087719298245614035087719298245614 0350877192982456140350877192982456140350877192982456140350877192982456 140350877192982456140350877192982456140350877, c[22] = .16447368421052 6315789473684210526315789473684210526315789473684210526315789473684210 5263157894736842105263157894736842105263157894736842105263157894736842 1052631578947368421052631578947368421052631578947368421052631578947368 4210526315789473684210526315789473684210526315789473684210526315789473 6842105263157894736842105263157894736842105263157894736842105263157894 7368421052631578947368421052631578947368421052631578947368421052631578 9473684210526315789473684210526315789473684210526315789473684210526315 7894736842105263157894736842105263157894736842105263157894736842105263 1578947368421052631578947368421052631578947368421052631578947368421052 6315789473684210526315789473684210526315789473684210526315789473684210 5263157894736842105263157894736842105263157894736842105263157894736842 10526315789473684210526316, b[16] = .215872690604931311708935511140681 1389654720741957730511230185948039919737765126474780500166257295923220 2016492575235361298903826153449363901394177460406917796783040505299207 8669982052455146035782801321313137767949074591269455791041697503428370 3022597690338692622293201473629209634625859522549621090258606958724429 6462640152384433300146746489870737390526002804891998443111867297186727 6160417304448437809883514648386657619983055336567584781023144212508129 4382139374274507036894676786295454588131947803477632867507290149186105 0063970929484567712363113225380146420632513293180389137808357906399072 6080355037642934675288441290975245879187341530774108001007493142192268 2070983841577607202516116856167252072478769116659141811423113550001271 9277472907856981743148250825169611915865697982134672594471719694518710 3145459, a[20,5] = 0., a[20,4] = 0., a[20,3] = 0., a[20,2] = 0., a[19, 17] = .168083047626617544413800091016034342342170737856338468723409187 9582812751321589095321949308605808852517114048963032684039155323111308 2934666082996540203631623060186991372631686896493534711978672578231293 1421321941678420585364637352774247472188088033416798067130752487939992 8589353862065750378360964322050954432965181926611159944794900572241714 8513362939102217391277137537861897506495106881465616289250953711475031 1750894641934241272964518660606652013963712901360754986802686400997802 0170015795082003558603947802947409663091232824922395227491482429486021 5362427709996616417887167680995179538183718999431696745946004034149378 1540603625783328513557209617338778824715290277237839920230699895902162 8100802469644737368459450828694197607038527925784734133447036868215307 70544834553505546618334249784161467995879744426e-4, a[19,16] = .778376 1260627937338705544952370637303437265442198231246348540202794422441986 8444950099111133419415682239882175719335492577624223591835842356100180 1718571952794079157539257935179904176768435930156271476220571530305934 6307282593041361518660731483506817833966239313170563482883799543680930 0244844453891250622365280109932823263130514218400439286771888494064015 4318752991863249286899165711007341534817752464205604788038879772507452 8850431404208507074049507931457093337860930756205849349967224590637537 5290976622582104064312166368182196794813165280660346191192741030300656 2735713938039886597575129458279505478916083366662382208408245221279530 8887263520197824187892441831291817886364487775847468371753592385996414 9668197273335828683913427596630912292758589187975883208856356727615468 1723846396608794388929371573809785e-2, a[19,5] = 0., a[19,3] = 0., a[1 9,4] = 0., a[19,2] = 0., a[18,16] = -.84618372967395396834367679731095 5438250124185720560824852062815217437244535068940074887868325793934774 3927391888827410293342878488834831844362723800333851972438258011854283 2770705399012984861147016928559521567486038351751841136975137458066910 0053883663894801634191282400775866565465345623168738882266663875486235 2202630423410498519106298610977193624419456003003550666107931086959250 3213164487569284028001840280920537704681985844611012697900850951370946 0986060135566102807275589570783693056348410510026887146118446291084269 4335758977326865438411214009594661027120258337858082562777073720996077 2473887733957858931714401992086632515518677771580952162827011026256921 0617338374327968865635613852623390838099851068209978887297066114565088 4891951071672994090284453531374585103235966237812345116513796485110662 08701304, a[18,6] = 0., a[18,4] = 0., a[18,5] = 0., a[18,3] = 0., a[18 ,2] = 0., a[17,16] = 1.74670196109933519996361608187240374933523258996 1167014444435282876535760443759733215479861610891290127526130746703806 5120277581487216069605467724739920021292292221918348736669195129902132 3163204391151576942689593096249358898698884467619548545207819780071951 7550480977580240227824040833520375416028900486547408259817448689107612 2790228911176608756984558680898461763957381826061762819750713314797247 5325452252038227026590110527934438357296102039140007271301613715603841 6794311024931213848124191710224641261375358293241567072993217176120392 2778122055818958252545118799432362987228016517878674695320468781889694 8883255694012502041382405290919341426699413884655821505407700634896685 4124279083111035576170733749638580081436347010137350367414451303424521 02367016745232921983216605386051286329469456881342090816270575, a[17,8 ] = 0., a[17,7] = 0., a[17,5] = 0., a[17,6] = 0., a[17,4] = 0., a[17,3 ] = 0., a[17,2] = 0., a[16,7] = 0., a[16,8] = 0., a[16,6] = 0., a[16,4 ] = 0., a[16,5] = 0., a[16,3] = 0., a[16,2] = 0., a[15,8] = 0., a[15,6 ] = 0., a[15,7] = 0., a[15,5] = 0., a[15,4] = 0., a[15,2] = 0., a[15,3 ] = 0., a[14,11] = -.2185380330434085597487818850115824768660155325792 3694392836453640448178157122399047651698423645286524461136236887451651 1779777732609951066883707971081814562713416569731771771812406056560947 7105239046123179748843078451716828094226597328686198789405933818830042 2682165247170390296205783911926919262752434834463956969938410830565903 4822635114013771815442535136874095048379739193653391040782862044104110 3739394917150917360447232930880540484570629801682703737738288366910033 3780393018135574176282007713041893852380764395477189734458834318905718 9907017458892317889543919569304054778302457288517286832858931888983307 8070915569881463897343879066983280402281675669826795157736374145116239 3500992149773651862490321759478066502402650085254180959625524106838820 9204151694855205516543172710213613488943352375151430674257730e-5, a[14 ,12] = .62357701380255660193517475314326685146473983104372771727058753 8024958043646831488528312741433774285748989886442286743669447112221606 5170570380927032719599863123144911748877127193577845623365962026779190 9893032635123839216848427863426926184814211720468709687877956660740741 5450126507145165917001485200363175548185666638570439182013596662168335 1569207115394532427654954393800602536738543781107883662852043071081066 6316437022491235089936109136487598793906847885855229085714234367928573 7037642361028158396741362442664599757027367321020758233644930531892278 7389278838020444119013411744026987001024314471301818710761730739571882 5733213210332030284474616876946759258583446184110462603897994575340829 0492077801524042362068484177755517069082241104685408838795439504528658 523720956123598928058251950917469202700685250969e-1, a[14,10] = .44442 5209019342141588047761342888994172858701614389915613543241626514167914 9665438354029420479936916182107402236805734994935404766842877955605196 5255366650106384308646896804704055065894112786589896766188075150467440 6739786869262171521429907028176672151886816980665523528820510828753592 5844646738291022481527594302236155265225342129561307224818047281826335 4248305327651929954185242654696095314789856136322764347063791838337092 1301161800131673693964426920008242267093151711826209060304859068288843 0733073820474765746845893869081355523777084492208178366923878048328450 7320540432297556815286835811002479946232950255104931433842238907665065 9971709124486221456449756508852448210099802039747308310416359133214351 2315821170816630253229886603772237450527264994555278347871939384088820 35376947726521492689913285002005614e-2, a[14,9] = .1299053198125687130 8060565613609452675634877137008486299603797862584722151186488286220148 9197032489281060599787233093993985910532023107624007004552796532921929 4386051489433848658606838605783097089295169905207070362544565452880589 8325162166927237991651876061537598985415385926839641482501895578792698 3211789594475112416911488097957330469766439661208356262258058493777144 6941765530558736109827440952196393602186356562772905519598443593142360 8792978892786326989927143983992733026706246087861815150999821597142363 1845512403194112731147031356351685539613879812130519102863619029346170 5323635195248812710193133092536215763272569465112160900655328149681706 0993195104056519279512157062226784800392884621698586102130729984559113 7597068684766290007024782682129412454979566784222877154771192969519462 731687295036827742777, a[14,7] = 0., a[14,8] = 0., a[14,6] = 0., a[14, 4] = 0., a[14,5] = 0., a[14,3] = 0., a[14,2] = 0., a[13,11] = .1833594 7773499287061307540924390753952406452680749165900651447204934857581906 5298105811457128281604008439318623557195547880582953833986055110401893 0617951172485237833822994665087219972657910052652358155513700711038243 6126038152835144397886162897227325430323725077960774546961091199585498 9347146107835869552835395500082206102373655379717670242426013588341225 0982675076879107030678776805071338828971748908235441432026226855374035 8543615388980193372103671843000812805205408441224219291044077977683126 1243276480021747112731230160686924413733793633149489623725586451104667 7125062594363510224781929066889764639333045911762230593096350442063524 7270238836582272077358585729554642289346076140729555512647092007716998 4213167130986686223368346076818193872643593608969562184872723696881109 988617203457252495685082226144207e-6, a[13,10] = -.9198044746158460357 2901462028287708116423431205518432174207711559689034041825887083379695 6729546338101521604964580684096968724429263761547295911007547404599229 7147068616378928428244974972580636006115374000128410504182440918702022 8649839645712024429372720986655249610299312343336888340419355075649987 1604336561849316707611851636806538590167523258796994868755669746947103 3845982266786584307569361585018720732540432681468833675948855413800332 9550617140012747890516745907969495535847178175319843048532859868886090 1215701575365777391699199461426004898937818094211622359186134610998616 1220017504825966714633035685978143726061901846836984983142714841985425 3668513965106517162743217822844045508709131139748986320538849258361193 0527407564193150370606254245551836366835021255707514234225712247554650 409027897738175873547e-4, a[13,8] = 0., a[13,9] = .8690027159880925811 7950932563304302201847194049029850107622379273041165026308854259160247 3844217528702685567350552160263509628448375519138446338033461304742760 3607810149313048282272698564748236989498581611516589754871397780034198 7108961185853569528768250318187061495311281431453711735449344259501892 4223300247543944101378910227445736486396364002335340905500224717005678 4568927678490457802821701945852149739507294802087537979053827642202998 0013852661745138877912183350178921651966062025992486092327319100648182 4115495294191981920348364652466535193472304693929506502577840369720522 7945577603629562726958433764279939972422955183746057940737232501092853 9579015122960647431441115080963914985185046910506510504379515989632089 9166460532912484502609556503553129744009170004134128821112896669649502 877222511738986929414e-3, a[13,7] = 0., a[13,6] = 0., a[13,4] = 0., a[ 13,5] = 0., a[13,3] = 0., a[12,4] = 0., a[12,5] = 0., a[12,3] = 0., a[ 12,2] = 0., a[11,6] = 0., a[11,4] = 0., a[11,5] = 0., a[11,3] = 0., a[ 11,2] = 0., a[10,5] = 0., a[10,4] = 0., a[10,2] = 0., a[10,3] = 0., a[ 9,5] = 0., a[9,3] = 0., a[9,4] = 0., a[9,2] = 0., a[8,4] = 0., a[8,2] \+ = 0., a[8,3] = 0., a[7,3] = 0., a[7,2] = 0., a[6,2] = 0., a[6,3] = 0., a[5,2] = 0., a[4,2] = 0., c[25] = 1., c[17] = .9151119481392834649360 1610698373256979358518243599804579540660601644226008634523763107485062 3039876863100621063087536103958631490416211115743986568715697781447948 2753979974550379403003213266567446387644355027768775794931510966606299 5385522531030237086066232139087812165027662544261358019752673225861783 0508789248613910091623773146579546856539103064178215506853921261305427 2624987427453101119435493023635799450923138509351387511868705280804913 1876877674174102786069790214538098422005342080836081967571956724924315 2128821797131431877132294414043373011435260805459923173530592218558665 1888457310933587178360962636830711529163691474664329396324917144152388 0787703010944534277182810730130294700331660000195706462105911691953890 7955128261507839684945864836119424086246075644820364707051659268247555 527246619208559689, c[18] = .94461160540655634968473757202373405913644 4084754527954863740762020332928013919911408755584417513113645839593777 8672048516296902865066982189544724498708808196748179191988962209325931 6859413418117511300876615221650454054201315900145968070950947806134595 1095002173861357052054240732026987421658586814222883169978558823732642 7547989327830906592206012030452762011517528366726532174021998699406919 8921469531852186849711638484379887705896583288755544763912470878424818 4919475516406704087588432069339286883453468341465461694397076314543903 7819663158577755348874956445232066411020357879199337873295523048330984 4591126515230211110133619799246048570267752125237175818192892052312462 5669541945012120018902471164175980492814142801262620434541214413010515 537766008412666878878714195169877275367772413252960806369607859954602, c[14] = .265575603264642893098114059045616835297201264164077621448665 2703185222349414361456016388321212052123359448912769064344638009737741 7208677881326929418701188883658375941300611876495807346143443167970670 9097438250444457119884961314051671896559239096751403219968602570886602 7668646640726581159257758254258977613773134235079528574750070920706748 6663810670913771750997206477656017617484116382435638956505406630574966 4013494620685206225057773056978489469685608068902341078370266867103340 1303719615188225577290435846102562498585177853403611762238102691528525 2347310125578886137066074787601995445274109609998007276032476391081724 4771328491616912426148908739319147327859311033284570385261362162308646 9739556234373392897468752169321645958285289066383022879459588401546561 06579946144665299888104065407840083783238739163999, c[12] = .517958468 0428461035835615460185207449756996828425551866437683588600807071775316 5232985519958993519364113695861613281622528618547958011209657151932935 9325316301362006917475669803158119351703687348754642535566825552673745 9845467968394567691856597948718039009349868207532745545777112651439591 5395492040990696454515392275038654853161370327690020306808638000869927 8825517833255614129068023549420893395181357975013478517434860546439103 6954166773562370142547233082192672291866256910712556714143130155257842 6002291295198656475371119314504711047438162625549674361887905516290058 0767346114617975728873460814763141033316850823904250290588603523446100 4929484977166996480190403865146755633922701488940118151007729071442259 2800284393208259044722364026926122691463910661943106608250074943908198 0463555855086857270392297737551e-1, c[13] = .8488805186071653506398389 3016267430206414817564001954204593393983557739913654762368925149376960 1231368993789369124638960413685095837888842560134312843022185520517246 0200254496205969967867334325536123556449722312242050684890333937004614 4774689697629139337678609121878349723374557386419802473267741382169491 2107513860899083762268534204531434608969358217844931460787386945727375 0125725468988805645069763642005490768614906486124881312947191950868123 1223258258972139302097854619015779946579191639180324280432750756847871 1782028685681228677055859566269885647391945400768264694077814413348111 5426890664128216390373631692884708363085253356706036750828558476119212 2969890554657228171892698697052996683399998042935378940883080461092044 8717384921603150541351638805759137539243551796352929483407317524444727 533807914403108e-1, c[19] = .33333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 333333333333333333333333333333333333333333333333333333333333333333333, c[10] = .333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333333333333333333333333333333 33333333333333333333333333333333333333333333333333, c[11] = .944611605 4065563496847375720237340591364440847545279548637407620203329280139199 1140875558441751311364583959377786720485162969028650669821895447244987 0880819674817919198896220932593168594134181175113008766152216504540542 0131590014596807095094780613459510950021738613570520542407320269874216 5858681422288316997855882373264275479893278309065922060120304527620115 1752836672653217402199869940691989214695318521868497116384843798877058 9658328875554476391247087842481849194755164067040875884320693392868834 5346834146546169439707631454390378196631585777553488749564452320664110 2035787919933787329552304833098445911265152302111101336197992460485702 6775212523717581819289205231246256695419450121200189024711641759804928 1414280126262043454121441301051553776600841266687887871419516987727536 7772413252960806369607859954602, c[24] = .4431818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 818181818182, c[2] = .443181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 8181818181818181818181818181818181818181818181818181818181818181818181 81818181818181818181818181818181818181818181818181818181818182, a[23,1 ] = .22159086425014402853738033793435455485316981161856508116895928530 2775607484748758986986133800538232671196660116327706355407186432117180 0396177127479066537238282390636961274080387653794856010922492916952750 7477645981801106454846067034432685402214488087064264349582909139695843 2969513301923274222443197512449590011916881722975905801390565933502221 5908642501440285373803379343545548531698116185650811689592853027756074 8474875898698613380053823267119666011632770635540718643211718003961771 2747906653723828239063696127408038765379485601092249291695275074776459 8180110645484606703443268540221448808706426434958290913969584329695133 0192327422244319751244959001191688172297590580139056593350222159086425 0144028537380337934354554853169811618565081168959285302775607484748758 986986133800538232671196660116327706355407186, a[17,9] = 18.2561062410 7715556037276346353731886721428374076065805698063141596163673691063345 8034258889973102584105311257029387264668229631701146021882849236316433 8263758230441515004348045774610267806398247326889744918864578534305249 0727671572255223333210328237550716247148083362089381492132814847362116 1858516722612198746784214724958689131253498980459371492182840891189411 2588526020600240440132641762805570047476075221500003385390744684394094 2790686541015707344361643147523274310693288768526559693941155607059417 6202096686620947444886211673008481084779138721652673931521696354040132 8612745141904050415096448836994825755388015639715279445137878723975611 6864849960201468639816628698464319151174582234381058937351484815122370 5336479801074600246877207269300826106353249722185644437737578284205733 7727253326176740535467438084, a[16,12] = 3.130326620946360027198193034 8661491976958055352980021448156894401961617504420051750959137488701900 6234169131782277800253602823139854465418997929192987926644187056976972 2774869499674857727007315873864789986460575883238289688926891489246985 5981579873526927656209048160592007874164344163552995208718354578892074 4014821250293039123145169843544711009098840532387688211784590949587784 5940935607652490945735437774422292110037599577112746328357459689758992 0151804891725208234402697753246429993265926883424854593643836981648166 4550767772228471622148209999943592135491823861263579302006330752874267 5367009116449503353363130272139039682392897663812749985573594892226414 9797703853935298721466101966502872039254317170435026967907876521826550 9290031331741189629753112432401941757121059053282617990437917457213970 113810647961, a[15,11] = .48343186094607047367852647497250192831856741 5947728768753244274146235334814700550833608385649613113630708485220068 2128347068493584544324687203081493648498576549702626583757802204190772 6343715850309207925616702783180343689051921808118175127715190850384765 8671115298188529628018035786758803748945446993463804323704431436173851 5216786250556665530750823961820289554067749517933906976830608383006088 5282995562534153049800872706968467470520327026193617600078400795609965 8164898378950290266326070751647022611540629536488653899375487110982938 5120760857568681503921827444287425223860822148241829046317472057650062 1642520021873036501667264632162774548241874655432642977749304833840037 6001989780306132895545840540718175300108032649836037540698601381037911 817817828129263242691217173694333906870834104738664037543638455146e-3, a[12,10] = -.32163742690058479532163742690058479532163742690058479532 1637426900584795321637426900584795321637426900584795321637426900584795 3216374269005847953216374269005847953216374269005847953216374269005847 9532163742690058479532163742690058479532163742690058479532163742690058 4795321637426900584795321637426900584795321637426900584795321637426900 5847953216374269005847953216374269005847953216374269005847953216374269 0058479532163742690058479532163742690058479532163742690058479532163742 6900584795321637426900584795321637426900584795321637426900584795321637 4269005847953216374269005847953216374269005847953216374269005847953216 3742690058479532163742690058479532163742690058479532163742690058479532 1637426900584795321637426900584795321637426900584795321637426900584795 321637426900584795321637426900584795321637426900584795e-1, a[17,12] = \+ -41.229469675875954352584102641355611052690610093250149138475118124824 7003211528516323192062569375442986836431495405165186763565907518213555 2575066963783968963708735057144042696064931997030679494322876370384182 6470307112350752833645386777075927921415326619053152256663194462869046 3170218339407268981540089351902871255338063287502308711177599176045374 1598457737998222128877210917126451353259815146295567514107295790863456 7570687560612133127213199487431988118168176583101709290182359192015072 9519114394637468671151038457056226091239620334202186663553310404273855 4461682234866546318101580500940805111438017690738562878480344731677169 5061496952893692893009194668005576054129863879090774389141433498891155 5551245531411628406776953930834759721696466295739120202444660198727535 159501026162939258174034918822714908698154, a[16,11] = .75217680717869 7812330966228966936488662975236250079075056955681279685566079805620012 3701757495905971759237032936045305007422659817597803087961808816525608 4679076525291677570924856538514759765390507619006630870293955945203382 7107940546812594642687021617850051394261606336826793924904111319165855 3138537601435320844886957220842415152547062594132898718541905718641574 6535553806828953350455915437774892708453310917177274918784674869144468 5242439533026397515478463387884985263852812371783393886418559807431411 3749404206319963602759923566728854407186032626135695442964482160949848 9215127242482938088460222878789508567588832930856598586291000471491217 9111233983593878970167522265914490054436346269133301556604276065455641 3827448521734377729983433411754122941761416923088388246649759499147823 26253498618944705623422145e-2, a[2,1] = .44318181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 1818181818181818181818181818181818181818181818181818181818181818181818 18181818182, a[18,9] = 5.145387528105505843576174175871990694127893492 0240624934639496226197915697531682934580293092142184840137916299666318 3561590229563093997630875530477123898293036548855826167706997613292914 4966700959540699161810604373942696810089518365218439507104265160926723 0111344858323356462113114628250788622263034436659383649973113068297467 9193138696066272505707252071715605980206034749216420096737018428750911 9505495752024082257806808868903192663531139243302819284426989776040401 9863219387706137235813619094673915097020009333145442071546661804825236 3507334719897392924946802546358534565147020566143332871200950746993951 8720251904097197417537857444201932085237891865424492673186297537655504 5962401168152793250091840160442740968742598020005868349004207405858174 0813987802409582591343529130918481044690647822285506369054900389, a[21 ,10] = -.6975023816048118989159659127206041759560640169551283418432821 4539664501353017181306744991398975395696125487605888342780392929671620 7858785232431344035183606524557013660305258449674413481589520970349333 4097580657911058529252920145626757793251868219740170424087194168612692 0513835855845288210995736168288626828916115454958676584486765900518102 4906702077345220682072950300414910370015809178002417727354316420405063 7284041626125324771723404490227361173692771228175270127505805725394090 2351724723056435280194659516777187065998563167442740625325429218539310 8538011362590724259070306138723921463347806749199651886586643553382756 2940413574317177986550019443014544039981966021401030065497420751854583 6925804098750589300193317274756368438377128558305159316650584402885197 7125070928547440218228478368667298747732173106670, a[18,15] = 5.342112 0942992607960035728102074953926242735104879487243387331659544272467919 4941698362671289383405844009092191075758497595889727535445613119547317 0535117183169808372436992113891553998891139241284254452066195287383231 4338850423088575554518384627076998749607140879082666432065340850947133 1333877122226881924249140613865502035169423803149485641054140266129456 7378143480083388215420115117927391157503909430431096410099502740786785 4755566838161700401420520908051848771587256367083838388365066261796736 4554439570033108655366648265489069808326615583269082940467249975695041 5728110371131440082848139930873556188710285480971531073826267246194313 7305806636952433157320648282874677649748886912832957638095708047273113 1679169086385794133230502087988780403926321057736813399304224965212518 595753998026123170098934395307875, a[24,23] = .43057675517113585969937 3747652338867218170740133541874736967732614744882255764725095947889774 5617175771508972933587100583539048121557025395569123309526950207599198 4974968750196292736854667998316593488734367246436894240614066494557194 9924937657426774957443734649907977964962531642389180972481328634870383 9800001256273515869875189226198327899950377196123139930025565166047951 9601007531359727639901759411058975760202511290758223880503263170457472 0008040150501567201211047669298559682414055188095552163617062706892544 6448200702256895371260230777444865296072260852632835220884290927820805 1456963210030087750705083510781967449953203811533847149201324112285726 8484494444130376065476975647137894862469456850145413659461938053152932 4564544192561604512534469004591679700504393816621754888474318628651830 07644424344068191, a[20,16] = -.38737655238428491600940087795351463282 0485142167336815579611780876137130711419531546293794139634759583953037 2233379951230141118508483237664453009570979613441969371877091222746001 2530809385662833419179273940677143894909832787358213502334030977946916 8334776645186463337226422169789096617790367673276160089845622696002534 3458383267216233486427156046265545910510199497107824661091343810891199 1344002686593575142332015398044299134199870684308991940644227049668256 5517104691649809299120498500234827833817223659691014582218771090755819 5022301321854355461662443896776872945934030049285245700699821313007712 7572247916301250298970930291012896112041268494649785357870315874185216 2362859955001353526130972860630082463988945150660512838654444997897437 4871389093985324985841358257066370056375985886346756277137061737736141 61e-2, a[19,9] = .1517928184840929633626316218223904803007006755867943 3435121147290860581539693730300046753991222451719204171832116614177217 5642117501202349640504577420196038172281704614223654965398977507091955 7606148514066173624797727088191024392278222454161916673342005333363941 3221438275256554193282570308900218806627105363496678814961316354313039 7490245478021603844396836441277320942301795057711020575841943510502867 5474976829007776685020179274588159960291363998961152571126347535686311 6572333400602711874836110570969019845996611722435813630966779272638051 7069790154736621742129405280575550004948632567354940516280089826759829 1600686684661120445315179810193675127692695490937867210075594586878643 0109781324927679138441734405737325745393507510730673383279332788451767 1751949904612235929986863320293828890513628769410817697044, a[12,9] = \+ .451620193219791006002184275632982063485229374702452876659059138782889 2817097594590705704628224621009449396173059970050954801841737610931773 8256013795373106382649101968041276584212192539734879830794404834213891 0282277459246826729579715237401785255753512469538207219493495064820282 3846897908172192840661273995613086859177679778592653021392838023208979 9377825649635033195080353791549144395806098809936180950369792965402668 2141604840124862715039132337375046844183648740069980627814101804936048 9688643889215240892604305842520621710067295816181498625129404627465345 4489794081562897661661097842705415763460231069215964632204896159202143 0174769519126784591549164411285521671962526326983363271268458043323866 2214434149533983864048251305783106334025883805054934406169713409884309 88196517666496653691700791235362298093158, a[4,3] = .49835526315789473 6842105263157894736842105263157894736842105263157894736842105263157894 7368421052631578947368421052631578947368421052631578947368421052631578 9473684210526315789473684210526315789473684210526315789473684210526315 7894736842105263157894736842105263157894736842105263157894736842105263 1578947368421052631578947368421052631578947368421052631578947368421052 6315789473684210526315789473684210526315789473684210526315789473684210 5263157894736842105263157894736842105263157894736842105263157894736842 1052631578947368421052631578947368421052631578947368421052631578947368 4210526315789473684210526315789473684210526315789473684210526315789473 6842105263157894736842105263157894736842105263157894736842105263157894 7368421052631578947368421052631578947368421052631578947368421052631578 94736842105263157894737, a[4,1] = .16611842105263157894736842105263157 8947368421052631578947368421052631578947368421052631578947368421052631 5789473684210526315789473684210526315789473684210526315789473684210526 3157894736842105263157894736842105263157894736842105263157894736842105 2631578947368421052631578947368421052631578947368421052631578947368421 0526315789473684210526315789473684210526315789473684210526315789473684 2105263157894736842105263157894736842105263157894736842105263157894736 8421052631578947368421052631578947368421052631578947368421052631578947 3684210526315789473684210526315789473684210526315789473684210526315789 4736842105263157894736842105263157894736842105263157894736842105263157 8947368421052631578947368421052631578947368421052631578947368421052631 5789473684210526315789473684210526315789473684210526315789473684210526 31579, a[15,13] = .690775666409550270905643638134390723241453375038799 8041128561404598405487435684376736960443873555173684597115050793488424 7167361990707521760875801483474209527600758487068262778330240430790528 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1867702149435539901521541407277278131557783767838497630475983558371263 2615769526791906032227939663255972917169392878388921281649281828552158 3152775771786673661672013470994861411689029990863077539672633221497520 1991241956199274464931370552924309676848222082034508769969690253868390 4942207212550512874066055891127153233258899486521224641379072255517859 8391348048249037817998138813146970410667506414070893404328199230937748 24437660067775700593718814975478021917682e-5, a[12,7] = -.483863054235 4663103593747561285591310313235391360244187778176821086097948658717661 2878429030989277704899657768693483689258030105041665855331533638210329 8974503871195253207016230837894136153825111066268073729519008530084581 2733862131158289661549159530157018716879111867199188724474179713715721 0615469219519948327337839676213574394206433757954980079602712180493841 2338301358778389404697180301390582683132919790903911202549919352876679 7343612526432195724630636021924840765356500149551049421665392160079154 3605852836981724736231028386019131922757739415711568327144583077752981 4842936443948859888396605522650925381492585554180264988848217380299081 2773755943509665754922947333672496897294459856020557838607553133854616 8067021740486103636972387632484895849279176375606156422604774655601561 9202447778033210203078689175e-3, a[6,1] = .385095150452495257524472652 0324404860898823844950484490138343541601000605579674908780271724180896 7330903623077073344832483568305638754564946176268562706534611227880106 8396713291549919669618537601640503196056287759635026052597261005895822 3449924579724009834994057716441936946854179898872946005206823673888745 4574924777712133715371217172909100220532371586222772013243557598441015 1141493928628185166940946214282282491567025843521802188217541958963133 5553968130775643553797069036610907931298648981839473341031111589802397 2930007766414139414261374945685258015483199255775841541920949413770364 3984619226600311606908571196265363218452704332983290330312826515414299 3886654702904300988409511243877547249205803639074513754366958052388185 5470781440646064286479839793289839302279292078943936626905371708183456 3235558878236e-1, a[11,1] = 6.3443269277336668736962630589807787713285 5837091415208438012268826311611318963187446606535807919511692725322509 3367128417193964594517126631168502295668738093817847345424523467547184 5350854404194367856191457166882346414019934926633096386766719068622445 0858874108055107772733429191122058159551718648430136887138603080110696 2932021113010144229182062585595637718029697452110243002029332091830957 3934887862388836743902387172728770073968530289626286682921441053903280 3168312257410823329536620521507527013170836349586403612804537357449044 5920965000900653343505280189524681163219213863338089948939663504868684 7140776364351471123551823017734481113982302607263924769396219857075791 4278069897586122058636965468689990435875489443414646766688766118098120 932443148799661265726622736994957921778614327032430297759105527621874, a[12,8] = .2472202336562846003574209446082032953844234432690202993740 3608886739721093370977026405447729718390926772637195635644658366109880 8395784961007759723833091431481644873933376494826749866665906259402107 0653907699983899159326580050671140097585661746979744562950255956385871 1947317772191864303645287792723789788576796701569037502273235038770299 5280382458030838429091105060872831418670421209727593443002573695638168 1279745105435622591621024620235193549808148382563685694844584853902071 5064773196902488574593985804984905925978624923957407054914136307038333 9209916235303911376506295226486901235415902885009666736476724360092490 7329236191842098519588315659199993425400161611104233426820289039600345 2487785052108858667359782668008891819686030723702386479856851976289344 5783246908857178569014996308217959873252663731192720, a[19,10] = -.232 0918712811526197475496307434660245851297260397453138111556440790410098 7214780070988805291289681122950921232498247682730252562800440420221936 3357198366064633017321683316648916878776989059750108910179269741754807 4270431025749065921320196785072023369321742856889730697620476148686651 6937474587394202532967469121123317528984375027819638835985031340540191 9899397046469300958640117013996485830570730179772727172137553309155588 2282670831098380590582335291036231226061867223628913744619910672685339 5235272599740514138621731680979931921882429850984057315596143313706012 2702494727410308722670336666081586158740222644899820750702199087745017 1652034077240967146462739969486903612732118921860317361898975776667784 4335318746559779156843824035644078018038442623951082116474566286956747 2163494398863524151760067706332851800, a[5,3] = .345698194816489700073 9691452644149026866903085576649936607708344667115529230058847942638248 3433851513624073228056595993624756451601680796832585655619801816213186 9811090499681139030623741132374029011616247352253804114103479678342465 8248127234962032932447450643505689628156880179041604932475669877930906 6965137453790743139573923728649805848630721542478981644454321285205748 3590559162022350115672535305306070774217798025541240092045113722916804 5532002749771822487650411968514819082000070456592565456916592230899493 5574274692900823162491315833602342640451683539309347544895111328702858 7666176747657771978237251566650281483111956148554144257783847558616461 8201471316058474770191105193143848792067276495319648270551818431469332 6028047987632782671709140260248685254775530206135646429695928426277344 5774424816297163287e-1, a[11,10] = -8.37099453687356232016824911619372 8530299734845668945403444620801647194262518201483006012307880227678749 6479839537041640126061850847038389848869556466316867195632074627720684 6486034603520493488010405716508611736741850308073644727995429870420588 3238037983829747089995310235506118149427961540000581617252599530114228 9431457164935831202671406314600114965385773602197979883685217443248996 1504988579829418429509349918699275722486872645855080589879682194696028 1200458021275948113969139973700446482575998449226757587272790970939976 5019151284524720031480888708389738047477033639337150383248932603470636 2963360736619692954645998936501406692668429212764305917076782825657921 8062466650714759094568245702782972718506645330716959201926318188797208 8378226526542504837924348505128609595542947652810426618543427602299676 7839904, a[20,17] = -.903324941053900048965709808112248216415342898529 3284160734504710049940707086205472023934269642310878358570734016376709 7967379243481525972615565534592052788579490802406383808554722005835262 4298607542894427122088208399006185886372016877121441814435662786975886 6589378519757859702443720163896487062297301297662776569003036457378554 3997963437370162208644416687914932385509488867562135337320940249656302 2102561733403634510645974120440585433462204258454182263523924552146552 5502056533485020870626902942849377908379969438124694057742652058531560 1884853881028894710472425374060825470105418980506279751850026028910551 5227776886824150452446285342764757511852935792697773060014251511606632 3689078630652037920540363815950570838615652670653506336269931862820677 94693992379119630406470882837504893338592759722093413379004489e-5, a[1 1,7] = 1.9752633196847668138509555007281881065578803925591277201356668 8911118913625381397572912234504847599572327686097718514102869531937605 4735657433180017989537436056423160775250466739811305803148792812283754 9694796535885709991094500495211567421622985151383354782338881603865033 2301052394460784053321571605811408098163725986977317471770015669006067 8667443669372051976769670050530866212130711740415124339696944996230467 4951609095361560296921951756795955911883591706651524673158313641333306 9000303853031900444223281195928209010379313290035931233634886972844780 9720578725435425093703436546396164259426447675739662588723123890476962 8271438193370276681485212649178327073114080222613975996268548014205701 8756631978408134819911361961314920376307784873178382054868478600104418 787141994284157704906914959769257941332699660807, a[6,4] = .9889604363 6513824628127989008705482822888021435539033928284235364900534244263455 7770889045003324618521559620561148584962419093276452577291177866404295 8996239241887732926696397704831809568916716842689953024175467689207173 9494115747051465188691884104843845252238687935200062469538879277531049 3752768633508853310033883697278470401404176016732192317986123617056460 6534971775791957542685003373713998761522028980614606268227784558854270 6768225117118941939644102429903274172918734666802951174503364107569373 3776947426890039259652032078612131783262733672546623760946311596347259 2289846151585194322308980384683884896107387720112047707258025416282961 4101812199535021272296572231961854823401968445891216323717947666303404 6939740166723784717515297990690225576988013294682554335761141138004887 082083510134774902928377917799e-4, a[15,10] = .93601599069101661597116 0494932065698611000072865875344529181178335240484036032783289676594221 8762772754199359092485093906355027518396879856339583667259212391608759 8363398375228396789309444489019072551002659045192920810540823917120994 1342259204830051999092411885269742799068181736913224906559274116416714 2291384203058293589024111122244046536666859459191953493073330281633342 9864006937217645552952898826015137273551499321180011485029517668473355 0663159739907595804566294207842766899846221414878885739044330028310249 8711326740084283767674053589001194997820004620685915914934202661494938 9849566282294020521904272250149752170481227997774751751385825370343085 5137268778170422827861828716572524642716181538020414733414210217985198 1228909619990944038493319166820129771934630981741295064426677869791208 03375817650932531, a[20,9] = .8017652875276497521699631452354949789805 6694411279125574348925191843349986558837748782747047179650990525559888 0270426239416544399826717251461518334289931242523953255315580023921944 3538457057568343046475232097697741267148134164582023420776040614554543 6830224874358596920685225819203855447139607979139123361654928107567322 5513950811570503799957247999991473082446769637017980840637961341780447 2363296458320576253372430606478021629153883057658413220043914259585150 4274333011958149108514986518250760825146457790970858225877981172825075 9122433198427350198735532819550925995032849226793072995407960701072866 3372147441992051237869405771137618994008809584563859404315431435010219 4751430190507730497800887853642109254004731882974807295203572670130238 3619142433733144816273389789414112459791043988598593173965697285800298 e-1, a[10,8] = .302665885173442858379928835836683483067049320005090375 7742452562250879940584583879948023877870591344192057677608649611599391 9110373576980816955337740045847018889725573242627623917673731163897093 7545305816098013949881893017019726292141981859675841016120584367444305 6808498402627218750287953480555619438821832082880268613228509912109106 1002896979747164488834004036517691481715149638527568352210336241279735 2848088799878131612767124312748068261521229912907466164814372286296756 5327611921539931870564138078278445003887802112569802099406742231815456 6168340474016846823470712602562343517808683822896464106336768188497297 4424224554950686447853745999272557887558388186480724809111539209847819 3170766127357742906391232576617336695824004026878118061959582671703915 17970417360439379480934204593026864613797826903489402411, a[5,1] = .86 8116319391850886508062971348367434861757015968588067387847640565978467 2427434203839647874085275779082485998725399445719006652106879728220423 0221773654497194530217511291341453472016474729243147274284852714695292 4012668121377383621805345502876938484117428195776897513883600452859186 1242150153658226078987104974076682006523928792904279692021921759251243 1575813897920358657064106242412062163023326723801152015956470893000270 7855099393501855430346887901151942386524506267092671702437122844235004 1764187612768491222908972704422640076715215908960185154886508122436896 8024916512301150805931739578391876763134470729302234791733429500292455 3832027433873921058842395434603887496872746921589633012742119783930113 2596120549909382267556837383777477726318913258508454851429314905519654 70857402513146417021719319764405539401e-1, a[18,10] = -35.962015538930 4662958673118000709823723780410021058959374385612679454436393953885988 4050762258999486870987947175471094971645698225423084399735725647270740 7256579878143934596477611372266111684218762218084168956554372170878707 4165733472160621039653045822144323089797305203519761302485700271733679 7140013365567826821604802348211097664839583394502374185168354209021030 0224786623572489569691406464059851777258999135730139115923580383097227 7579756764328012770397933625534524749647610348849136168900458165347202 3789131386523839206111055151583873955843471363807854129776772568279356 3955242211224622249733342506588016635217527065745298370387080202182621 6811850563069608700072916083865383896423105001308859290307525382747423 1819820192522692451444150571118185013039814680969216310469711200781251 99509716398673210185285523, a[9,6] = .83811983297409393837584672686841 2810663829046917352897587294440924486336538798165765122086593212078253 5728708153550053728022822191921120803954894795680707602482990292490278 4557320917634367207824781483790632117541963028390908705099975572532001 9540548152880201467293523239217105596267593586934821503881219169663748 1882847593997214116048706867926571526230108642643647327749542040835339 7062378301606641971656696329405895518557226426228697388079678451595152 8466152662587022605746832942647876387956777773722094942405182731790156 4513669135361999094071545638186072051450293883546997081076434802920763 3162062078549524068924458347606551126198809016382742445384810261200148 9521285350806487665007566078438763765782669383919783112756088970784654 1485529177752466657648671296786926018572439905200116555647936229364629 44612901e-1, a[18,14] = 36.4904433216429590412962072205288231186595519 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2321884803148483573950179061246125784185294256563735312959231865038268 5674197437184129125171519861078689438476232512162116970166902677346013 6389615736859224568261263885983521497101970071057425721288203640033074 7138050470872439248027322010083359619666685795354055438040539064573936 8554866605326236111146385434260066889898359695169017588528042036832804 6162916656572905403025340199719657658417097989875001975376511675871366 7973172125897418552816900467596346936685456177383409841665181567208425 4112382336135091143770476464209715765756861141016615966257820695268036 7508081187134907821887767921468207966162557707828e-1, a[19,18] = -.213 5549195295375067495229039525877007339581782873806009604806379884885927 8859930146668644191059875812291346655472654729359940927433218841799436 7215732059486296027205544002408287623907606138846605678960980633024157 3307976548624218690950384117601313112104523472324498407887669870313243 6141917779953276750550378077006104542812694811173118521021269834731577 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0984495557740314430512102813833368346645578734171547604451612337804581 1234138401135378697630369443539124106668204847239853074625763236281644 85005726623603637417338594487233832184147606703302639, a[25,15] = -14. 4632469865853858907314210714528496473041335986842389594106035484523144 4731889608147930984082850721619438115998438744752012982391140138820942 1131337571448201396339907699604697824827062606639516314625909476853836 8819236995316721979913182462154179126908474941722545380914307007457346 5570232419832691740909898365578531587376152567413553809541158318289571 1687437086534804984134101641564622687692487870473604360481311725993441 6876910185174162550848270637541035346171408977624777036655312694024106 2834501458672400139920173952311525788078135912031394804279085431243266 7830680854932013264520297778716744138292919731847593329979181853372909 4334525632600951645107258999295724372621722959523038304912226659564466 1318771303283277169831550065713066435604961666786889490652062097486231 92336197025752899846071150053248047458, a[19,15] = -.11578999193032480 0785527884513702628718501695274638270045923074011157144698422444960121 1420765941432854492858987177904408864759990780165136342173490832088047 3002958648107024251409312909806543540447165836857891295581898553580832 6305408626494717048435853952193663756402649482756875871104400747591524 5440566463536728655766301508898203519232878340626274839800535627220123 9024660766530664786627603445786475079105441240620595822662359328135573 5796868793237980272643823383995824893794786276772523585265747549981857 9819785980825701716912402379015770488097817489634507880765941542652630 1016599349895981676299214378063344854409317650655082075158872676522150 1986242187038655189076177062972853442735479828867137982663089350955093 4072935141477009633370637826644589425452110980041489627197159019584469 16748320218745500901127e-1, a[20,1] = .6919958468547614013916962778553 9119261241339034127690305506136513492643189698364906669042010230684577 7391344315728541145380057874914370567394783590167404824033200535834314 6299246839222349758175885837444075437338866888133677424093614825699785 6329507129172488361904052757587458785062648665936967384756520198847565 8516448259607649249803689344330574833797367237012971718422670667095565 0648294314343093279726011325593110333620562378843668857263805950388244 0970612386978957061252213466112541228437308063970710633098751284734355 7233861636600542026592274030276936221001607496815083554463152503389668 4729386120400840488005548453499116555200028684407239761806806307490488 4176595794080052138489722945375939893056731178257473563386800729411933 4181803989948684884019789341186391627774375753714469392865174033808569 725155159e-1, a[25,10] = 73.423305232063616355688827073282001106845297 0882916694382608214612956422910519680535847758662299228515572512937947 1606290899146827779827225605053313192447225417834650674478647820103837 9791697840164390445049344662864272875928217953298809954761847297510693 1099795041040078115131339789674245112073296382228706888158672101509775 3773616639599268911328922744820038807944179944061448991866409493874945 0887059424031460387761841929178361545808093398307637334214730100712842 9033062561725644851551811465388007798472179159488288746746088483531570 6740205955800905016495623052270323588529911677597741029671156115966575 0258986140554071176747741751927859659240029736743688281837853559195815 7694791038621035850522287238908558635167569126017527888421515285100900 721344186456974272307630151057680370937581562688966741290696819409, a[ 18,11] = .831177632461941356986857095120265129366860189349844351751586 3951228397327718531856230683990243812008381160006187243863312079701068 8634755558367368722087518847196736331222813882902262581968669424745345 0742785305854699854561999829851429426444237473946148693436492857469559 2477830747137734538884934536846468595339571479143164801250987299746614 2294716101200158321611719770872438220551551441609222747096946297913117 8758608781979480044172096504069992459023373661027086769097871826838622 7610849687000884410503728743345620706215344618890510584286238975735322 5283900335302807461733741172214055102623782751494363214134476136137041 4629027038188871010262732288542129652361566580040581854822088960432915 5414815136342652970829414152890573655885874568287067204871725770125592 18774816844782021176643610391178501030519627977279e-1, a[22,4] = .9889 6043636513824628127989008705482822888021435539033928284235364900534244 2634557770889045003324618521559620561148584962419093276452577291177866 4042958996239241887732926696397704831809568916716842689953024175467689 2071739494115747051465188691884104843845252238687935200062469538879277 5310493752768633508853310033883697278470401404176016732192317986123617 0564606534971775791957542685003373713998761522028980614606268227784558 8542706768225117118941939644102429903274172918734666802951174503364107 5693733776947426890039259652032078612131783262733672546623760946311596 3472592289846151585194322308980384683884896107387720112047707258025416 2829614101812199535021272296572231961854823401968445891216323717947666 3034046939740166723784717515297990690225576988013294682554335761141138 004887082083510134774902928377917799e-4, a[18,7] = 1.97526331968476681 3850955500728188106557880392559127720135666889111189136253813975729122 3450484759957232768609771851410286953193760547356574331800179895374360 5642316077525046673981130580314879281228375496947965358857099910945004 9521156742162298515138335478233888160386503323010523944607840533215716 0581140809816372598697731747177001566900606786674436693720519767696700 5053086621213071174041512433969694499623046749516090953615602969219517 5679595591188359170665152467315831364133330690003038530319004442232811 9592820901037931329003593123363488697284478097205787254354250937034365 4639616425942644767573966258872312389047696282714381933702766814852126 4917832707311408022261397599626854801420570187566319784081348199113619 6131492037630778487317838205486847860010441878714199428415770490691495 9769257941332699660807, a[17,14] = -65.9160188937621362955572571885880 2097202183020833072991281523970217032211602024447115823451024206048383 3078796239312016558986084279436889758588713500445890307328422327546590 2559027654427882759264053341204922712297239189717229601316611006191040 2374702661047794669166701151005708204836707379762379821072635519091004 7913723283492388288423346867017121255498357601793670189259127272794675 6100561201617934631178642699929367607291701607242732137849662533143508 9223293146426218340604869498673341197947982255598905139004357973315571 7167434657305057856433432954507106640765167309204370395710397404824795 2045871556292217960427750581894913342771071503952943603162085160310337 4201398976614207624440446612726590977933439443134604155122295587879612 9165986728757111581014193698341496482521623277837503851128011369760645 6242589, a[20,18] = .1115698445067510405466371509721845171708543487949 2275580235967414563978734964877174926100264703329685471890357973708346 9262925762094430747302467690414293803163770095451535542573888650502551 0137413291646248370779600443813575012318869729860126587828673636422540 0331289826571808276387228095833985962027462804694701152407617282153215 8501278979937192919193031609348959965246251155127420387275229893943022 1960503827238712344424722439881619004643900499776147421685545863816035 2062118127090630778486653708385875087125697653291705916110359684048929 1042578925434299700768563748976646901548989732555682953910739028320541 8919909049167589584815412681607348286811379621268975890375002518886708 4731696985917898462794631621819018352985342357147439222480929448788223 7548416046092293369760758994637388068848484704262561135026524e-1, a[15 ,14] = -1.440752035387565146330805115717291554387888099255603740582340 9545402794616444767824066928555006306750522973472377078079780531217614 4806863431161033401233041403048628612962947697356337330202195005525188 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5727797882691765241148988841109504789098052442639650372199663502394858 6957757988677593626876539917418234410061574588112175943164140520213597 6446103100229906535452062235535040278913750202537104440593333769627992 0029668439783804844567778080671495425976290832236164124645638170708162 6291622995436405395372025249795187934201207366238998244039091380026352 2629349164266105070184317365670650563101221549523591069656408446398989 280528109610152363712874143270030, a[17,10] = 49.516175274311346082774 2122263756240234949092572345633745438761211782383225536920832572817430 0178155647453249927462687287966172985135474567001392819527668702951014 6774637579896996941737680265972135408813418326107502917640972833868463 5514218488874009403340720964930141229191795330419095436564118706885578 0564795243841200297461880536930067451202881935570105289834338295932419 1022960381723483677004172294892422250850283494392648662317948553843231 7343736732201100266517256690062830811596415945289470874681186679843391 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1397331226765982535813002328356588992372156360623247363814466580519557 1312533134691708831030771765125604948115372948229965176280395080936187 61e-1, a[22,1] = .3850951504524952575244726520324404860898823844950484 4901383435416010006055796749087802717241808967330903623077073344832483 5683056387545649461762685627065346112278801068396713291549919669618537 6016405031960562877596350260525972610058958223449924579724009834994057 7164419369468541798988729460052068236738887454574924777712133715371217 1729091002205323715862227720132435575984410151141493928628185166940946 2142822824915670258435218021882175419589631335553968130775643553797069 0366109079312986489818394733410311115898023972930007766414139414261374 9456852580154831992557758415419209494137703643984619226600311606908571 1962653632184527043329832903303128265154142993886654702904300988409511 2438775472492058036390745137543669580523881855470781440646064286479839 7932898393022792920789439366269053717081834563235558878236e-1, a[21,1] = 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0771210401761137118880211272015288642405330449260348379199372291213244 9270901190838097621833344924449755746983335830581848900110421262517485 1889388244994843521394235956140871823609346980121144862195563449775134 0885897114599625202695393298060161764143188452959635576630429560296913 8800061630317087640950323377532087534739933423832799835544861786330834 9708342949492827301464329223301204509185244945893917353903287596365201 3123545525581533658986825015269681362311785139775719374666333755038648 5789105038288431604974690771806678545389374567010217543086343985024366 9821151981347835521752319954858110305934024931898700382985111773958418 03219038482571343474185193990, a[25,14] = -97.338667056040756087824403 4308210666791637907879165691687639897247784501742727178916738966730109 7594689368293595475876590911348818099718000827710683964566120100300101 9754813863274760494785464124282861183767586835454813774144708886759549 8383545088749323505377207371286824125017107375315382316268441658499092 2746194082533133422080259685644900424935501109939139546623697471878617 3681344835463566771108440544779780656960106353367847183031201126008854 2787812150265678742890524416531158181303265076744974928242225998653276 5965556630913373192768204069152010590640039070590813878818775620831851 9301598436701726153758952631179736427403753428556135686680833300732577 1262187842199545365212923858467360249470357001064590372516788432666998 0903505881516118984537468616823498376385805791810629868915022992216666 21996021103836, a[20,6] = .8381198329740939383758467268684128106638290 4691735289758729444092448633653879816576512208659321207825357287081535 5005372802282219192112080395489479568070760248299029249027845573209176 3436720782478148379063211754196302839090870509997557253200195405481528 8020146729352323921710559626759358693482150388121916966374818828475939 9721411604870686792657152623010864264364732774954204083533970623783016 0664197165669632940589551855722642622869738807967845159515284661526625 8702260574683294264787638795677777372209494240518273179015645136691353 6199909407154563818607205145029388354699708107643480292076331620620785 4952406892445834760655112619880901638274244538481026120014895212853508 0648766500756607843876376578266938391978311275608897078465414855291777 5246665764867129678692601857243990520011655564793622936462944612901e-1 , a[19,8] = .302665885173442858379928835836683483067049320005090375774 2452562250879940584583879948023877870591344192057677608649611599391911 0373576980816955337740045847018889725573242627623917673731163897093754 5305816098013949881893017019726292141981859675841016120584367444305680 8498402627218750287953480555619438821832082880268613228509912109106100 2896979747164488834004036517691481715149638527568352210336241279735284 8088799878131612767124312748068261521229912907466164814372286296756532 7611921539931870564138078278445003887802112569802099406742231815456616 8340474016846823470712602562343517808683822896464106336768188497297442 4224554950686447853745999272557887558388186480724809111539209847819317 0766127357742906391232576617336695824004026878118061959582671703915179 70417360439379480934204593026864613797826903489402411, a[21,4] = .7610 6514299659419909319461902963736605339221302089006430925606619206521462 6881542488488228022893194285670672661875940487302656120909083690847289 1512461993828204892729413394785548385626744403782401791567650378799255 3441032071928876677466885999707358161956421842527499827867419992045452 7802549504671846812453917793707135021816777478145720923159097637315801 6369792643435731080852508863780248225694583314483485716772142531930085 8365715090959776325527527515541881612622245734692511829765661307194300 8491333899769587512727684142101454781106133545792341185327882301739266 7192723192940989913748435914759138297663648788728817483232107042009846 5262469454249785028606591854617118909712598488751190210251637018744853 7688409627734413919487798333504810775736908849836359743826355680912476 519650776885349556430137763486756551e-1, a[20,8] = .972972170255080855 7495179062390587533407132090396133365371679075124515126725605224923794 0578521555005668500918929203438933984137649746038359760343774821593074 9052736036422373989450436017366635225815108207870420012714725950512891 4576346746479854021102991389948181241814255040848858009723468511485474 5155785770142202375047138106130459568897979715710665100416898379983072 4483098260378813021352350820541814803905351138679151813216811539346816 6277875784869958061342034070050583327206855201364453454327256578145382 2395750853203773827608703155868854616554181639250378503854688570284694 8547630469932439731864391231115836597016897151663467397330210111424228 3738933323696965314271082266197594040695609865338403001151847975783074 6234729767041172652861715402082057948691975323503483179237900140844456 4157516324086595077814e-1, a[20,7] = .90315852414364506593396268777865 5672713424456919245602077150967097143544457382512132790309274808192992 2971054329154863895955263554351812845725963146383859138517039931997161 4818032827711236282731258191104661846023468706410035763469249332283671 3749636366402954318461195499009614661358704629823378558175606063060293 4882002984439232167410221143010716820186770214943553990152154140727727 8131557783767838497630475983558371263261576952679190603222793966325597 2917169392878388921281649281828552158315277577178667366167201347099486 1411689029990863077539672633221497520199124195619927446493137055292430 9676848222082034508769969690253868390494220721255051287406605589112715 3233258899486521224641379072255517859839134804824903781799813881314697 0410667506414070893404328199230937748244376600677757005937188149754780 21917682e-5, a[23,6] = -.700367647058823529411764705882352941176470588 2352941176470588235294117647058823529411764705882352941176470588235294 1176470588235294117647058823529411764705882352941176470588235294117647 0588235294117647058823529411764705882352941176470588235294117647058823 5294117647058823529411764705882352941176470588235294117647058823529411 7647058823529411764705882352941176470588235294117647058823529411764705 8823529411764705882352941176470588235294117647058823529411764705882352 9411764705882352941176470588235294117647058823529411764705882352941176 4705882352941176470588235294117647058823529411764705882352941176470588 2352941176470588235294117647058823529411764705882352941176470588235294 1176470588235294117647058823529411764705882352941176470588235294117647 05882352941176470588235294117647058823529411764705882352941176471, a[2 2,20] = .1311351243636448823913476512658127321487191165570692263018637 1051381380316757702092802167917465224943729444111001742177645599995676 2481073416500441754517734466502202073433799201397636085524378658629891 1307004655266257066216286987033043855995205615422629257733111355949758 6374486574289038928923701276859430465038157033705772746175807991797396 5040988995688164348456356533265600920834993646726517919129866011427922 5246900311870047815405510597569270664151638836103908851639893676703359 9401491227440491503622019683416598728998748525215931899420505536952979 1487082031426004976881972063783997471225940242375389200133818688468281 6770250217033090057651395310397504331499069596027675402622149841105614 1941666164102139619846189689466150470860282018354466634309134653945851 5941371527493796344671975293243949893548331316330, a[15,1] = .26477088 9480913257265228390838955864038635174152680771154637320982965396118995 1799080795477347329808804678620589677921836034722660434098556117293908 0872135615082461042314662261587933012290579354172081743600274761719799 7059458829071170012164724838709484729421605992983059774715902106425227 1631215858187741319545842300439531930588204310009059428554794638516252 2297992907381915539287883459048524885337935386240467352151238587292713 7471281419249092173463142512393895043669386376231361175525339036386203 1001214428521221138357449091266312389203769133410659322211686911223354 8603733057394249715287356185073532177912813724118919877652706830107246 3013576362917712795222977313875636211737374418981154628332459987235981 1439721925056230384683383999898293504069153620310619843235214293816049 53562314399685147675043327894891, a[25,23] = 1.98784065426724706503099 8549004089170294156443740931275557314338477773380820472233214615486083 6301279514575913467880226882996966099459174251418018731038121619839071 3626170689882601239941960163566811766257749637251022292573539110935232 8188893285846194433452051180583036538715209075319878643978366970056720 7492415248647935628545046827595304049597678406542672470650309985490040 8917029415644374093127555731433847777338082047223321461548608363012795 1457591346788022688299696609945917425141801873103812161983907136261706 8988260123994196016356681176625774963725102229257353911093523281888932 8584619443345205118058303653871520907531987864397836697005672074924152 4864793562854504682759530404959767840654267247065030998549004089170294 1564437409312755573143384777733808204722332146154860836301279514575913 4678802268829970, a[25,20] = 1.351106761096203756653323362512646813003 1232129503365151981700611445915629261426120617604363700347512426868429 1558527251132714555931905159899705274270883737298200853384946993357673 8661857211982580389741785070162319095587911846214753882021730523907975 1902520564817665948181058373289930937403774248889279901464830862622619 0999868033255619583864866053754453877622839044560770685787181630229182 2460739893546826199797650991950028592794615756829279021686534993181718 2070118330180794439801170105133506356398187656710508951744160471561166 5860203228786345840848106277218141028460827871376413143887740289447059 3410460563937887652311617472396955967096291734482910306602736110500153 9612017771521576562706198038094400211146791008665816214313992873795803 4575287027669027405093916333084062816170325078080323758412879954251529 , a[20,19] = -.3180238880138175203633411085576241668890208668657812665 8690606324040362308237434118602295842980636535520724846001784571774501 2810471017923183079275727815506264650272706667470010873837661350249467 0671911356704911589100914377852947842877349423170168386683995948849926 6948410956467351230694634521422034010150827874896354251842670775198469 0554680484187030007775224290833555687533532447933253572729907070289749 0410078526896250964730320218739151266845123844116794771376845898497459 4239448217293131693937333413667754050432801691613373385780233715782557 6758104451961450954401608983683505335066261551659336150776231340178973 6130118808870067681749454156302091850933744186513572213471508536966744 4189095750022235420019911459992023939657373695641570767451935629176313 9698688540581793351179051078346143804351256516412609061, a[21,20] = .6 1252096898887869408885741274219587125284560509345250101053940237587010 0657210063597803465285288015674342802383954161191899543626498452316909 2147088537540936479936387306721483895814288186285119868190166657528483 8757724186480235482922324049719977448044802724177952056204926320619528 1025051371254393330910124989440907760586092536829701367202846974191640 5747927744816132944255835351383728352947762205725762361230590251679819 8402596923930223513159867613061213458853443589096902115042388330504384 3493553083015550817739257030595545529604312062806779443621744089604672 0873237728121484316181702156705331513890004220372096347359184353591757 2013911992091293190390985722535509775691086473133472439245456386068374 8824052568840498435258476306809763645687960284790744256357154605955143 987680665109620902117425818671743036327, a[18,12] = 22.990747960231592 8008496275200202081831157308734562367723552955285930598734510349669011 0154767554849993347126200478380703953819251265767251157119859285584199 8247732414083198338725078921727383171597739307561034717535931774626109 8510234147829242600519908086243035512984758895116221580397010120058890 4565053345319021445390923843124343642523139894255130582090712957535705 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3167442740625325429218539310853801136259072425907030613872392146334780 6749199651886586643553382756294041357431717798655001944301454403998196 6021401030065497420751854583692580409875058930019331727475636843837712 8558305159316650584402885197712507092854744021822847836866729874773217 3106670, a[23,21] = .3841432262252079340469853296205751191707035527677 1988807172417492658403676091206722607826412174980489589355627976304213 2216292133499321927752640199468920987439763383023261760732177928310397 7418534383067412122806571884816001928655351049279320888565246746222470 2071410271795194651876434553085610391900167861683204498929311088574073 5135590491742036640621555420158668844189862607221741476981315962863127 1442660089843417656610480435593324050589704902350036993386876025952310 7326858688499900196551243649432497314137022565075484000026921283655196 1743726460381769852499203198080742198343342319494785585829197100266376 9297929833436599579183434781014753517720222237820524515088004696247723 9741691646186035429440536462590525700080881656705653220102665725850999 4620950850490748972589154266122020823135826558478315015290222, a[22,19 ] = .36645915985809166386214566220898593631445830043565606082290759843 3723972028481796019297050029735114435345256193129169757781421531593728 2703345010564272438256262972862073486546771250891606516379395544001789 9093767358686604698019897607888668071148376856737127605857585960669615 5969189622822205357136465512645509809181778807687817811842277899467121 3107724946672515445262523273398952251407214611773901040094699696965994 4539698163337633500860609423315176630733429157310394651872083658352743 4122726942794974793406181517272886229272355304193784731862352101237107 9532409543664720772203648550670073753394166380623409722490200892472648 2027966733111404524569090773562995987176353380231069715134626623057626 8712214065133892211718280772374447438151702377135610483545689149207475 762954502784870598643207295325788701479999730, a[25,19] = 1.5844178067 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7162630574791885842855667351665419218170528609184648092975444306805547 8989342536724816508312508340025648659435406304020455196040040854996626 2426045331441162252122648865534169545978396699007645776671932771772573 8217515488542697291838782897580311966101113259662641185789310844053938 1143993742081935102810749620716739839361500216507748365755787380872109 5021279775624136090329626027186980461587171865356100679206682410985526 4900595923459780902335595847214214706682413654551767044167575976639714 7070735835718322752438566643829878357716596668613924113546960846166921 2331449375353244138305531104940868468461428291186931477068832625662237 6515892030681, a[18,8] = -13.82822337504897849061527131653164721900473 6953958582107261645711543833577186619165081267663813588801649581590858 4437385876227420299010259012685294973952345844995354908950270170655649 2593264605482338017776851157965101626601017254315006770440846367062972 3882219333877836713102412130440642163132313698083211964771002624498517 6713761258537941646792530223530077977229614531433145306529825467534370 8182248955853119026939892209735224770877324202273946131616692693890943 6932180757714634219528698092877692602215299259099838628334888071480624 9741071641306091569654646503992989796155653635690636901914099268861336 9231870961925709844955577403144305121028138333683466455787341715476044 5161233780458112341384011353786976303694435391241066682048472398530746 2576323628164485005726623603637417338594487233832184147606703302639, a [19,1] = -.15372671963960878439381104565445971886121401118135009026258 2474116867410004264764791722290819354367391275477938417684401787827275 6274089090108672050331234254654932063707669211587711863607646003962168 5826643759793087879956723889131844230257281004203941640028062201579996 5532373345632753754175530012327674638321949610202159071476508310940451 6941372372374147316524058569949972232884625940606656551100973340364789 7510572983906545149556029163336388911565357202538216131860948823819692 0914653375496118433017135026327925728799867541875445628911791742175700 1500909510344636886784828518727329067795490521941322304500108106485256 6538704008907546804563929230524679830351431048691079198027976255540636 7197640828558021462642749932122336375122197860429635212179311915115638 020191277514656829946050492553309461809311233875715, a[23,22] = .70036 7647058823529411764705882352941176470588235294117647058823529411764705 8823529411764705882352941176470588235294117647058823529411764705882352 9411764705882352941176470588235294117647058823529411764705882352941176 4705882352941176470588235294117647058823529411764705882352941176470588 2352941176470588235294117647058823529411764705882352941176470588235294 1176470588235294117647058823529411764705882352941176470588235294117647 0588235294117647058823529411764705882352941176470588235294117647058823 5294117647058823529411764705882352941176470588235294117647058823529411 7647058823529411764705882352941176470588235294117647058823529411764705 8823529411764705882352941176470588235294117647058823529411764705882352 9411764705882352941176470588235294117647058823529411764705882352941176 47058823529411764705882352941176471, a[21,18] = .253955213538338723195 8801508125376392283214913791972115937570999107334296436460657367696398 8633938896494690653336274793196006271764825842829016641160529210014646 9077346274183850323575916751087698032232114533178399518121353886795640 0777060277787823396455447276370448390454107369802356387838302728318367 8673220418716780474161798160070181664881580395447582953217067865046345 1521821887789227904805988155681410430518030682948893007875856508037949 9546541276204282093943884820837846881166772223650378563173128908062076 2661579833925397926384407684323934843303329623758897631798179173239950 4277676908472464416524949545930769653976063027033829958257870195370327 2997130842033313085134583749571111213629600698398791736295445104482278 3906104151046988283762861998688872582068412003961744529853961571389739 5981043920382457542e-1, a[16,15] = .9314731355465816846944233097416818 4338584615681147859377016774173144531770845160210790126748890337341153 7968578900091362333831895469348843179057521412931910720206803023151827 1609710023948983626872759995890539353139509460252986155626299982601880 2308255632037361959283071840638790581835814437769879939210175416644092 8838230265800973369632393712545459931035489646742906568108429783914725 8951432019664436521339365162473521801508858399440334246264874761447669 2634020116561379515917114881315644499216670133173014662858063060294669 2137102355178946292293609560530908582446986495910163528547795460323614 3366860906068403030154733036610090783810893192334139230683876790414362 5175971525031541667258252592765334141501923395153501324505152952090252 9405231572079476149746624015146987021700931456244088717393574349202246 798703, a[18,1] = 1.96981413175550498373701759389253905767850963408909 8924321089268809484155587799041727151464523471714630066749596884442738 5980474438753401257987143482778439005149282100373722869778290973687026 0972193007118030928235864480135883134145551737628240652843564408915972 9259169311045847052988170387827796901748623887744523705446347942611434 0516870516132223453429376665859522056333367391225337503056415536049359 2493788472120543719497714529133422690310800934541971386551061286969865 8938564639161957517855006223721382518332328409215320560069075162884683 6466640087401409571699283895095427071438202619461686833288529773639994 7141028546477386523689657239257377700496593458533195204578728717963850 2355966241077131173140928859750025218460927081514683066638182202259940 62588013698360512235790191868211899976534447362724543053220, a[8,6] = \+ -.18000169577132198123767815132152610663544725820051257593445157446446 5594713570707049315525047342512506404132375837455693277141029653437785 1855620203547962968447711480184095400751741153366702967117036603854038 5494387761837607392700761074234646530411709501776310343775120004799066 9848894277205770860826536443803892202587279094396094221496967501876034 9597706457314553027466278912273905299391410621088917246029375455973078 2000372778718700034161448636390746725899548270772430219148031142966418 5882520640438481002657591512917794171840074665194943037469677762835657 0236925248524536607504631433854146263081666974302779618629715118085101 0940410361707962789328418504363844715364449199047471486345790499834834 8804914344472096003344825282458392372490704890993471918412420488170004 667156645724687278904574161705464522281563e-1, a[19,11] = .19949120123 0293390792150497220768116171729430320198435912424933327585082354295854 7431236552636963125164070975885760842163581120899053639087552110852694 0752648772716605060335028528545774386028869174391167501068371192940934 0269398461646434045146399503836628889384483723429131846253980855418284 9707089024329587166135361630553634664411449978136752476077629182024478 3533783946643109364392212112884709572737742022387103479162669779788618 4840012294770253108629483782301868560540681658754176668336931010638805 9689078444229744990709495470285698884393663475137490530985870341267041 8131627969846299968158913699659052051630095374006549068411305607927097 9723735894466477385967454690604823471050882940415379836664623629429712 6423974847023925553048178530496530601293363350587373066774177085477069 57475391272245420808692916781e-1, a[20,10] = .334709166165543107030536 9273125762189258904188524372300396968657301428871536166082549564652407 1032623724776735441099747988958037178020248435968700627171657355592625 2356388247687022564753713730302897636691646905253996174233491658451021 3377078992382505835919944698720634311093199708572815768747889178585596 1110395287773123217227080401630696792814717621159468619087789929110087 4332750810906206029550652984040974748858057061892745261806377447329706 7641532570619305379638543168034410837983066538793762087959317570315830 3341002228032834528604366333041545926070848925527463230124847928658936 3063249366870965910595911305123717894760104292316301075556491167771325 2530460676091441798590688727969834081591515001182410350382879543503923 2502989497815309457207737884515881280905429612347423929818079577212811 2297958266674368, a[10,7] = .15852226583679018230928102899956603655582 7711428336394628411304404714126967580974148123749987833680994321230647 2720125061621514296468396459149860879976719000601617906209547224032707 0122640611958915169208257269448753326160310733691879549510352826447052 3835848591280223349121426681318826854897434532101014959286926792306560 0257868830448998202613374079887231233158125754037410284538596267284719 4876979123306112907695785977417405664916773415123720312091518311426027 3454955928148716141303464267496185469482966588478740641781846203004217 4844311053205054463147165457036434436564485866034595897919452011018778 7401852652957987335473742476633750340963799906741000458791410005135320 2019382268140241533815360846020588206208295856387112252573823596084709 723857585353036844875437468779917926622450084425256920689512263732223e -2, a[9,8] = .97297217025508085574951790623905875334071320903961333653 7167907512451512672560522492379405785215550056685009189292034389339841 3764974603835976034377482159307490527360364223739894504360173666352258 1510820787042001271472595051289145763467464798540211029913899481812418 1425504084885800972346851148547451557857701422023750471381061304595688 9797971571066510041689837998307244830982603788130213523508205418148039 0535113867915181321681153934681662778757848699580613420340700505833272 0685520136445345432725657814538223957508532037738276087031558688546165 5418163925037850385468857028469485476304699324397318643912311158365970 1689715166346739733021011142422837389333236969653142710822661975940406 9560986533840300115184797578307462347297670411726528617154020820579486 919753235034831792379001408444564157516324086595077814e-1, a[24,3] = - .430576755171135859699373747652338867218170740133541874736967732614744 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0366045882412428682127592457424878827255393227362814217726078121752527 6154823089888855277472530930654481255464408153735541604969193544707830 2941810644251235792051012440856350357069997020432824653427700546655238 1785345511481955401962417283696865941506064751638084565028166469859059 3079952208400444843172145355059789374712725063745373357088812242632279 6105079423928613572697609587500484616539138075143439422800883551125303 83082851, a[21,11] = -.25395521353833872319588015081253763922832149137 9197211593757099910733429643646065736769639886339388964946906533362747 9319600627176482584282901664116052921001464690773462741838503235759167 5108769803223211453317839951812135388679564007770602777878233964554472 7637044839045410736980235638783830272831836786732204187167804741617981 6007018166488158039544758295321706786504634515218218877892279048059881 5568141043051803068294889300787585650803794995465412762042820939438848 2083784688116677222365037856317312890806207626615798339253979263844076 8432393484330332962375889763179817917323995042776769084724644165249495 4593076965397606302703382995825787019537032729971308420333130851345837 4957111121362960069839879173629544510448227839061041510469882837628619 986888725820684120039617445298539615713897395981043920382457542e-1, a[ 18,13] = -22.446815218056352853649269632932188566004431954884118489110 3304731666371550724112027497291414611689602969528756778767938175503181 6037035473801231862010233809624917880820490875665385968513141602127941 9701594326332858660890825889986768462890006739940735232347628330929204 4699459692551787553299834329146755107569015063244504236862451885522654 9458565843044021760090123174201889559018891907356379816967485269061471 7652590859110216128019118805207718560152618599222635331614827743538806 4335316254613461806254188700065336644587312442237276986934910738529084 7664123193184500374882889495848352190690813614699444064053983335029555 5885201239904021083605877069771521808214420700787310614653706201853644 0354355449094857128005380835243830136309043517957211545434753329786540 354898747975154296800246394812013144112552188570924, a[8,5] = .4596205 6415093051618634833116753608862930266326929610863743141754105289451299 3142903060614254205382784244353573386201706260713523447213929203478851 6134202219668242991097607511113599899422902727386530159441585395710413 7566251546688953454049481314405565668207353728108227347606413242777946 3461991766218135417602094574285604684943129604443341236218186810468065 0703374096300185363158098076189739595782542270956293463221508187070477 4972413419892357789046796772544122811536685523278379630974132136537975 7177118072986744245134965280011095056101913112746333041534171130232916 3421789884494571138915441523697353697918543342942491927069712331713257 4653775256743286636736451627665026477215443718672333250258891730292759 0031312885477012859433547076980406226671739737131043185590407136681659 861644413576629361102358835807524e-1, a[7,1] = .5247404461891304721708 3656398443563543866244707379976473973291164376169590598547017775131902 4091604586209580717945589840688774916454807247639756837738406676150595 3176333033000179938400630720870593393281445162017198680399014072433051 2035775736133874814916171826278152250428277013189921446469513369020228 3896305496699006752410834709324442300253567835923678298396663147020523 0775687768539356618941784610988962237461864619568921008857607007364564 3131783109927463597211256050806216588904706058797399594090883144958381 3871990901725600072016404449917240749854539158280037557910518066280769 9230998300517969110783908505146406539049450566856849246244164340405529 7816612943793905401650508606793724082758991081629610583738194805173832 8725035686163822514819635461454290167097970016236119121429951461010842 335945871072320623, a[7,4] = .7610651429965941990931946190296373660533 9221302089006430925606619206521462688154248848822802289319428567067266 1875940487302656120909083690847289151246199382820489272941339478554838 5626744403782401791567650378799255344103207192887667746688599970735816 1956421842527499827867419992045452780254950467184681245391779370713502 1816777478145720923159097637315801636979264343573108085250886378024822 5694583314483485716772142531930085836571509095977632552752751554188161 2622245734692511829765661307194300849133389976958751272768414210145478 1106133545792341185327882301739266719272319294098991374843591475913829 7663648788728817483232107042009846526246945424978502860659185461711890 9712598488751190210251637018744853768840962773441391948779833350481077 5736908849836359743826355680912476519650776885349556430137763486756551 e-1, a[20,13] = .11548114578237683893912764212739312829591958690542449 3812037464400028266833472382652843295263581241579125038173210836389549 5036209962859347006585426134872218910709332911748937136703858607559221 2441055950636194779210604357812011475928733609437745285255420679579077 9160871528701253556892128967774383071871635565637309783012379704812060 1700689071412395695345394269100035261900771955673783141869816953973205 5826387104644482569128567694490562583619357317475145929225454165884944 6436101890350911098501972939272873000986618288543712064511570041145324 9323312236119111506780442649268077713667332489307753735987909189347938 6983366141392068415961400299778518335364332242561403453067589873779406 6818753161505883025817074724858910871398669123088493222833563518368383 241972465072258937265178872712779888945454147885118252970, a[15,9] = . 9041935089428447305690438956504513790551652375543511519398980421133820 6973188998959935710637994028055815003765602232345539876053763196041664 3238249407879014987320865315409184072783172267748334294881055174807236 5599319589914193371163476768434219294592097164129808284318433615396448 1880933540758629611314663049660189879504724007102726099885226923046498 7132048906086691084387577761167299706997027259365296269103755819013217 8524049198178809993641409122479590481006685524438609092394211436377584 2582897398211586518684362131384689042909672161867727443867261750795604 2757300123683843343610458164335114956046831631101738954258121834079534 7874800267891831350815651151644242146398465574944049124999878448647439 5804147133367561613831382894929430347299468564709246064316401304112729 6371597277452540045869025403174845539369, a[17,1] = 7.9791382043879970 5717660603121190341779125883967239943345518387485229714307428239463525 5497699728443665535709089773246161527590660162424980248241089846758338 4088852037212390123479349871010126499890699180576160710101608686027513 0389207251230389599641068098721405510225256277108017217314621152955956 7387259531105576031724057332423990321377833481158592424373161237931583 8230881891155744435020860958764928548104652272326069043030066293364032 9404534586861948702373696728470296306365503826355413373569142065086673 9441261416506478818119366315561075337599050084005435128037471660352001 1145397094509076957508051233188198805085967790186627651365525090955560 5755786863594701051533020033851076544013036975506805405210915979897183 0190680616419931822437842550717709773839248788776317875916753506520015 73338600975674447454881, a[17,15] = -9.2752181806776299867165418890926 1516337116209376416419825956882792778794725657520042271094394186608614 1521766576530875689860624466074997263819627610778804053072607926375127 5030574471036837361662392574957524687934051955373877559306061062947481 4869339475619033934337018781075564058407625286901533325167378379372323 5672640877719531414236299669286083565512421255553946191122744070359672 4077918506588320451200798345307173770447039159651456842488643850566960 3822373670257258022694760275973278251381986036402808684744571950330800 7934254850825401779933450365276528307242339117803513744424821053560848 6929306123000274636120184958318346998161496634413737896291092394028644 6113450684235300904418599462572367498371276632794132457398621710629966 5311781797552860300617250365458620832035937501472891812772805477470627 57371838, a[16,14] = 5.62732560715304973783049770124068683749936764113 8337449203708496857213243120336093718529970632540430806857041110135115 7640439604226094015959765087788487300648437226582282696368584507773390 1846308275846229176317443868251266688318928695300821313385346327959669 0218520244921590156858010773246037983745106270519209506058810419624578 1787093847477235716041528940840295168763047433233254476069613146822744 1583833785279809657264037739967417412751509427408505039909042933701458 8095346459383022710647346783654992215637922282528604927171902566036344 1382119002691265984192254678534016522762029525621318301655201285932752 5053043823938128682499695432056683494774131557767496897882198870105047 2525215534897214729839108029920824062200278370536637802922845303222161 36881110203570830651666740230224858183444222324389438226879607, a[12,6 ] = -.5710877887064641318405871297712412642428984822852290337451166027 2611017516323278325422716659684638509564268992098155700908104399181568 8473333007522955768130428370243685278103339851087719828895239584415025 7585785314601825910955537718552952811560910148432444067409768892534610 4423442955701474313243348682533541022959061088897743099903973241335033 4083057478038692142796492985441579853926142461939502848640572897515677 6859498563428305609182180404060084524007389234808284126471833267576861 9120315456728477478059202871266286234275090824885557511595806621281357 4118737528611934858346168209984850169616317794022946095486245503668179 3241612342102493858692061850607194352497191579453176022324937435996932 9214643227398801621503899060705354965518006246745100571649310939189168 2209788204471461476567431815029237975654955169, a[25,1] = 11.659923056 3222564606167426450194625210741528630266843655533999933347421381950439 4046770844052820024390704505011336615020263016895563727544312128062890 1790986270235278763362849057200096641044066384748379688576637119240940 8102892155658852608007550930233045780548524647846610036330270413183501 9826108153757873384694416023522940100288677828406159180015836216904017 9763504466614555451291105086730788743847636010143434190176665076662831 8213379119234578453286576056755227855787155828281281568576611294322077 0778317854812871188066967682202089479813452473521613373496349085387683 3717457284642950728456831352868270064734376606934081061848730027769797 1574272629941294368059169717185513721486550145331566536136554697827297 9188836130567061210742813524793743841009234241475566995406724586347999 37827244669552425133224355156, a[16,13] = -2.6187575408242655484134529 2193845520688268375438558954920923846353589715422273069896584457223793 6370185879767263223819161762352416262482430368854193007268205469474577 8672998746057887564765028130747026960626594937505528511369981920032863 4132379724401957457134537856776483290600907900705194329134080174130343 5641380411388699665147053474068951258771577142345144043897627267075812 3189976850180210022375977961838911868630057614999980450941800372673243 9387258134560831405462027322993328392014890513435110514821697546983428 9813616667954935718254176422865005545978599724419518838295958658179357 8246332399907228484099675468739389028326065587116327331908694619679559 7636854698579682089494150191328715507228205741845255499491442670791280 4164029680060958862379973223013950666612778398473965009967992696291317 46927929477804, a[19,7] = .1585222658367901823092810289995660365558277 1142833639462841130440471412696758097414812374998783368099432123064727 2012506162151429646839645914986087997671900060161790620954722403270701 2264061195891516920825726944875332616031073369187954951035282644705238 3584859128022334912142668131882685489743453210101495928692679230656002 5786883044899820261337407988723123315812575403741028453859626728471948 7697912330611290769578597741740566491677341512372031209151831142602734 5495592814871614130346426749618546948296658847874064178184620300421748 4431105320505446314716545703643443656448586603459589791945201101877874 0185265295798733547374247663375034096379990674100045879141000513532020 1938226814024153381536084602058820620829585638711225257382359608470972 3857585353036844875437468779917926622450084425256920689512263732223e-2 , a[19,6] = -.60055393086467113941736174172826357846030297816943460620 8000905080616491959031826915632426683855648333650145005906926825095880 6114959205658025622131315259181505088015471626305747918858428556673516 6541921817052860918464809297544430680554789893425367248165083125083400 2564865943540630402045519604004085499662624260453314411622521226488655 3416954597839669900764577667193277177257382175154885426972918387828975 8031196610111325966264118578931084405393811439937420819351028107496207 1673983936150021650774836575578738087210950212797756241360903296260271 8698046158717186535610067920668241098552649005959234597809023355958472 1421470668241365455176704416757597663971470707358357183227524385666438 2987835771659666861392411354696084616692123314493753532441383055311049 408684684614282911869314770688326256622376515892030681, a[20,11] = -.1 0520833628375086637066557974886809620823481850445980604211639188666053 9467944992876535918993145067203556769718905272955418823679229173235944 3661205683760875545646350577557404991013727823107615302945827458055744 5479579407396097010767354464315890601002531296780954740630672418316802 6655425179478999217834490825678877331689325797294824898858208579130396 7400323242802407700500445928415048802453493048110110663049339536688021 2002984559026241312804557837490062131217500338181416647680891775175811 8337205636346858002713612196814086280724452503470555053702988401119786 2409824329558492911386280013544301083813126422996652419395046407028643 5528010771069103969522437896698849669945186447081842948551304622837322 7816712586390243763074837595319598281137659580760139636166848083015907 075938615062856692778160018346648643706e-1\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "The Butcher tab leau with entries represented by 3 digit rational approximations is gi ven below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 148 "evalf[20](subs(e26,[seq([c[i],seq(a[i,j],j=1..i-1) ,``$(26-i)],i=2..25),[`b`,seq(b[i],i=1..25)]])):\nconvert(convert(eval f[30](%),rational,3),matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7;7<#\"\"%\"\"*F(%!GF+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F +F+7F5F5#FM\"##*#!#:FM#\"# rF+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+7<#F3\"\"&#F3\"#`F5F5F5F5#F3FC#F3 \"'A26#F1\"#JF+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+7<#F3F1#!\"#\"#\\F5F5F5 F5#!\"$F]o#F3\"$I'#F1FOF0F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+F+7<#FV\"#=#FPF 1F5F5F5F5F5#\"#z\"#S#!#pF]o#\"#uF]o#!#DF1F+F+F+F+F+F+F+F+F+F+F+F+F+F+F +7<#F1\"#e#F?F;F5F5F5F5#!\"%FM#F?\"%m?#\"#A\"#*)#F]o\"#6#F?Fdo#F3\"%) \\&F+F+F+F+F+F+F+F+F+F+F+F+F+F+7<#F)FY#F3F^qF5F5F5F5F5F5F5#F3\"%]6#F? \"&r3\"#F3\"(nPX&#F3\"#:F+F+F+F+F+F+F+F+F+F+F+F+F+7<#F)Ffr#F3\"#nF5F5F 5F5F5F5F5#F1F;#F3\"$D##F?\"''ed%#F3\"#;#F.\"#PF+F+F+F+F+F+F+F+F+F+F+F+ 7 \+ " 0 "" {MPLTEXT 1 0 158 "[seq(subs(c[1]=0,eval(subs(Sum=add,SimpleOrde rConditions(12,25)[ii]))),ii=2039..2048)]:\nmap(u_->simplify(expand(su bs(e26,u_))),%);\nmap(u_->lhs(u_)-rhs(u_),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,/#\"\"\"\"#%)F%/#F&\"#gF)/#F&\"#'*F,/#F&\"#[F//#F&\"$ 3\"F2/#F&\"#OF5/#F&\"$?\"F8/#F&\"#CF;/#F&\"$K\"F>/#F&\"#7FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"!F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "[seq(subs (c[1]=0,eval(subs(Sum=add,SimpleOrderConditions(12,25)[ii]))),ii=1500. .1509)]:\nmap(u_->simplify(expand(subs(e26,u_))),%);\nmap(u_->lhs(u_)- rhs(u_),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,/#\"\"\"\"&?f#F%/#F& \"&+_#F)/#F&\"&!oAF,/#F&\"&g,#F//#F&\"&S%>F2/#F&\"&W\"=F5/#F&\"&_b\"F8 /#F&\"'g,QF;/#F&\"'SELF>/#F&\"'!o&HFA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"!F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "[seq(subs(c[1]=0,eval(subs(Sum=add,SimpleOrderConditions(12,25)[i i]))),ii=2039..2048)]:\nmap(u_->simplify(expand(subs(e27,u_))),%):\nma p(u_->lhs(u_)-rhs(u_),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,$\"\")! $E)$\"\"\"F&$\"\"(F&$\"#AF&$\"#n!$F)$\"#_F&$\"#sF/$\"$<\"F&$\"#pF/$\"# 9F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "[seq(subs(c[1]=0,eval(subs(Sum=add,SimpleOrderCondit ions(12,25)[ii]))),ii=1500..1509)]:\nsubs(e27,%):\nmap(u_->lhs(u_)-rhs (u_),%):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,$!#W!$H)$! #`F&$!$3#F&$!#TF&$!$<\"F&$!$\"=F&$!$p\"F&$\"%@G!$I)$\"$t$F5$\"$(yF5" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "[seq(subs(c[1]=0,eval(subs(Sum=add,SimpleOrderConditions(12,25) [ii]))),ii=1025..1027)]:\nsubs(e27,%):\nmap(u_->lhs(u_)-rhs(u_),%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"%R!*!$L)$\"&()e\"F&$!%aXF&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "#-------- --------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 51 "#-------------------------------- ------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "absolute st ability region" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 10:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "coefficients of the scheme correct to 85 digits" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20329 "e85 := \{c[2]=.443181818181818181818181818181818181818181818181 8181818181818181818181818181818181818,\nc[3]=.442982456140350877192982 4561403508771929824561403508771929824561403508771929824561404,\nc[4]=. 6644736842105263157894736842105263157894736842105263157894736842105263 157894736842105,\nc[5]=.1069403994175161223216143124609943831911795298 522987310172664863740378614520490950697,\nc[6]=.1644736842105263157894 736842105263157894736842105263157894736842105263157894736842105,\nc[7] =.58432510885341074020319303338171262699564586357039187227866473149492 01741654571843251,\nc[8]=.63823582358235823582358235823582358235823582 35823582358235823582358235823582358235824e-1,\nc[9]=.2,\nc[10]=.333333 3333333333333333333333333333333333333333333333333333333333333333333333 333333333,\nc[11]=.944611605406556349684737572023734059136444084754527 9548637407620203329280139199114088,\nc[12]=.51795846804284610358356154 60185207449756996828425551866437683588600807071775316523299e-1,\nc[13] =.84888051860716535063983893016267430206414817564001954204593393983557 73991365476236893e-1,\nc[14]=.2655756032646428930981140590456168352972 012641640776214486652703185222349414361456016,\nc[15]=.5,\nc[16]=.7344 2439673535710690188594095438316470279873583592237855133472968147776505 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8138031675770209280,\na[22,10]=-.3664591598580916638621456622089859363 144583004356560608229075984337239720284817960193,\na[22,11]=0.,\na[22, 12]=0.,\na[22,13]=0.,\na[22,14]=0.,\na[22,15]=0.,\na[22,16]=0.,\na[22, 17]=0.,\na[22,18]=0.,\na[22,19]=.3664591598580916638621456622089859363 144583004356560608229075984337239720284817960193,\na[22,20]=.131135124 3636448823913476512658127321487191165570692263018637105138138031675770 209280,\na[22,21]=-.11485466578247081368022416497764467440254061005826 55444423341878411637297467351936831,\na[23,1]=.22159086425014402853738 03379343545548531698116185650811689592853027756074847487589870,\na[23, 2]=.221391591890206848655602118205996322339812644521785796024023170837 5752697082336971534,\na[23,3]=0.,\na[23,4]=0.,\na[23,5]=0.,\na[23,6]=- .700367647058823529411764705882352941176470588235294117647058823529411 7647058823529412,\na[23,7]=-.38414322622520793404698532962057511917070 35527677198880717241749265840367609120672261,\na[23,8]=0.,\na[23,9]=0. ,\na[23,10]=0.,\na[23,11]=0.,\na[23,12]=0.,\na[23,13]=0.,\na[23,14]=0. ,\na[23,15]=0.,\na[23,16]=0.,\na[23,17]=0.,\na[23,18]=0.,\na[23,19]=0. ,\na[23,20]=0.,\na[23,21]=.3841432262252079340469853296205751191707035 527677198880717241749265840367609120672261,\na[23,22]=.700367647058823 5294117647058823529411764705882352941176470588235294117647058823529412 ,\na[24,1]=.4431818181818181818181818181818181818181818181818181818181 818181818181818181818181818,\na[24,2]=0.,\na[24,3]=-.43057675517113585 96993737476523388672181707401335418747369677326147448822557647250959, \na[24,4]=0.,\na[24,5]=0.,\na[24,6]=0.,\na[24,7]=0.,\na[24,8]=0.,\na[2 4,9]=0.,\na[24,10]=0.,\na[24,11]=0.,\na[24,12]=0.,\na[24,13]=0.,\na[24 ,14]=0.,\na[24,15]=0.,\na[24,16]=0.,\na[24,17]=0.,\na[24,18]=0.,\na[24 ,19]=0.,\na[24,20]=0.,\na[24,21]=0.,\na[24,22]=0.,\na[24,23]=.43057675 5171135859699373747652338867218170740133541874736967732614744882255764 7250959,\na[25,1]=11.6599230563222564606167426450194625210741528630266 8436555339999333474213819504394047,\na[25,2]=-2.5725,\na[25,3]=-1.9878 4065426724706503099854900408917029415644374093127555731433847777338082 0472233215,\na[25,4]=0.,\na[25,5]=0.,\na[25,6]=-1.666875,\na[25,7]=-1. 8400101376090497154805510286039927807518541848125827303198932782114806 04766702319850,\na[25,8]=0.,\na[25,9]=22.69942286392168856359381664830 269209988914122916681587352946080752209500479602829761,\na[25,10]=73.4 2330523206361635568882707328200110684529708829166943826082146129564229 105196805358,\na[25,11]=-.82366882628750614624496805918614715644427311 65837003579459833918663154920598010472804,\na[25,12]=-62.6187074169504 8664691990216349645376057743804324083598067464982459275125872198241326 ,\na[25,13]=62.3161177148076402411653884871269604690654608447378405145 7579384530744899667445210231,\na[25,14]=-97.33866705604075608782440343 082106667916379078791656916876398972477845017427271789167,\na[25,15]=- 14.4632469865853858907314210714528496473041335986842389594106035484523 1444731889608148,\na[25,16]=2.5097164340865699853630779566742382867242 67256774032757480361264001771307725305753519,\na[25,17]=.1162475886937 0883605356664760650141313264439016112750968360798073110573867358326658 99,\na[25,18]=.5840328281597421751021178306186155103094570763962374837 158281176580253234011237044692,\na[25,19]=1.58441780671270839748383407 3813399023446091584920600008797782958694409008929749703075,\na[25,20]= 1.35110676109620375665332336251264681300312321295033651519817006114459 1562926142612062,\na[25,21]=1.8400101376090497154805510286039927807518 54184812582730319893278211480604766702319850,\na[25,22]=1.666875,\na[2 5,23]=1.98784065426724706503099854900408917029415644374093127555731433 8477773380820472233215,\na[25,24]=2.5725,\n\nb[1]=.2380952380952380952 380952380952380952380952380952380952380952380952380952380952380952e-1, \nb[2]=-.11,\nb[3]=-.17,\nb[4]=0.,\nb[5]=0.,\nb[6]=-.19,\nb[7]=-.21,\n b[8]=0.,\nb[9]=-.23214285714285714285714285714285714285714285714285714 28571428571428571428571428571429,\nb[10]=-.27,\nb[11]=-.29,\nb[12]=0., \nb[13]=.1384130236807829740053502031450331467488136400899412345912671 194817223119377730668077,\nb[14]=.215872690604931311708935511140681138 9654720741957730511230185948039919737765126474781,\nb[15]=.24380952380 9523809523809523809523809523809523809523809523809523809523809523809523 8095,\nb[16]=.21587269060493131170893551114068113896547207419577305112 30185948039919737765126474781,\nb[17]=.1384130236807829740053502031450 331467488136400899412345912671194817223119377730668077,\nb[18]=.29,\nb [19]=.27,\nb[20]=.2321428571428571428571428571428571428571428571428571 428571428571428571428571428571429,\nb[21]=.21,\nb[22]=.19,\nb[23]=.17, \nb[24]=.11,\nb[25]=.2380952380952380952380952380952380952380952380952 380952380952380952380952380952380952e-1\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "The stability functi on R for the 25 stage, order 12 scheme is given (approximately) as fol lows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "Digits := 28:\nA := subs(e85,Matrix([seq([seq(a[i,j] ,j=1..i-1),seq(0,j=i..25)],i=1..25)])):\nB := subs(e85,Matrix([[seq(b[ i],i=1..25)]])):\nid := Matrix([seq([1],i=1..25)]):\nadd(z^j/j!,j=0..1 2)+add((B.A^(j-1).id)[1,1]*z^j,j=13..25):\nmap(convert,%,rational,24): \nR := unapply(%,z):\n'R(z)'=taylor(R(z),z,26);\nDigits := 10:" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG+WF'\"\"\"\"\"!F)F)#F)\" \"#F,#F)\"\"'\"\"$#F)\"#C\"\"%#F)\"$?\"\"\"&#F)\"$?(F.#F)\"%S]\"\"(#F) \"&?.%\"\")#F)\"'!)GO\"\"*#F)\"(+)GO\"#5#F)\")+o\"*R\"#6#F)\"*+;+z%\"# 7#!)\\NM;\"1Ld=!eM7l&\"#8#!)&HWo$\"2-U5%e0)QH#\"#9#!(ej`$\"1P\\.)\\w?i (\"#:#!((y]6\"25+xtOHMX)\"#;#\"(6G-$\"2p7oG=/EI#\"#<#!(&)z/\"\"21)GDG9 _qr\"#=#!'*pR\"\"3fbBiK\")*f2\"\"#>#\"&l!f\"3To)zFiy$y@\"#?#!&$**>\"4! G;JM6AYP;\"#@#!%B'*\"5Lof8c!GZ3'p\"#A#\"%j=\"5=ujBc23@=&)\"#B#!$`$\"6m w.!pHJl " 0 "" {MPLTEXT 1 0 29 "z0 := newton(R(z)=-1,z=-3.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#z0G$!+dDR3Q!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 308 "z0 := newton(R(z)=-1,z=-3.8):\np1 := plot([ R(z),-1],z=-4.19..0.49,color=[red,blue]):\np2 := plot([[[z0,-1]]$3],st yle=point,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0 ,0],[z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1 ,p2,p3],view=[-4.19..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 391 276 276 {PLOTDATA 2 "6+-%'CURVESG6$7Y7$$!3Q++ ++++!>%!#<$!3wo,*)=$Hp4&F*7$$!33+]iHt\\kTF*$!3CFB4\"R$\\*f%F*7$$!3l++D fY**QTF*$!3kq%*)Qvyt9%F*7$$!3M+]()))>\\8TF*$!3Q-j_fctOPF*7$$!3-++]=$*) z3%F*$!3;[3s>c,kLF*7$$!3O+vBW)4O/%F*$!31W')HqnD'z#F*7$$!3E+](*p.B**RF* $!3\"z;f.cu\"=BF*7$$!3W+Dm45K\\RF*$!3'[*\\=)4s6(=F*7$$!3;++N\\;T**QF*$ !3])o1Y&Gj/:F*7$$!3A++Sd9<\\QF*$!3%[;]3j3J?\"F*7$$!3E++Xl7$*)z$F*$!3yi &zbB#ev&*!#=7$$!37+]FZ%G*)p$F*$!3Y.H@)H@/)fFin7$$!3X+]dLI@1OF*$!3Q&pZ) R7\"*oPFin7$$!3=+](Hf6-^$F*$!3_g&p@Wr$[AFin7$$!3?+]xln#4T$F*$!3[ad\"3* )zpA\"Fin7$$!3D+]FY.'>J$F*$!3f^+d=4![t&!#>7$$!3?++I>7;5KF*$!3SNhT+X>L9 Fcp7$$!33++&G\\'\\?JF*$\"3i(Hl,)o()o5Fcp7$$!32++!zCb&>IF*$\"3cP#GR++]G&*>=HF*$\"3C()*>!ppqmVFcp7$$!3******47X_?GF*$\"39ZJB35)* )Q&Fcp7$$!3<+]xRi#=t#F*$\"3CPs\\))px(='Fcp7$$!3\"******[Tbji#F*$\"3QR8 Cz)y]2(Fcp7$$!3y****f\"z2q`#F*$\"3NA7/X.NDyFcp7$$!3$)**\\F*$\"3La^IBBO?9Fin7$$!31++D***4B&=F* $\"3mlCI`oto:Fin7$$!3/+]xH!G\"\\Fin7$$!3!****\\p`*Hi:F*$\"3yrK_ok`'4#Fin7$$!33++?'[!3i9F *$\"3$H=%3iu`p5F*$\"3x&QS@\\KGV$Fin7$$!3)o****\\Cu9o*Fin$\"314:Q,(fyz$Fin7$$!3g, +v$H$zl()Fin$\"3'y^**>xT?;%Fin7$$!3+&****4:7Zw(Fin$\"3O%[71&[E+YFin7$$ !3%>+]<\"GxAoFin$\"35lQ/dfka]Fin7$$!37&**\\Zzu\"QeFin$\"3I-$)*fS]wd&Fi n7$$!3M*****\\E]b([Fin$\"3`?J&3#3EThFin7$$!3!z**\\Pfrx'QFin$\"3;&fI(3W 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l-F`al6#%&CROSSGFcalFeal-F$6&F[al-F`al6#%(DIAMONDGFcalFeal-F$6%7$7$F]a lF_]lF\\al-%&COLORG6&F[]lF_]l$\"\"&Ff]lF_]l-%*LINESTYLEG6#\"\"$-%%FONT G6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Ficl-Facl6#%(DEFAULTG-%%V IEWG6$;$!$>%!\"#$\"#\\Fddl;$!$Z\"Fddl$\"$Z\"Fddl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The following picture shows the stability regio n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1548 "R := z ->add(z^j/j!, j=0..12)-\n 16343549/5651234580185733*z^13-36844295/22938805584104 202*z^14-\n 3536358/7622076498034937*z^15-1150787/8453429367377001 0*z^16+\n 3022811/23026041828681269*z^17-1047985/71705214282528806 *z^18-\n 139699/107599813262235559*z^19+59065/217837862277986841*z ^20-\n 19993/1637462211343116280*z^21-9623/69608472805613596833*z^ 22+\n 1863/85182108075623637418*z^23-353/867917653129690037666*z^2 4+\n 29/4787690070573159957798*z^25:\npts := []: z0 := 0: tt := 0: \nwhile tt<=441/20 do\n zz := newton(R(z)=exp(tt*Pi*I),z=z0):\n z 0 := zz:\n if (4/5<=tt and tt<=6/5) or (57/20<=tt and tt<=16/5) \n \+ or (94/5<=tt and tt<=383/20) or (104/5<=tt and tt<=106/5) then\n \+ hh := 1/80 \n else \n hh := 1/20\n end if;\n tt := tt+ hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,col or=COLOR(RGB,.48,.1,.13)):\np2 := plots[polygonplot]([seq([pts[i-1],pt s[i],[-1.9,0]],i=2..46),[pts[31],pts[96],[-1.9,0]],\n seq([pts[i -1],pts[i],[-1.9,0]],i=97..435),[pts[435],pts[501],[-1.9,0]],\n \+ seq([pts[i-1],pts[i],[-1.9,0]],i=502..nops(pts)),[pts[31],pts[96],[-0. 06,3.44]],\n seq([pts[i-1],pts[i],[-0.06,3.44]],i=32..96),[pts[4 35],pts[501],[-0.06,-3.44]],\n seq([pts[i-1],pts[i],[-0.06,-3.44 ]],i=436..501)],style=patchnogrid,color=COLOR(RGB,.95,.2,.25)):\np3 := plot([[[-4.29,0],[0.79,0]],[[0,-4.29],[0,4.29]]],color=black,linestyl e=3):\nplots[display]([p||(1..3)],view=[-4.29..0.79,-4.29..4.29],font= [HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scal ing=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 884 649 649 {PLOTDATA 2 "6+-%'CURVESG6$7`\\m7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******R l#fTJF-7$F($\"3<+++!)*)Q7ZF-7$F($\"3))*****pI&=$G'F-7$F($\"3C+++N;)R&y F-7$$\"3_+++%z]zM&!#F$\"3J+++tzxC%*F-7$$\"3-+++?j3XU!#E$\"3%******zVd& *4\"!#<7$$\"3!******\\([3XC!#D$\"33+++6rjc7FF7$$\"3'******\\YG`3\"!#C$ \"3/+++Cpr89FF7$$\"33+++RETbQFP$\"31+++hszq:FF7$$\"3#******>;1[7\"!#B$ \"3&******4Xzys\"FF7$$\"33+++<7TvFFen$\"35+++kz'\\)=FF7$$\"3E+++f;jCiF en$\"3,+++0s2U?FF7$$\"3$******zWB\\_\"!#A$\"31+++y5D*>#FF7$$\"3))***** 4wG!**\\Feo$\"3;+++9YhcBFF7$$\"32+++6CkB?!#@$\"3>+++q'*\\9DFF7$$\"3E++ +2!\\A*GF`p$\"3'******\\]JTb#FF7$$\"3\")******oI?FTF`p$\"3;+++:I'Qf#FF 7$$\"3w*****z`)RteF`p$\"3&)******='>Pj#FF7$$\"36++++]EI$)F`p$\"3))**** *>uKPn#FF7$$\"3'******H7js<\"!#?$\"3%)*****\\7TRr#FF7$$\"3#********ob! 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"Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stabilit y" }{TEXT -1 89 " (or stability interval) is the intersection of the \+ stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "F or this scheme the stability interval is (approximately) " }{XPPEDIT 18 0 "[-3.8084, 0];" "6#7$,$-%&FloatG6$\"&%3Q!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see t hat the largest interval on the nonnegative imaginary axis that contai ns the origin and lies inside the stability region is " }{XPPEDIT 18 0 "[0, 3.87];" "6#7$\"\"!-%&FloatG6$\"$(Q!\"#" }{TEXT -1 18 " approxi mately. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "Digits := 2 5:\nz0 := 3.87*I:\nfor ct from 241 to 244 do\n newton(R(z)=exp(ct*Pi /100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\":s6'*4x%[q3lFNp!#F$\":F.;n&z/!)[>TwQ!#C" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$\":IMq>C*QG:tu_M!#F$\":GZ^c&Rl6F6#\\(Q!#C" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!:]X))H%))=\\Yf8$G$!#H$\":$=c*=tX%[ L$fL(Q!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!:)po8=>$)p_5W@N!#F$\" :cQY-#\\z2?bsrQ!#C" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 107 "Then we apply the bisection method to calculate the pa rameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "Digits := 15:\nreal_part := proc(u)\n \+ Re(newton(R(z)=exp(u*Pi*I),z=3.87*I))\nend proc:\nu0 := bisect('real_p art'(u),u=2.41..2.44);\nnewton(R(z)=exp(u0*Pi*I),z=3.87*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0n:'[e!*HC!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"07U@PuL(Q!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "larg est interval on the nonnegative imaginary axis that contains the origi n and lies inside the stability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 3.8734];" "6#7$\"\"!-%&FloatG6$\"&M(Q!\"%" }{TEXT -1 18 " ( approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#---------- -----------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{PARA 0 "" 0 "" {TEXT -1 91 "#================================== ========================================================" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "#================= =====" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Abreviated calculations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 41 "#--------------------------- -------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "required procedu res and data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 83 "Calculat ion of the data required to shorten the calculation of principal error norm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 " We use Hiroshi Ono's scheme to determine relationships between the p rincipal error terms of order 12 schemes constructed in the same manne r." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficient s of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 146033 "ee := \{c[2]=.25,\nc[3]=.44444444444444444 4444444444444444444444444444444444444444444444444444444444444444444444 4444444444444444444444444444444444444444444444444444444444444444444444 4444444444444444444444444444444444444444444444444444444444444444444444 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4177460406917796783040505299207866998205245514603578280132131313776794 9074591269455791041697503428370302259769033869262229320147362920963462 5859522549621090258606958724429646264015238443330014674648987073739052 6002804891998443111867297186727616041730444843780988351464838665761998 3055336567584781023144212508129438213937427450703689467678629545458813 1947803477632867507290149186105006397092948456771236311322538014642063 2513293180389137808357906399072608035503764293467528844129097524587918 7341530774108001007493142192268207098384157760720251611685616725207247 8769116659141811423113550001271927747290785698174314825082516961191586 56979821346725944717196945187103145459,\nb[15]=.2438095238095238095238 0952380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095238095238095238095238095238 0952380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095238095238095238095238095238 0952380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095238095238095238095238095238 0952380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 523809523809523810,\nb[16]=.215872690604931311708935511140681138965472 0741957730511230185948039919737765126474780500166257295923220201649257 5235361298903826153449363901394177460406917796783040505299207866998205 2455146035782801321313137767949074591269455791041697503428370302259769 0338692622293201473629209634625859522549621090258606958724429646264015 2384433300146746489870737390526002804891998443111867297186727616041730 4448437809883514648386657619983055336567584781023144212508129438213937 4274507036894676786295454588131947803477632867507290149186105006397092 9484567712363113225380146420632513293180389137808357906399072608035503 7642934675288441290975245879187341530774108001007493142192268207098384 1577607202516116856167252072478769116659141811423113550001271927747290 78569817431482508251696119158656979821346725944717196945187103145459, \nb[17]=.1384130236807829740053502031450331467488136400899412345912671 1948172231193777306680766426908855612196369412078853336067272524745275 1220646700343939681645107746455309232722207044303661830568249931484392 9725765177652111230162584946725882108573439834545166804164520563941383 5136475082312833345932360525985358984184274965931276186995271281824938 7006911809025686236794301283167041313847009553868409837276475830771065 8990566657312303777181295046911415017775848475003482868350105962466070 8474025547251950536652242753498527079567521364600499086860859065458203 1912821507960581353910394336202135221723559633930676642779610425844423 3104684065269441835116062847067925720934460786159015565249940341026000 9756050703783737404837153314340293071415852151098520714446828280320603 4018166555628734464367559769568517340527182568827,\nb[18]=.29,\nb[19]= .27,\nb[20]=.23,\nb[21]=.21,\nb[22]=.19,\nb[23]=.17,\nb[24]=.11,\nb[25 ]=.2380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095238095238095238095238095238 0952380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095238095238095238095238095238 0952380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095238095238095238095238095238 0952380952380952380952380952380952380952380952380952380952380952380952 3809523809523809523809523809523809523809523809523809523809523809523809 5238095238095238095238095238095238095238095e-1\}:" }}}{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 21 "e6 0 := evalf[60](ee):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "Digits := 50:\nA := Matrix([seq([seq(a[i ,j],j=1..i-1),seq(0,j=i..25)],i=1..25)]):\nB := Matrix([[seq(b[i],i=1. .25)]]):\nC := LinearAlgebra[DiagonalMatrix]([0,seq(c[i],i=2..25)]):\n A_ := subs(e60,A): B_ := subs(e60,B): C_ := subs(e60,C):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "errt erms12 := PrincipalErrorTerms(12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "kernelopts(stacklimit=7907) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 913 "errfact12 := []: errkey := []: vals := []:\nfor jj f rom 1 to 12486 do\n dd := convert(errterms12[jj],'Matrix_form',A_,B_ ,C_);\n val := modz(expand(eval(dd)));\n flg := 0;\n if jj>=2 th en\n for ii to nops(errkey) do\n if abs(val)>10^(8-Digit s) then\n rr := val/vals[ii];\n rat := convert (rr,rational,15);\n if rat<>0 and rat=convert(rr,rational, 20) and length(rat)<8 then\n flg := 1;#don't add jj to \+ errkey\n ##print(errkey[ii],jj,rat);\n e rrfact12[ii] := errfact12[ii]+rat^2;\n break;\n \+ end if;\n end if;\n end do; \n end if;\n if \+ flg=0 then\n errkey := [op(errkey),jj];\n errfact12 := [op (errfact12),1];\n vals := [op(vals),val];\n end if;\n if ` mod`(jj,100)=0 then print(jj) end if;\n save errkey,errfact12,vals, jj, \"C:\\\\Maple/RK_data/error_info.m\"; \nend do:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 89 ": T he calculation had to be performed in two stages due to the stacklimit being reached. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Recover the data for the 2nd stage after kernel aborts." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "read \"C:\\\\Maple/RK_data/error_info.m\"; \njj;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%/r" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 890 "for jj from 7105 to 12486 do\n d d := convert(errterms12[jj],'Matrix_form',A_,B_,C_);\n val := modz(e xpand(eval(dd)));\n flg := 0;\n if jj>=2 then\n for ii to nop s(errkey) do\n if abs(val)>10^(8-Digits) then\n r r := val/vals[ii];\n rat := convert(rr,rational,15);##prin t('rat'=%);\n if rat<>0 and rat=convert(rr,rational,20) an d length(rat)<8 then\n flg := 1;#don't add jj to errkey \n ##print(errkey[ii],jj,rat);\n errfact 12[ii] := errfact12[ii]+rat^2;\n break;\n e nd if;\n end if;\n end do; \n end if;\n if flg=0 \+ then\n errkey := [op(errkey),jj];\n errfact12 := [op(errfa ct12),1];\n vals := [op(vals),val];\n end if;\n if `mod`(j j,100)=0 then print(jj) end if;\n save errkey,errfact12,vals,jj, \" C:\\\\Maple/RK_data/error_info.m\"; \nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "errkey;\nnop s(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7T\"\"\"\"\"#\"\"$\"\"%\"\"& \"\"'\"\"(\"\")\"#5\"#6\"#@\"#F\"#K\"#U\"#V\"#X\"#Y\"#Z\"#\\\"#^\"#_\" #`\"#b\"#c\"#e\"#f\"#g\"#i\"#m\"$.\"\"$9\"\"$C\"\"$O\"\"$f\"\"$r\"\"$ \"=\"$)=\"$m#\"$n#\"$q#\"$r#\"$!H\"$.$\"$/$\"%!4\"\"%M6\"%l6\"%!>\"\"% 77\"&'[7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#]" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "errfact12;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7T\"\"##\"#?\"\"*#\"#R\"\"%#\"%_?\"# D#\"%'G$\"\"$#\"0btAA$)[3*\")OftI#\"%m()F-#\"$x#\"\")#\"#YF'#\"\"&F*\" \"\"F=F$#\"%[>F<#\"%&Q\"\"#O#\"$g%\"#\")#\"'P4<\"$+\"#F-\"#=#\"&.3\"\" #k#\"(;R**)\"$D'#\"$*H\"#7#\"$&>\"#K#\"',@9FH#\"(Y3![F-#\"%W_F-#\"'6^X \"#C#\"$8&\"#5#\"&yb(\"#F#\"%:#)FS#F\\oF'#F)F8#\"%E5F-#\"%V;F0FboFcoFd oFfoF4F6F9F;F=F=F=F=FboFcoFdoFfoF=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 252 "The fol lowing subsection contains the data needed to compute the principal er ror morm of similar schemes by a shortened method which only requires \+ the calculation of 50 principal error terms in conjunction with approp riate rational multiplying factors." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "errterms_12 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 342 "errfact12 := [2,20/9,39/4,2052/25,3286/3,908493124717355/3073 5936,8766/25,277/8,46/9,5/4,1,1,2,1948/5,1385/36,460/81,170937/100,25/ 18,10803/64,8993916/625,299/12,195/32,142101/100,4800846/25,5244/25,45 5111/24,513/10,75578/27,8215/12,10/9,39/8,1026/25,1643/3,10/9,39/8,102 6/25,1643/3,8766/25,277/8,46/9,5/4,1,1,1,1,10/9,39/8,1026/25,1643/3,1] :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3838 "errterms_12 := [b*``(a*``(a*``(a*``(a*``(a*``(a*``( a*``(a*``(a*``(a*``(a*c)))))))))))-1/6227020800, b*``(a*``(a*``(a*``(a *``(a*``(a*``(a*``(a*``(a*c*``(a*c))))))))))-1/2075673600, b*``(a*``(a *``(a*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*c))))))))))-1/1556755200, b *``(a*``(a*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a*c))))))))))-1/124 5404160, b*``(a*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a*``(a*c)))))) ))))-1/1037836800, b*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a*``(a*`` (a*c))))))))))-1/889574400, b*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a*``( a*``(a*``(a*c))))))))))-1/778377600, b*``(a*``(a*``(a*c*``(a*``(a*``(a *``(a*``(a*``(a*``(a*c))))))))))-1/691891200, b*``(a*``(a*c*``(a*``(a* ``(a*``(a*``(a*``(a*``(a*``(a*c))))))))))-1/622702080, b*``(a*c*``(a*` `(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c))))))))))-1/566092800, b*``(a *``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c)))))))))*``(a*c)-1/94348800 , b*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c))))))))*``(a*``(a*c))-1/ 28304640, 1/2*``(b*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(c^2 *a))))))))))-1/3113510400), b*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a*``( a*c*``(a*c)))))))))-1/259459200, b*``(a*``(a*``(a*c*``(a*``(a*``(a*``( a*``(a*c*``(a*c)))))))))-1/230630400, b*``(a*``(a*c*``(a*``(a*``(a*``( a*``(a*``(a*c*``(a*c)))))))))-1/207567360, b*``(a*``(a*``(a*``(a*c*``( a*``(a*``(a*c*``(a*``(a*c)))))))))-1/194594400, b*``(a*c*``(a*``(a*``( a*``(a*``(a*``(a*``(a*c*``(a*c)))))))))-1/188697600, b*``(a*``(a*``(a* c*``(a*``(a*``(a*``(a*c*``(a*``(a*c)))))))))-1/172972800, b*``(a*``(a* ``(a*``(a*c*``(a*``(a*c*``(a*``(a*``(a*c)))))))))-1/155675520, b*``(a* ``(a*c*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*c)))))))))-1/155675520, b* ``(a*c*``(a*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*c)))))))))-1/14152320 0, b*``(a*``(a*``(a*c*``(a*``(a*``(a*c*``(a*``(a*``(a*c)))))))))-1/138 378240, b*``(a*``(a*``(a*``(a*c*``(a*c*``(a*``(a*``(a*``(a*c)))))))))- 1/129729600, b*``(a*``(a*c*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a*c))))) ))))-1/124540416, b*``(a*``(a*``(a*c*``(a*``(a*c*``(a*``(a*``(a*``(a*c )))))))))-1/115315200, b*``(a*c*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*` `(a*c)))))))))-1/113218560, b*``(a*``(a*c*``(a*``(a*``(a*c*``(a*``(a*` `(a*``(a*c)))))))))-1/103783680, b*``(a*c*``(a*``(a*``(a*``(a*c*``(a*` `(a*``(a*``(a*c)))))))))-1/94348800, b*``(a*``(a*``(a*``(a*``(a*``(a*` `(a*c*``(a*c))))))))*``(a*c)-1/31449600, b*``(a*``(a*``(a*``(a*``(a*`` (a*c*``(a*``(a*c))))))))*``(a*c)-1/23587200, b*``(a*``(a*``(a*``(a*``( a*c*``(a*``(a*``(a*c))))))))*``(a*c)-1/18869760, b*``(a*``(a*``(a*``(a *c*``(a*``(a*``(a*``(a*c))))))))*``(a*c)-1/15724800, b*``(a*``(a*``(a* ``(a*``(a*``(a*c*``(a*c)))))))*``(a*``(a*c))-1/9434880, b*``(a*``(a*`` (a*``(a*``(a*c*``(a*``(a*c)))))))*``(a*``(a*c))-1/7076160, b*``(a*``(a *``(a*``(a*c*``(a*``(a*``(a*c)))))))*``(a*``(a*c))-1/5660928, b*``(a*` `(a*``(a*c*``(a*``(a*``(a*``(a*c)))))))*``(a*``(a*c))-1/4717440, 1/2*` `(b*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a*``(a*``(c^2*a)))))))))-1/3891 88800), 1/2*``(b*``(a*``(a*``(a*c*``(a*``(a*``(a*``(a*``(a*``(c^2*a))) ))))))-1/345945600), 1/2*``(b*``(a*``(a*c*``(a*``(a*``(a*``(a*``(a*``( a*``(c^2*a)))))))))-1/311351040), 1/2*``(b*``(a*c*``(a*``(a*``(a*``(a* ``(a*``(a*``(a*``(c^2*a)))))))))-1/283046400), 1/2*``(b*``(a*``(a*``(a *``(a*``(a*``(a*``(a*``(c^2*a))))))))*``(a*c)-1/47174400), 1/2*``(b*`` (a*``(a*``(a*``(a*``(a*``(a*``(c^2*a)))))))*``(a*``(a*c))-1/14152320), 1/2*``(b*``(c^2*a)*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c))))))))- 1/14152320), 1/4*``(b*``(c^2*a)*``(a*``(a*``(a*``(a*``(a*``(a*``(c^2*a )))))))-1/7076160), 1/2*``(b*``(c^2*a)*``(a*``(a*``(a*``(a*``(a*``(a*c *``(a*c)))))))-1/4717440), 1/2*``(b*``(c^2*a)*``(a*``(a*``(a*``(a*``(a *c*``(a*``(a*c)))))))-1/3538080), 1/2*``(b*``(c^2*a)*``(a*``(a*``(a*`` (a*c*``(a*``(a*``(a*c)))))))-1/2830464), 1/2*``(b*``(c^2*a)*``(a*``(a* ``(a*c*``(a*``(a*``(a*``(a*c)))))))-1/2358720), 1/479001600*``(b*c^12- 1/13)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "#-------------------------------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "errkey := [1,2,3,4,5,6,7,8, 10,11,21,27,32,42,43,45,46,47,49,51,52,53,55,56,58,59,60,62,66,103,114 ,124,136,159,171,181,188,266,267,270,271,290,303,304,1090,1134,1165,11 90,1212,12486]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "errterms_12 := [seq(errterms12[i],i=errkey)]: " }}}{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 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 94 "#------- ---------------------------------------------------------------------- ----------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Set \+ up order conditions etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2453 "eqA := c[11]=(12*c[17]^3+9*c[17]^ 2-15*c[10]*c[17]^2-15*c[9]*c[17]^2-10*c[9]*c[17]+6*c[17]-10*c[17]*c[10 ]+20*c[10]*c[9]*c[17]+3-5*c[9]-5*c[10]+10*c[10]*c[9])/(5*(3*c[17]^2-4* c[17]*c[10]+2*c[17]-4*c[9]*c[17]-2*c[9]-2*c[10]+1+6*c[10]*c[9])):\neqB := c[12]=(15*c[9]*c[13]*c[11]-20*c[9]*c[11]*c[10]-12*c[9]*c[13]^2+15* c[9]*c[13]*c[10]-12*c[11]*c[13]^2+15*c[11]*c[13]*c[10]+10*c[13]^3-12*c [10]*c[13]^2)*c[13]/(20*c[9]*c[13]*c[11]-30*c[9]*c[11]*c[10]-15*c[9]*c [13]^2+20*c[9]*c[13]*c[10]-15*c[11]*c[13]^2+20*c[11]*c[13]*c[10]+12*c[ 13]^3-15*c[10]*c[13]^2):\nRSeqs := RowSumConditions(13,'expanded'):\nS Oeqs := [op(StageOrderConditions(2,13,'expanded')),op(StageOrderCondit ions(3,5..13,'expanded')),\n op(StageOrderConditions(4,7..13,' expanded')),op(StageOrderConditions(5,10..13,'expanded'))]:\ncdns := [ op(RSeqs),op(SOeqs)]:\ncdns2 := [seq(add(b[i]*a[i,j],i=j+1..25)=b[j]*( 1-c[j]),j=[$14..24]),\n seq(add(b[i]*c[i]*a[i,j],i=j+1..25)=b[j ]/2*(1-c[j]^2),j=[$14..23]),\n seq(add(b[i]*c[i]^2*a[i,j],i=j+1 ..25)=b[j]/3*(1-c[j]^3),j=[$14..21]),#omitting 22\n seq(add(b[i ]*c[i]^3*a[i,j],i=j+1..25)=b[j]/4*(1-c[j]^4),j=[$14..19]),#omitting 20 \n seq(add(b[i]*c[i]^4*a[i,j],i=j+1..25)=b[j]/5*(1-c[j]^5),j=[$ 14..16])]:#omitting 17\ncdns3 := [a[18,7]=a[11,7],a[18,8]=a[11,8],a[19 ,6]=a[10,6],a[19,7]=a[10,7],\n a[19,8]=a[10,8],a[20,6]=a[9,6] ,a[20,7]=a[9,7],a[20,8]=a[9,8],a[21,4]=a[7,4],\n a[21,5]=a[7, 5],a[21,6]=a[7,6],a[23,2]=a[3,2],a[22,4]=a[6,4],a[22,5]=a[6,5],\n \+ \n a[25,2]+a[25,24]=0,a[25,3]+a[25,23]=0,a[25,6]+a[25,22 ]=0,a[25,7]+a[25,21]=0,\n a[24,3]+a[24,23]=0,a[23,7]+a[23,21] =0,a[23,6]+a[23,22]=0,\n a[22,7]+a[22,21]=0,a[22,9]+a[22,20]= 0,a[22,10]+a[22,19]=0,\n a[21,9]+a[21,20]=0,a[21,10]+a[21,19] =0,a[21,11]+a[21,18]=0,\n seq(add(a[i,j],j=1..i-1)=c[i],i=21. .24)]:\nSOeqs2 := [seq(add(a[i,j]*c[j],j=2..i-1)=1/2*c[i]^2,i=[$14..20 ,25]),\n seq(add(a[i,j]*c[j]^2,j=2..i-1)=1/3*c[i]^3,i=[$14.. 20,25]),\n seq(add(a[i,j]*c[j]^3,j=2..i-1)=1/4*c[i]^4,i=[$14 ..20,25]),\n seq(add(a[i,j]*c[j]^4,j=2..i-1)=1/5*c[i]^5,i=[$ 14..20,25]),\n seq(add(a[i,j]*c[j]^5,j=2..i-1)=1/6*c[i]^6,i= [$14..17,25])]:\ncdns5 := [add(a[18,j]*c[j]^5,j=2..17)=add(a[11,j]*c[j ]^5,j=2..10),\n add(a[19,j]*c[j]^5,j=2..18)=add(a[10,j]*c[j] ^5,j=2..9),\n add(a[20,j]*c[j]^5,j=2..19)=add(a[9,j]*c[j]^5, j=2..8),\n seq(add(a[i,j],j=1..i-1)=c[i],i=[$14..20,25])]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 69 "#-------------------------- ------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "procedure for chopping \"small\" coefficients: " }{TEXT 0 5 "modz " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 427 "modz := proc(u)\n local nm,n;\n \n nm := op(1,'procname');\n if type(nm,posint) then n := nm els e n := 10 end if;\n\n if type(u,\{list,set\}) then return map(modz[ n],u) end if;\n if type(u,`=`) then return map(modz[n],u) end if; \+ \n if type(u,`*`) then return map(modz[n],u) end if;\n if type(u ,`+`) then return map(modz[n],u) end if;\n if type(u,float) and abs (u)<10^(n-Digits) then return 0 end if;\n u;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "#-------------- ---------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 8 "Examp le:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 65:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "0.8*1e-56*a*b+0.98*1e-57*x^3 *y*z=0.8*1e-56*a*b+0.98*1e-57*x^3*y*z;\nmodz[8](%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,&*($\"\")!#d\"\"\"%\"aGF)%\"bGF)F)**$\"#)*!#fF))%\" xG\"\"$F)%\"yGF)%\"zGF)F)F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*($ \"\")!#d\"\"\"%\"aGF)%\"bGF)F)F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 69 "#-------------------------------------------------------------- ------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 19 "convert/Matrix_form" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 19 "convert/Matrix_form" }{TEXT -1 128 " converts an order condition given in abreviated form into an \+ expression involving matrices which evaluates the order condition." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5000 "`convert/Matrix_form` := proc(ord,A::Matrix,B::Matrix,C::Matrix) \n local fact,flag,LS,RS,L,is_simple,G,H,nA,mA,nB,mB,nC,mC,dim,id,i, j;\n\n if type(ord,`=`(algebraic,rational)) then\n flag := 0;\n LS := op(1,ord);\n RS := op(2,ord);\n elif type(ord,`&+`( algebraic,rational)) then\n flag := 1;\n LS := op(1,ord);\n \+ RS := -op(2,ord);\n elif type(ord,`&*`(rational,specfunc(`&+`(a lgebraic,rational),``))) then\n flag := 2;\n fact := op(1,or d);\n LS := op([2,1,1],ord);\n RS := -op([2,1,2],ord);\n e lse\n error \"the 1st argument is invalid, it must be an order co ndition, or an error term, in abreviated form\"\n end if;\n\n ## a llow for order conditions that involve `b*`\n if has(LS,`b*`) then L S := subs(`b*`=b,LS) end if;\n\n ## check if we have a genuine order condition (probably not foolproof!)\n L := eval(subs(``=(u_->u_),LS ));\n if not (type(L,`*`) and indets(L) minus \{a,b,c,e\}=\{\} and c oeffs(L)=1 and type(1/RS,posint)) then\n error \"the 1st argument is invalid, it must be an order condition, or an error term, in abrev iated form\"\n end if;\n\n nA,mA := LinearAlgebra[Dimension](A);\n if nA<>mA then\n error \"the 2nd argument must be a square mat rix\"\n end if;\n dim := nA;\n nB,mB := LinearAlgebra[Dimension] (B);\n if nB<>1 or mB<>dim then\n error \"the 3rd argument must be a row vector with row dimension %1\",dim;\n end if;\n nC,mC := LinearAlgebra[Dimension](C);\n if nC<>mC or nC<>dim then\n err or \"the 4th argument must be a diagonal matrix with dimension %1\",di m;\n end if;\n for i to dim do\n for j to dim do\n if i<>j and C[i,j]<>0 then\n error \"the 4th argument must be a diagonal matrix with dimension %1\",dim;\n end if;\n e nd do;\n end do;\n id := Matrix([seq([1],i=1..dim)]);\n\n is_sim ple := proc(tt) local v;\n v := ListTools[Flatten](eval(subs(``=( u_->[u_]),subs(\{b=1,a=1,c=1,e=1\},[tt]))));\n if v=[1] then retu rn true else return false end if;\n end proc:\n\n ## We need to re place ordinary multiplication with matrix multiplication\n ## The \" hidden\" functions are replaced by the following procedure G \n if is_simple(LS) then\n G := proc(ff) local la,E,n,nn,i;\n \+ if patmatch(ff,b*E::algebraic,'la') then b.subs(la,E)\n elif f f=a*c then a.c\n elif patmatch(ff,a*c^n::posint,'la') then sub s(la,a.c^n)\n elif patmatch(ff,a*c^n::posint*E::algebraic,'la' ) then subs(la,a.c^n.E)\n elif patmatch(ff,c^n::posint*E::alge braic,'la') then subs(la,c^n.E) \n elif patmatch(ff,a*E:: algebraic,'la') then subs(la,a.E) \n else ff;\n end \+ if;\n end proc:\n else ## non-simple order condition\n\n # # procedure to form a diagonal matrix from row sums of a square matrix \n H := proc(U)\n LinearAlgebra[DiagonalMatrix](Lis tTools[Flatten](convert((U.id),listlist)))\n end proc;\n\n G := proc(ff) local la,E,n,gg,hh,nn,i;\n if patmatch(ff,b*E::al gebraic,'la') then\n gg := b.'H'(subs(la,E))\n elif ff=a*c then\n gg := 'H'(a.c)\n elif patmatch(ff,a* c^n::posint,'la') then\n gg := subs(la,'H'(a.c^n))\n \+ elif patmatch(ff,a*c^n::posint*E::algebraic,'la') then\n \+ hh,nn := op(subs(la,[E,n]));\n if type(hh,`*`) then\n \+ gg := op(1,hh);\n for i from 2 to nops(hh) do\n gg := gg.op(i,hh);\n end do;\n \+ gg := 'H'(a.c^nn.gg);\n else \n gg := \+ 'H'(a.c^nn.hh)\n end if;\n elif patmatch(ff,c^n::po sint*E::algebraic,'la') then\n hh,nn := op(subs(la,[E,n])); \n if type(hh,`*`) then\n gg := op(1,hh);\n \+ for i from 2 to nops(hh) do\n gg := gg.o p(i,hh);\n end do;\n gg := 'H'(c^nn.gg);\n else \n gg := 'H'(c^nn.hh)\n end \+ if; \n elif patmatch(ff,a*E::algebraic,'la') then\n \+ hh := subs(la,E);\n if type(hh,`*`) then\n \+ gg := op(1,hh);\n for i from 2 to nops(hh) do\n \+ gg := gg.op(i,hh);\n end do;\n \+ gg := 'H'(a.gg);\n else \n gg := 'H'(a.hh) \n end if;\n elif type(ff,`*`) then\n gg := op(1,ff);\n for i from 2 to nops(ff) do\n \+ gg := gg.op(i,ff);\n end do;\n gg := 'H'(gg);\n elif patmatch(ff,(E::algebraic)^n::posint,'la') then \+ \n gg := subs(la,'H'(E)^n); \n else\n \+ gg := 'H'(ff)\n end if;\n gg;\n end proc:\n end if;\n\n if flag=0 then\n subs(\{a=A,b=B,c=C\},(op(1,LS).eval(s ubs(\{b=1,``=G\},``(LS))).id))[1,1]=RS;\n elif flag=1 then\n su bs(\{a=A,b=B,c=C\},(op(1,LS).eval(subs(\{b=1,``=G\},``(LS))).id))[1,1] -RS;\n else ## flag=2\n fact*(subs(\{a=A,b=B,c=C\},(op(1,LS).ev al(subs(\{b=1,``=G\},``(LS))).id))[1,1]-RS);\n end if; \nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 48 "#-------------------------- ---------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Set up s pecial error terms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 342 "errfact12 := [2,20/9,39/4,2052/25,3286/3,90 8488322227355/30735936,8766/25,277/8,46/9,5/4,1,1,2,1948/5,1385/36,460 /81,170937/100,25/18,10803/64,8993916/625,299/12,195/32,142101/100,480 0846/25,5244/25,455111/24,513/10,75578/27,8215/12,10/9,39/8,1026/25,16 43/3,10/9,39/8,1026/25,1643/3,8766/25,277/8,46/9,5/4,1,1,1,1,10/9,39/8 ,1026/25,1643/3,1]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3838 "errterms_12 := [b*``(a*``(a*``(a*``(a*` `(a*``(a*``(a*``(a*``(a*``(a*``(a*c)))))))))))-1/6227020800, b*``(a*`` (a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c*``(a*c))))))))))-1/2075673600, b*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*c))))))))))-1/1 556755200, b*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a*c)))) ))))))-1/1245404160, b*``(a*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a* ``(a*c))))))))))-1/1037836800, b*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a* ``(a*``(a*``(a*c))))))))))-1/889574400, b*``(a*``(a*``(a*``(a*c*``(a*` `(a*``(a*``(a*``(a*``(a*c))))))))))-1/778377600, b*``(a*``(a*``(a*c*`` (a*``(a*``(a*``(a*``(a*``(a*``(a*c))))))))))-1/691891200, b*``(a*``(a* c*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c))))))))))-1/622702080, b*` `(a*c*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c))))))))))-1/56609 2800, b*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c)))))))))*``(a*c )-1/94348800, b*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a*c))))))))*``(a *``(a*c))-1/28304640, 1/2*``(b*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``(a *``(a*``(c^2*a))))))))))-1/3113510400), b*``(a*``(a*``(a*``(a*c*``(a*` `(a*``(a*``(a*c*``(a*c)))))))))-1/259459200, b*``(a*``(a*``(a*c*``(a*` `(a*``(a*``(a*``(a*c*``(a*c)))))))))-1/230630400, b*``(a*``(a*c*``(a*` `(a*``(a*``(a*``(a*``(a*c*``(a*c)))))))))-1/207567360, b*``(a*``(a*``( a*``(a*c*``(a*``(a*``(a*c*``(a*``(a*c)))))))))-1/194594400, b*``(a*c*` `(a*``(a*``(a*``(a*``(a*``(a*``(a*c*``(a*c)))))))))-1/188697600, b*``( a*``(a*``(a*c*``(a*``(a*``(a*``(a*c*``(a*``(a*c)))))))))-1/172972800, \+ b*``(a*``(a*``(a*``(a*c*``(a*``(a*c*``(a*``(a*``(a*c)))))))))-1/155675 520, b*``(a*``(a*c*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*c)))))))))-1/1 55675520, b*``(a*c*``(a*``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*c)))))))) )-1/141523200, b*``(a*``(a*``(a*c*``(a*``(a*``(a*c*``(a*``(a*``(a*c))) ))))))-1/138378240, b*``(a*``(a*``(a*``(a*c*``(a*c*``(a*``(a*``(a*``(a *c)))))))))-1/129729600, b*``(a*``(a*c*``(a*``(a*``(a*``(a*c*``(a*``(a *``(a*c)))))))))-1/124540416, b*``(a*``(a*``(a*c*``(a*``(a*c*``(a*``(a *``(a*``(a*c)))))))))-1/115315200, b*``(a*c*``(a*``(a*``(a*``(a*``(a*c *``(a*``(a*``(a*c)))))))))-1/113218560, b*``(a*``(a*c*``(a*``(a*``(a*c *``(a*``(a*``(a*``(a*c)))))))))-1/103783680, b*``(a*c*``(a*``(a*``(a*` `(a*c*``(a*``(a*``(a*``(a*c)))))))))-1/94348800, b*``(a*``(a*``(a*``(a *``(a*``(a*``(a*c*``(a*c))))))))*``(a*c)-1/31449600, b*``(a*``(a*``(a* ``(a*``(a*``(a*c*``(a*``(a*c))))))))*``(a*c)-1/23587200, b*``(a*``(a*` `(a*``(a*``(a*c*``(a*``(a*``(a*c))))))))*``(a*c)-1/18869760, b*``(a*`` (a*``(a*``(a*c*``(a*``(a*``(a*``(a*c))))))))*``(a*c)-1/15724800, b*``( a*``(a*``(a*``(a*``(a*``(a*c*``(a*c)))))))*``(a*``(a*c))-1/9434880, b* ``(a*``(a*``(a*``(a*``(a*c*``(a*``(a*c)))))))*``(a*``(a*c))-1/7076160, b*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a*c)))))))*``(a*``(a*c))-1/56609 28, b*``(a*``(a*``(a*c*``(a*``(a*``(a*``(a*c)))))))*``(a*``(a*c))-1/47 17440, 1/2*``(b*``(a*``(a*``(a*``(a*c*``(a*``(a*``(a*``(a*``(c^2*a)))) )))))-1/389188800), 1/2*``(b*``(a*``(a*``(a*c*``(a*``(a*``(a*``(a*``(a *``(c^2*a)))))))))-1/345945600), 1/2*``(b*``(a*``(a*c*``(a*``(a*``(a*` `(a*``(a*``(a*``(c^2*a)))))))))-1/311351040), 1/2*``(b*``(a*c*``(a*``( a*``(a*``(a*``(a*``(a*``(a*``(c^2*a)))))))))-1/283046400), 1/2*``(b*`` (a*``(a*``(a*``(a*``(a*``(a*``(a*``(c^2*a))))))))*``(a*c)-1/47174400), 1/2*``(b*``(a*``(a*``(a*``(a*``(a*``(a*``(c^2*a)))))))*``(a*``(a*c))- 1/14152320), 1/2*``(b*``(c^2*a)*``(a*``(a*``(a*``(a*``(a*``(a*``(a*``( a*c))))))))-1/14152320), 1/4*``(b*``(c^2*a)*``(a*``(a*``(a*``(a*``(a*` `(a*``(c^2*a)))))))-1/7076160), 1/2*``(b*``(c^2*a)*``(a*``(a*``(a*``(a *``(a*``(a*c*``(a*c)))))))-1/4717440), 1/2*``(b*``(c^2*a)*``(a*``(a*`` (a*``(a*``(a*c*``(a*``(a*c)))))))-1/3538080), 1/2*``(b*``(c^2*a)*``(a* ``(a*``(a*``(a*c*``(a*``(a*``(a*c)))))))-1/2830464), 1/2*``(b*``(c^2*a )*``(a*``(a*``(a*c*``(a*``(a*``(a*``(a*c)))))))-1/2358720), 1/47900160 0*``(b*c^12-1/13)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 63 "#====================================================== ========" }}{PARA 0 "" 0 "" {TEXT -1 84 "The list of error terms used \+ (in abreviated form) can be obtained from the computed " }{TEXT 0 6 "e rrkey" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "errterms12 := PrincipalErrorTerms(12):\nn ops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&'[7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "errkey := [ 1,2,3,4,5,6,7,8,10,11,21,27,32,42,43,45,46,47,49,51,52,53,55,56,58,59, 60,62,66,103,114,124,136,159,171,181,188,266,267,270,271,290,303,304,1 090,1134,1165,1190,1212,12486]:\nerrterms_12 := [seq(errterms12[i],i=e rrkey)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 48 "#----------------- ------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 25 " calc_RKcoeffs_with_params" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2882 "calc_RKcoeffs_with_params := proc ()\n local eq1,eq2,eqns,eqns2,eqns3,eqns4,eqns5;\n global e1,e2,e3 ,e4,e5,e6,e7,e8,e9,e10,e11,e12,e13,e14,e15,e16,e17,e18,e19,e20,e21,e22 ,e23;\n\n e1 := \{c[2]=convert(c_2,rational,Digits+4),c[4]=convert(c _4,rational,Digits+4),\n c[9]=convert(c_9,rational,Digits+4), c[10]=convert(c_10,rational,Digits+4)\}:\n\n e2 := simplify(subs(e1, \{c[3]=2/3*c[4],c[5]=(2*c[4]+3)*(2*c[4]-1)/(8*(c[4]+1)),c[6]=c[4]-1/2 \}));\n e3 := `union`(e1,e2):\n e4 := `union`(e3,subs(e3,\{c[7]=(3 *c[9]^2-4*c[6]*c[9]+2*c[9]-2*c[6]+1)/(4*c[9]+2-6*c[6])\})):\n e5 := \+ `union`(e4,subs(e4,\{c[8]=(20*c[6]*c[7]-15*c[7]*c[9]+12*c[9]^2-15*c[6] *c[9])*c[9]/\n (30*c[6]*c[7]-20*c[7]*c[9]+15*c[9]^2-20*c[6] *c[9])\})): \n e6 := \{b[4]=0,b[5]=0,b[8]=0,b[12]=0,c[25]=1,c[14]=1/ 2-1/66*(495-66*15^(1/2))^(1/2),\n c[16]=1/2+1/66*(495-66*15^( 1/2))^(1/2),b[1]=1/42,b[13]=31/175-1/100*15^(1/2),\n b[25]=1/ 42,b[16]=31/175+1/100*15^(1/2),b[15]=128/525,c[13]=1/2-1/66*(495+66*15 ^(1/2))^(1/2),\n c[15]=1/2,b[14]=31/175+1/100*15^(1/2),c[17]= 1/2+1/66*(495+66*15^(1/2))^(1/2),b[17]=31/175-1/100*15^(1/2),\n \+ seq(a[i,2]=0,i=4..13),seq(a[i,3]=0,i=6..13),\n seq(a[i ,4]=0,i=8..13),seq(a[i,5]=0,i=9..13),seq(a[13,j]=0,j=6..8),a[11,6]=0\} :\n e7 := `union`(e5,e6);\n e8 := `union`(e7,\{simplify(expand(rat ionalize((subs(e7,eqA)))))\}):\n e9 := `union`(e8,\{simplify(expand( rationalize((subs(e8,eqB)))))\}):\n e10 := evalf(e9):\n eqns := su bs(e10,[op(RSeqs),op(SOeqs)]):\n e11 := solve(\{op(eqns)\},indets(eq ns) minus \{a[12,10],a[12,11]\}):\n e12 := `union`(e10,e11):\n e13 := \{b[2]=b_2,b[3]=b_3,b[6]=b_6,b[7]=b_7,b[9]=b_9,b[10]=b_10,b[11]=b_ 11,\n seq(a[i,2]=0,i=[$14..20]),seq(a[i,3]=0,i=[$14..20]),seq(a [i,4]=0,i=[$14..20]),\n seq(a[i,5]=0,i=[$14..20]),seq(a[i,6]=0, i=[$14..18]),seq(a[i,7]=0,i=[$14..17]),\n seq(a[i,8]=0,i=[$14.. 17]),seq(a[21,j]=0,j=[2,3,7,8,$12..17]),seq(a[22,j]=0,j=[2,3,6,8,$11.. 18]),\n seq(a[23,j]=0,j=[3,4,5,$8..20]),seq(a[24,j]=0,j=[2,$4.. 22]),seq(a[25,j]=0,j=[4,5,8])\}:\n e14 := `union`(e12,evalf(e13)):\n e15 := `union`(e14,evalf(subs(e14,\{c[18]=c[11],c[20]=c[9],c[21]=c[ 7],c[19]=c[10],c[22]=c[6],c[23]=c[3],\n c[24]=c[2],b[19]=-b[10 ],b[22]=-b[6],b[23]=-b[3],b[24]=-b[2],b[18]=-b[11],b[20]=-b[9],b[21]=- b[7]\}))):\n eqns2 := subs(e15,cdns2):\n e16 := solve(\{op(eqns2) \},indets(eqns2) minus \{a[19,14],a[20,14],a[20,15]\}):\n e17 := `un ion`(e15,e16):\n eqns3 := subs(e17,cdns3):\n e18 := solve(\{op(eqn s3)\}):\n e19 := `union`(e17,e18):\n eqns4 := subs(e19,SOeqs2):\n \+ e20 := solve(\{op(eqns4)\},indets(eqns4) minus \{a[18,11],a[19,11],a [20,11],a[20,15],a[20,14],a[19,14]\}):\n e21 := `union`(subs(e20,e19 ),e20):\n eqns5 := subs(e21,cdns5):\n e22 := solve(\{op(eqns5)\},i ndets(eqns5) minus \{a[20,15],a[20,14],a[19,14]\}):\n e23 := `union` (map(u_->lhs(u_)=modz(subs(e22,rhs(u_))),e21),e22):\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 48 "#------------------------------ -----------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "calc_RKcoeffs " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3923 "calc_RKcoeffs := proc()\n local eq1,eq2,eqns,eqns2,eqns3,e qns4,eqns5,A,A_,B,B_,C,C_,id,Rz,stb12,ct,dd,sm;\n global e1,e2,e3,e4 ,e5,e6,e7,e8,e9,e10,e11,e12,e13,e14,e15,e16,e17,e18,e19,e20,e21,e22,e2 3;\n\n e1 := \{c[2]=convert(c_2,rational,Digits+4),c[4]=convert(c_4, rational,Digits+4),\n c[9]=convert(c_9,rational,Digits+4),c[1 0]=convert(c_10,rational,Digits+4)\}:\n\n e2 := simplify(subs(e1,\{c [3]=2/3*c[4],c[5]=(2*c[4]+3)*(2*c[4]-1)/(8*(c[4]+1)),c[6]=c[4]-1/2\})) ;\n e3 := `union`(e1,e2):\n e4 := `union`(e3,subs(e3,\{c[7]=(3*c[9 ]^2-4*c[6]*c[9]+2*c[9]-2*c[6]+1)/(4*c[9]+2-6*c[6])\})):\n e5 := `uni on`(e4,subs(e4,\{c[8]=(20*c[6]*c[7]-15*c[7]*c[9]+12*c[9]^2-15*c[6]*c[9 ])*c[9]/\n (30*c[6]*c[7]-20*c[7]*c[9]+15*c[9]^2-20*c[6]*c[9 ])\})):\n e6 := \{b[4]=0,b[5]=0,b[8]=0,b[12]=0,c[25]=1,c[14]=1/2-1/6 6*(495-66*15^(1/2))^(1/2),\n c[16]=1/2+1/66*(495-66*15^(1/2)) ^(1/2),b[1]=1/42,b[13]=31/175-1/100*15^(1/2),\n b[25]=1/42,b[ 16]=31/175+1/100*15^(1/2),b[15]=128/525,c[13]=1/2-1/66*(495+66*15^(1/2 ))^(1/2),\n c[15]=1/2,b[14]=31/175+1/100*15^(1/2),c[17]=1/2+1 /66*(495+66*15^(1/2))^(1/2),b[17]=31/175-1/100*15^(1/2),\n a[ 12,10]=convert(a12_10,rational,Digits+4),a[12,11]=convert(a12_11,ratio nal,Digits+4),\n seq(a[i,2]=0,i=4..13),seq(a[i,3]=0,i=6..13 ),\n seq(a[i,4]=0,i=8..13),seq(a[i,5]=0,i=9..13),seq(a[13,j ]=0,j=6..8),a[11,6]=0\}:\n e7 := `union`(e5,e6);\n e8 := `union`(e 7,\{simplify(expand(rationalize((subs(e7,eqA)))))\});\n e9 := `union `(e8,\{simplify(expand(rationalize((subs(e8,eqB)))))\});\n e10 := ev alf(e9);\n eqns := subs(e10,[op(RSeqs),op(SOeqs)]):\n e11 := solve (\{op(eqns)\},indets(eqns)):\n e12 := `union`(e10,e11):\n e13 := \+ \{b[2]=b_2,b[3]=b_3,b[6]=b_6,b[7]=b_7,b[9]=b_9,b[10]=b_10,b[11]=b_11, \n seq(a[i,2]=0,i=[$14..20]),seq(a[i,3]=0,i=[$14..20]),seq(a[i, 4]=0,i=[$14..20]),\n seq(a[i,5]=0,i=[$14..20]),seq(a[i,6]=0,i=[ $14..18]),seq(a[i,7]=0,i=[$14..17]),\n seq(a[i,8]=0,i=[$14..17] ),seq(a[21,j]=0,j=[2,3,7,8,$12..17]),seq(a[22,j]=0,j=[2,3,6,8,$11..18] ),\n seq(a[23,j]=0,j=[3,4,5,$8..20]),seq(a[24,j]=0,j=[2,$4..22] ),seq(a[25,j]=0,j=[4,5,8]),\n a[19,14]=a19_14,a[20,14]=a20_14,a [20,15]=a20_15\}:\n e14 := `union`(e12,evalf(e13)):\n e15 := `unio n`(e14,evalf(subs(e14,\{c[18]=c[11],c[20]=c[9],c[21]=c[7],c[19]=c[10], c[22]=c[6],c[23]=c[3],\n c[24]=c[2],b[19]=-b[10],b[22]=-b[6],b [23]=-b[3],b[24]=-b[2],b[18]=-b[11],b[20]=-b[9],b[21]=-b[7]\}))):\n \+ eqns2 := subs(e15,cdns2):\n e16 := solve(\{op(eqns2)\}):\n e17 := \+ `union`(e15,e16):\n eqns3 := subs(e17,cdns3):\n e18 := solve(\{op( eqns3)\}):\n e19 := `union`(e17,e18):\n eqns4 := subs(e19,SOeqs2): \n e20 := solve(\{op(eqns4)\},indets(eqns4) minus \{a[18,11],a[19,11 ],a[20,11]\}):\n e21 := `union`(subs(e20,e19),e20):\n eqns5 := sub s(e21,cdns5):\n e22 := solve(\{op(eqns5)\}):\n e23 := `union`(map( u_->lhs(u_)=modz(subs(e22,rhs(u_))),e21),e22):\n if nargs>0 and args [1]='give_characteristics' then\n A := Matrix([seq([seq(a[i,j],j= 1..i-1),seq(0,j=i..25)],i=1..25)]):\n A_ := subs(e23,A);\n B := Matrix([[seq(b[i],i=1..25)]]):\n B_ := subs(e23,B);\n C \+ := LinearAlgebra[DiagonalMatrix]([0,seq(c[i],i=2..25)]):\n C_ := \+ subs(e23,C);\n id := Matrix([seq([1],i=1..25)]):\n Rz := add (z^j/j!,j=0..12)+add((B_.A_^(j-1).id)[1,1]*z^j,j=13..25):\n Rz := map(convert,Rz,rational,24):\n stb12 := evalf[14](max(fsolve(Rz= 1,z=-8..-1e-7),fsolve(Rz=-1,z=-8..-1e-7)));\n print(infinity*`-no rm of linking coeffs`=evalf[10](max(seq(seq(subs(e23,abs(a[i,j])),j=1. .i-1),i=2..25))));\n print(`order 12 stability interval` = evalf[ 8]([stb12,0]));\n sm := 0:\n for ct from 1 to nops(errterms_ 12) do\n dd := convert(errterms_12[ct],'Matrix_form',A_,B_,C_) ;\n dd := modz(expand(eval(dd)));\n sm := sm+errfact12 [ct]*dd^2;\n end do:\n print(`2-norm of principal error` = e valf[10](sqrt(sm)));\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 48 "#-----------------------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 41 "#----------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "We use the nodes and wei ghts of Hiroshi Ono's scheme to obtain a coefficients scheme expressed in terms of the parameters " }{XPPEDIT 18 0 "a[12,10]" "6#&%\"aG6$\" #7\"#5" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[12,11]" "6#&%\"aG6$\"#7\"# 6" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[19,14]" "6#&%\"aG6$\"#>\"#9" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[20,14]" "6#&%\"aG6$\"#?\"#9" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "a[20,15]" "6#&%\"aG6$\"#?\"#:" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "Digits := \+ 95:\nc_2 := 1/4: c_4 := 2/3: c_9 := 1/5: c_10 := 1/3:\nb_2 := -11/100: b_3 := -17/100: b_6 := -19/100: b_7 := -21/100:\nb_9 := -23/100: b_10 := -27/100: b_11 := -29/100:\ncalc_RKcoeffs_with_params():" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "We set up an ex pression for the square of the principal error norm in terms of the 5 \+ parameters." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "Digits := 75 :\nA := Matrix([seq([seq(a[i,j],j=1..i-1),seq(0,j=i..25)],i=1..25)]): \nB := Matrix([[seq(b[i],i=1..25)]]):\nC := LinearAlgebra[DiagonalMatr ix]([0,seq(c[i],i=2..25)]):\nA_ := subs(e23,A): B_ := subs(e23,B): C_ := subs(e23,C):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "sm := 0:\nfor ct from 1 to nops(errterms_12) do\n dd := convert(errterms_12[ct],'Matrix_form',A_,B_,C_);\n dd \+ := modz(expand(eval(dd)));\n sm := sm+errfact12[ct]*dd^2;\nend do:\n sm := modz(expand(sm)):" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "sm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6083 "sm := -.8987788241115143026055575268852362906063190 84145231591770195146405845851141e-8*a[20,15]*a[20,14]*a[12,10]+.204687 230224408902391081377334214086039027404178389850434178612055414723785e -6*a[20,15]*a[19,14]*a[12,11]+.537268610174292539375031201715682286422 707133854348509617979923094184984463e-6*a[20,15]*a[20,14]*a[12,11]-.34 2414473147904090419898950813233528058322510478985498132036151967310935 960e-8*a[20,15]*a[19,14]*a[12,10]+.70775380662613305675920849954339939 1477092567959544926194340666922890648731e-7*a[20,14]*a[12,10]^2*a[19,1 4]-.455422433910016927816815145521716744243892479065086137567277453096 924811553e-2*a[20,15]*a[12,11]^2*a[20,14]-.420367786562516297334823807 716976685550438574196896627042120553356701884266e-6*a[20,15]*a[12,10]^ 2*a[19,14]-.4674445063777264455796921278886211431049587330291822475990 41946793638729261e-3*a[20,15]^2*a[12,10]*a[12,11]+.2921234946422336263 24534984511904305860502153002798611151714445153235927650e-3*a[19,14]*a [12,11]^2*a[20,14]-.16828415474654281898070612822213511501255640826549 7775777057050491098966687e-5*a[19,14]^2*a[12,11]*a[12,10]-.17350568191 3669843524313839265490432709409387454765624196718732176960883785e-2*a[ 19,14]*a[12,11]^2*a[20,15]-.106655430558321496061985631966123773007874 614013918487841207652618517124413e-4*a[20,14]^2*a[12,10]*a[12,11]+.229 7214485398637823613594436167751242506204835453716775553340825486915003 08e-5*a[12,11]*a[20,14]*a[12,10]+.954456911124564843623667537783857197 501990284908743229244673364468184936273e-6*a[12,11]*a[19,14]*a[12,10]- .111774421886528741968640498557304065112047947022287996420145914184054 810507e-4*a[12,11]*a[20,15]*a[12,10]-.53384270160013725879603497988856 0415848628255749205204619905854934129393004e-11*a[20,15]*a[19,14]-.121 9752408256286979919960586354643075441693176916820405398005707719596245 17e-12*a[20,14]+.40922035879231039030660934718078807344716158856154717 2251381836564918484471e-11*a[20,15]-.538280021370968815163471141861033 617422102667105669664279015835667559135700e-13*a[19,14]-.1111803514326 68971979055699354242373837583287953969978220008716169537023785e-8*a[12 ,11]+.1815853466138528800716182601580688199459948986291501422407015391 73826222128e-10*a[12,10]+.33745561670162771801552745660322184841511582 3023438864438947137148092233508e-12*a[20,14]^2+.2496192457895807701155 77745698381877617034112751459682252722048650390109064e-9*a[20,15]^2+.5 3244764869289627850729223268428482185370607641735725101505592732959124 2198e-13*a[19,14]^2+.5827318283427436598693767705975816028675676384351 22723295630047996027134614e-4*a[19,14]^2*a[12,11]^2+.89479636991824935 3960469355781672657288894013266752781042043834445402661475e-7*a[20,14] ^2*a[12,10]^2-.8007511430471753845147003306117109443245384060371317028 58179948987984546890e-8*a[19,14]*a[12,10]^2+.9377426581114056285402843 26941409270125955098341254411019813345620256628383e-7*a[20,15]*a[12,10 ]^2-.19272709994211753514137529936653066705862354288134547456499600372 8656690744e-7*a[20,14]*a[12,10]^2+.13394858002156815938190104876351390 8811034647908042056514859906758782347824e-8*a[12,10]^2-.79547596099464 8708500180455350146752114386740930238421522307116255118030301e-4*a[20, 14]*a[12,11]^2+.387050779237004673647563965472018904109317621051378644 438810236902933188960e-3*a[20,15]*a[12,11]^2-.330507897033848053492587 829666147384255116829881020921766629420769150912007e-4*a[19,14]*a[12,1 1]^2-.1596602752890176115866269659551501773668911922819491894797284887 95439385059e-6*a[12,11]*a[12,10]+.552869188860969556483224207366051393 614369586117653032378582045972212094874e-5*a[12,11]^2+.369324813411822 668118910161192245763300426832034499291583238348656875107562e-3*a[20,1 4]^2*a[12,11]^2+.14118366968630539421852628298917849033499289118815017 1045615013302031613111e-7*a[19,14]^2*a[12,10]^2+.161865977376494402759 501267836576311039796144769327220195023707337968942729e-1*a[20,15]^2*a [12,11]^2+.39216722979361073086027588246957919883338762253857741674820 2259806234829494e-5*a[20,15]^2*a[12,10]^2+.173809626265933741850215165 816838040254281532459119001002907166774390685276e-9*a[20,14]^2*a[12,10 ]+.2669160328732299827638253048696704572212326447261229020794544095980 12805320e-12*a[20,14]*a[19,14]-.14972408733763589060383630184772456915 8036862743406659841060744768797673385e-9*a[20,14]*a[12,10]-.1401244845 73826859212460857599593294972873539094527767151837592447893196014e-10* a[20,15]*a[20,14]+.380723978556172671098582059532244643108436411579279 716473491651411087209297e-8*a[19,14]*a[12,11]+.13747756333568832665806 2980039490373193830418984493607604419743285035213179e-9*a[20,14]*a[12, 10]*a[19,14]-.78256579728927048096001919470536661217714485281743136447 7975439657607587999e-8*a[19,14]*a[12,11]*a[20,14]-.3385493314494484204 87998878412111848729852390668053762285356259683600607294e-5*a[20,15]^2 *a[12,11]+.13123328128845020274328958111114941748969668472321772810098 2591192576942780e-8*a[20,15]*a[12,10]-.8073866519551353698026134978309 11097817858476774050794006200234528715737837e-7*a[20,15]*a[12,11]+.274 2420699054883972911935563107246016539512818421315370899161875026132305 60e-10*a[19,14]^2*a[12,10]-.989379397300106727350106016568919251862036 446618148391964566587800298172237e-8*a[20,14]^2*a[12,11]-.156107264980 987851926878353605010834991720979723127744022446492816712682104e-8*a[1 9,14]^2*a[12,11]-.6217521966595879881519000695659547656798341881360941 33544305921966591074969e-10*a[19,14]*a[12,10]-.11033928017926453477647 9228715169099568325757444238720588994668143677932381e-5*a[20,14]*a[12, 10]^2*a[20,15]+.919037817920868175379300678562293599789347073455092665 790039963442435035000e-8*a[20,14]*a[12,11]+.56271041490304600372232299 3075709607545044210694565681694492542123384024890e-7*a[20,15]^2*a[12,1 0]+.695220289813829015848461296396466265616522971711449512695173336954 985279097e-13+.5010582158795520409773214201510585704391480356443814139 18517767827610289255e-4*a[19,14]*a[12,11]*a[20,15]*a[12,10]+.131519123 575455557070594229110578244126693484002303034623737970140530455271e-3* a[20,14]*a[12,10]*a[20,15]*a[12,11]-.843608551763545810719481872541373 093862532978248259013591621525653854655979e-5*a[20,14]*a[12,10]*a[19,1 4]*a[12,11]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 95 "We construct an expression for the norm of the lin king coefficients in terms of the parameters." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "smL := expand(subs(e23,add(add(a[i,j]^2,j=1..i-1 ),i=2..25))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "smL" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1635 "smL := -218405854.1114221726741457677674323829721259094393319434 9799226651638894325202367592819095105052*a[20,15]*a[19,14]-1220772.235 2156097233669934288521481360531264364607311496745082574051747219620404 957315483408239*a[20,14]+7503676.8468431533382510328691762650761042679 250971490126683236236523886781385425785813791331835*a[20,15]-403749.57 0182774085552806148674643109876354399471589793106095419324469212712158 70044845246836448*a[19,14]+72.0460894274950156925213753219807000501640 97191154937818405117694074607361721269379131615327482*a[12,11]-.918302 3571336602775807050340780283992170767279013230008230984674932220156191 2180410745114780476*a[12,10]+38905669.15498986190408739541100746502506 8527024214087201470827314878403812650307446505440623667*a[20,14]^2+321 6112502.44831115364218868811860638261270081552604913302291371709658646 05476225117458502365076*a[20,15]^2+4223977.964254973144958996354432193 6422740449823785855387591924196662348790942051042596299862506*a[19,14] ^2+304.907676185794154294022485013404569104083688280375985857881711382 22463657753893456378983914513*a[12,10]^2-43514.92771310287480875410375 8129677712902236624092636873601263791520594688591782816500954551548*a[ 12,11]*a[12,10]+1591844.9714006443289522204538524579204825746337298797 672300331394705515466240488257841372153434*a[12,11]^2+25636981.2014361 8488838117520103410247785977900396199754593227110044936369755984619667 4214690603*a[20,14]*a[19,14]-663078171.5912351396485309949862686100114 3916074209610322968334558158743433542166848906094271468*a[20,15]*a[20, 14]+19962.348795926816190354220199010275858245572608526835074674889560 340493238333130375026757205493:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "expand(subs(e23,add(add(a[i, j]^2,j=1..i-1),i=2..25)));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 92 "We form a weighted sum of the princip al error norm and the norm of the linking coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "smP := sm+smL*3*10^(-20):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We set ou t to try to minimize the weighted sum with respect to the pair of para meters " }{XPPEDIT 18 0 "a[12,10];" "6#&%\"aG6$\"#7\"#5" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[12, 11];" "6#&%\"aG6$\"#7\"#6" }{TEXT -1 32 " \+ and the triple of parameters " }{XPPEDIT 18 0 "a[19,14];" "6#&%\"aG6 $\"#>\"#9" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[20, 14];" "6#&%\"aG6$\" #?\"#9" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[20,15];" "6#&%\"aG6$\"#?\" #:" }{TEXT -1 39 " alternately using the Maple procedure " }{TEXT 0 8 "minimize" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 106 "This proces s initially minimzes both quantities. Eventually the balance is lost, \+ so we stop at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 833 "Digits := 50:\neA := \{a[19,14]=0. ,a[20,14]=0.,a[20,15]=0.\}:\nfor ct to 1000 do\n minimize(eval(smP,e A),a[12,10]=-.5e-1..0,a[12,11]=0..0.2e-3,'location'):\n eB,ss := op( 1,op(op(2,[%]))),sqrt(op(2,op(op(2,[%])))):\n minimize(eval(smP,eB), a[19,14]=-1.5..1.5,a[20,14]=-0.5..0.5,a[20,15]=-.2..0.2,'location'):\n eA,ss := op(1,op(op(2,[%]))),sqrt(op(2,op(op(2,[%])))):\n if `mod `(ct,50)=0 then\n print(ct);\n print(a[12,10]=evalf[30](subs (eB,a[12,10])),a[12,11]=evalf[30](subs(eB,a[12,11])));\n print(a[ 19,14]=evalf[30](subs(eA,a[19,14])),a[20,14]=evalf[30](subs(eA,a[20,14 ])),\n a[20,15]=evalf[30](subs(eA,a[20,15])));\n e AB := `union`(eA,eB):\n print(`principal error norm`=evalf[30](sq rt(eval(sm,eAB))));\n print(`norm of linking coefficients`=evalf[ 30](sqrt(eval(smL,eAB))));\n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"#7\"#5$!?o&p8'yg'fx?ZW)oj:!#J/&F%6$F'\"#6$\" ?Rs/\"e?66f\"*ov\"f^q!#M" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$ \"#>\"#9$\"?/6_(45k#zuXxM.>n!#I/&F%6$\"#?F($!?@J1,3^%R\\O0cDk_\"F+/&F% 6$F/\"#:$\"?H#4(p%*okGm>()G0S\"*!#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%5principal~error~normG$\"?#H]OW8Dp'*=sK'H2_!#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%=norm~of~linking~coefficientsG$\"?62u'zz%=XlH7;I/?!#F " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"#7\"#5$!?>nY7#f'4X!pfVL%)H#!#J/&F%6$F'\"#6$ \"?BjVo3sfol-;oZn5!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"#> \"#9$\"?ihdmWAAE)4'*QWi'o!#I/&F%6$\"#?F($!?+SYp$\\O2a>]PqdK\"F+/&F%6$F /\"#:$\"?[z;A#e%R:\"z&[vy.6!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5p rincipal~error~normG$\"?\"RV2+,>3(4S-J7sY!#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%=norm~of~linking~coefficientsG$\"?I%G_f%QC2OmN&==*=!# F" }}{PARA 261 "" 0 "" {TEXT -1 3 " : " }}{PARA 260 "" 0 "" {TEXT -1 3 " : " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"#7\"#5$!?(yz)RPS) y<#p=TjkR!#J/&F%6$F'\"#6$\"?Lz)\\7l_nhB_m9v*=!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"#>\"#9$\"?!R#e%G)oB\"\\]--$[$*p!#I/&F%6$\"#? F($!?gA$egW;iYlXy)og6F+/&F%6$F/\"#:$\"?t\"*pz2l8$3%QhX`57!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"?a(p\"y)eWS>:W\"f emW!#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%=norm~of~linking~coefficie ntsG$\"?\")[1%)[,**HIPnfL>=!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$+ (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"#7\"#5$!?dHKp:\\cy\\w% QZ^)R!#J/&F%6$F'\"#6$\"?70aR^qKwaN1L!y!>!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"#>\"#9$\"?s?+u'pM7sui\"fY&*p!#I/&F%6$\"#?F($ !?ji@(G\")yMO\"Htz4g6F+/&F%6$F/\"#:$\"?:2*4L?7#=C,0([4@\"!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"?vjqI3`CU_:qRcmW! #P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%=norm~of~linking~coefficientsG $\"?b.4AYZ3KmwK.C>=!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$](" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"#7\"#5$!?\\5)o&**=8ua96s3,S !#J/&F%6$F'\"#6$\"?$*37@RdO$[*)H!*)z:>!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"#>\"#9$\"?-beaBS/B35m\"3q*p!#I/&F%6$\"#?F($! ?k%QAb%eg>:>86lf6F+/&F%6$F/\"#:$\"?:y_lFv\")e#*)Q(QE67!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"?u_h#oga;>?f#\\cmW!# P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%=norm~of~linking~coefficientsG$ \"?_:)e(z!>q!32'er\">=!#F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "[a[12,10] = -.39851473847649785649 1569322957e-1, a[12,11] = .190780330635547632705139540512e-3,\na[19,14 ] = .699546591627472123469674002072, a[20,14] = -.11600979732913634788 1287216263,\na[20,15] = .121094870501241821220330990715e-1]:\nconvert( %,rational,5);\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/&% \"aG6$\"#7\"#5#!#6\"$w#/&F&6$F(\"#6#\"\"\"\"%U_/&F&6$\"#>\"#9#\"$\\\" \"$8#/&F&6$\"#?F8#!#]\"$J%/&F&6$F?\"#:#F(\"$\"**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/&%\"aG6$\"#7\"#5$!+Ys]&)R!#6/&F&6$F(\"#6$\"+H)ow!>!# 8/&F&6$\"#>\"#9$\"+k^I&*p!#5/&F&6$\"#?F8$!+2G4g6F;/&F&6$F?\"#:$\"+$3)* 3@\"F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "Digits := 30:\nc_2 := 1/4: c_4 := 2/3: c_9 := 1/5: c _10 := 1/3:\nb_2 := -11/100: b_3 := -17/100: b_6 := -19/100: b_7 := -2 1/100:\nb_9 := -23/100: b_10 := -27/100: b_11 := -29/100:\na12_10 := - 11/276: a12_11 := 1/5242:\na19_14 := 149/213: a20_14 := -50/431: a20_1 5 := 12/991:\ncalc_RKcoeffs('give_characteristics'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+ w3^%e*!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% " 0 "" {MPLTEXT 1 0 314 "Digits := 30:\nc_2 := 1/4 +1/100: c_4 := 2/3: c_9 := 1/5: c_10 := 1/3:\nb_2 := -11/100: b_3 := - 17/100: b_6 := -19/100: b_7 := -21/100:\nb_9 := -23/100: b_10 := -27/1 00: b_11 := -29/100:\na12_10 := -11/276: a12_11 := 1/5242:\na19_14 := \+ 149/213: a20_14 := -50/431: a20_15 := 12/991:\ncalc_RKcoeffs('give_cha racteristics'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\" \"%8-norm~of~linking~coeffsGF&$\"+w3^%e*!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% " 0 "" {MPLTEXT 1 0 315 "Digits : = 30:\nc_2 := 39/88: c_4 := 101/152: c_9 := 1/5: c_10 := 1/3:\nb_2 := \+ -11/100: b_3 := -17/100: b_6 := -19/100: b_7 := -21/100:\nb_9 := -221/ 958: b_10 := -27/100: b_11 := -29/100:\na12_10 := -11/276: a12_11 := 1 /5242:\na19_14 := 149/213: a20_14 := -50/431: a20_15 := 12/991:\ncalc_ RKcoeffs('give_characteristics'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ *&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+S8hQ&*!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/% " 0 "" {MPLTEXT 1 0 472 "Dig its := 30:\nA := subs(e23,Matrix([seq([seq(a[i,j],j=1..i-1),seq(0,j=i. .25)],i=1..25)])):\nB := subs(e23,Matrix([[seq(b[i],i=1..25)]])):\nid \+ := Matrix([seq([1],i=1..25)]):\nadd(z^j/j!,j=0..12)+add((B.A^(j-1).id) [1,1]*z^j,j=13..25):\nmap(convert,%,rational,24): R := unapply(%,z):\n pts := []: z0 := 0:\nfor ct from 0 to 220 do\n zz := newton(R(z)=exp (ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]] :\nend do:\nplot(pts,color=COLOR(RGB,0,.35,.35),style=point);" }} {PARA 13 "" 1 "" {GLPLOT2D 417 427 427 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2#Gx#H(p,R\"zr$Fgu7$$!?p4^.tthXjo3;\"4&yF@$\"?TQ7&prZQ#)*pdID6PFgu7$$! ?foa\\RPB'))H(3*>W)yF@$\"??vG'*)o)Hga))[YG/PFgu7$$!?4rI)H<3eKyEZ:p\"zF @$\"?7NEaq(Q]!Rf0@)pp$Fgu7$$!?Sh)z(4qgg4yd%[&[zF@$\"?j]XrjH/J'p\"fZJ*o $Fgu7$$!?\\B?fpgy$G9`!)p%zzF@$\"?`=6:x&3eP^`%pC\"o$Fgu7$$!?]]bBd(ewX&Q ))p$)4!)F@$\"?1Q9m\"oR7Ru.&otsOFgu7$$!?ZPf?jTo^ny'4@)R!)F@$\"?a->noOn( =C)3]tjOFgu7$$!?*)z:\"Q*GIB*46<;'p!)F@$\"?%*\\7\"fxfFu3`K#=aOFgu7$$!?_ $R%R&G&*pQ#H6AX*4)F@$\"?=B\"z.\"\\MEW(*Ru+WOFgu7$$!?`jDLzNsdK-(>9'H\") F@$\"?TzkzxvLyl&z=BJj$Fgu7$$!?e0h#>Z0B_\\H6r/;)F@$\"?rS0**>TT(*)z3#=U@ OFgu7$$!?3&4eI)y6ZO'H0AD>)F@$\"?b()*[*yz0kJD1'o(3OFgu7$$!?CA6mj76=wVTB ZE#)F@$\"?bEQs'y5#*)QsrN*\\f$Fgu7$$!?g)3!z)o?Hw(*H\"QOj#)F@$\"?)*H1)=D 6c=+d;#))zNFgu7$$!?;Aq(4S0:Wvge.[I)F@$\"?za'3V;QO.UX')pJc$Fgu7$$!?$pc' *p)\\S04\"\\NhLN)F@$\"??:kf1')eX*oz?iXa$Fgu7$$!?3;3IgTzL+,b)eJT)F@$\"? %31r0v=@dS*4A'Q_$Fgu7$$!?f9O(33E]'4)R')H+\\)F@$\"?\"*[i:`9(p]m4\")o9]$ Fgu-%%FONTG6$%*HELVETICAG\"\"*-%&COLORG6&%$RGBG$\"#l!\"#F]jpF(-%+AXESL ABELSG6$Q!6\"Fcjp-%*THICKNESSG6#\"\"#-%%VIEWG6$%(DEFAULTGF\\[q" 1 2 0 1 10 2 2 6 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "#-------------- -------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 71 "We recalculate the linking coefficients w ith the new nodes and weights." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "Digits := 95:\nc_2 := 39/88: c_4 := 101/152: c_9 := 1/5: c_10 : = 1/3:\nb_2 := -11/100: b_3 := -17/100: b_6 := -19/100: b_7 := -21/100 :\nb_9 := -221/958: b_10 := -27/100: b_11 := -29/100:\ncalc_RKcoeffs_w ith_params():" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "Digits := 75:\nA := Matrix([seq([seq(a[i,j],j=1 ..i-1),seq(0,j=i..25)],i=1..25)]):\nB := Matrix([[seq(b[i],i=1..25)]]) :\nC := LinearAlgebra[DiagonalMatrix]([0,seq(c[i],i=2..25)]):\nA_ := s ubs(e23,A): B_ := subs(e23,B): C_ := subs(e23,C):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "sm := 0: \nfor ct from 1 to nops(errterms_12) do\n dd := convert(errterms_12[ ct],'Matrix_form',A_,B_,C_);\n dd := modz(expand(eval(dd)));\n sm \+ := sm+errfact12[ct]*dd^2;\nend do:\nsm := modz(expand(sm)):" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "sm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6084 "sm := -.64961936 9597915212766055114719494668140782015315073841840440211785452633350e-8 *a[20,15]*a[20,14]*a[12,10]+.14128237099339864060626417085930706438426 4813909701140576237853500332911791e-6*a[20,15]*a[19,14]*a[12,11]+.3801 9296369464906345943265199111256435805184649781273693602731263645923459 5e-6*a[20,15]*a[20,14]*a[12,11]-.2414031124830155285809195193561790260 50714185386771764654259333657736053932e-8*a[20,15]*a[19,14]*a[12,10]+. 4114484329220356835792270771958324053316451079058106483673536819826263 15608e-7*a[20,14]*a[12,10]^2*a[19,14]-.3039758972298967869011448139798 02882844703153177493369857669813235645552162e-2*a[20,15]*a[12,11]^2*a[ 20,14]-.26699142357508376768397552461316656924937531060370777662624183 7422945626616e-6*a[20,15]*a[12,10]^2*a[19,14]-.32295379424973871249298 6281643719780665188991850644827755018509897324065458e-3*a[20,15]^2*a[1 2,10]*a[12,11]+.174076902450028942064969290955324645346093619303955079 866779280150718635448e-3*a[19,14]*a[12,11]^2*a[20,14]-.941869165165630 166648799993684662571494506336326579219765960272056010430837e-6*a[19,1 4]^2*a[12,11]*a[12,10]-.1129595747068528283495254800607257922588734237 02350666318677146677744253369e-2*a[19,14]*a[12,11]^2*a[20,15]-.6273810 53800845090560707028908123256761899667857205853906377202699296938860e- 5*a[20,14]^2*a[12,10]*a[12,11]+.11741441266399340443744403991478746011 1755099902642742066326960682839247324e-5*a[12,11]*a[20,14]*a[12,10]+.4 8105658266876034313361722010502985807256847976870960172024095318513272 5460e-6*a[12,11]*a[19,14]*a[12,10]-.6060113158532467419192739143835767 71399700847844064745281414628264481854659e-5*a[12,11]*a[20,15]*a[12,10 ]-.5100338403392135321186400832704033822469242649381641773444513206681 55993564e-11*a[20,15]*a[19,14]-.90358622274979758861741615056921685488 5676793242039208659563959429185802585e-13*a[20,14]+.212713159421648142 093040287710448414588249079972968234750445941434302834020e-11*a[20,15] -.40013216660730077924970045456597286087640261571959798738922317601934 6003894e-13*a[19,14]-.568092048664447026158859601789240419587154091844 371917360725951901896598433e-9*a[12,11]+.93358711773965720782842309663 2138855329827997078472216752458798238969325286e-11*a[12,10]+.329118350 407135277973430112917701334915430608869092879139514646340818878087e-12 *a[20,14]^2+.198109730511861868308391612108974623754708296524653145125 049112820748522657e-9*a[20,15]^2+.494095931110249102209220169533745433 002457944262346105301360839119406372138e-13*a[19,14]^2+.33887573723174 7224485269086346414125256159026850005200237909045365576321689e-4*a[19, 14]^2*a[12,11]^2+.5335259624434696874981223736710110743347596850675737 28341904302851737142282e-7*a[20,14]^2*a[12,10]^2-.40909137230591196177 0562207748072128067275708190549346566148475498948819060e-8*a[19,14]*a[ 12,10]^2+.515353099338049292600314590445019246783019057145570318606706 544129933139733e-7*a[20,15]*a[12,10]^2-.998494250691499204297475631181 609134259626108162725902103065770631137585111e-8*a[20,14]*a[12,10]^2+. 6552274601405560410648803358078306803650541335250233252069431407098795 57991e-9*a[12,10]^2-.4224461010584881046560336391755505432047769850739 21238731618648144286235029e-4*a[20,14]*a[12,11]^2+.2180372168722995336 62086608532508380313805731081056429856269555740985449290e-3*a[20,15]*a [12,11]^2-.17307966980041638952801771803834788150987851086940327914914 4772720777279570e-4*a[19,14]*a[12,11]^2-.77049164119316762388982309033 2421238127395390260251202175955313347813183780e-7*a[12,11]*a[12,10]+.2 7721570319622710456580628532688157707255058143353743770076802258074274 4444e-5*a[12,11]^2+.22572584918904828908105989858150632734479230744535 7315047897723755705349760e-3*a[20,14]^2*a[12,11]^2+.800967210910287865 802084299286338526770504416572305657507651271927677525522e-8*a[19,14]^ 2*a[12,10]^2+.11619576175309305334866374206548099274785512054073577084 8396612072992050945e-1*a[20,15]^2*a[12,11]^2+.274640480228078632753728 953533176234599691291196051681908919620159380719012e-5*a[20,15]^2*a[12 ,10]^2+.16501863360598116763944120563999256878319810923965986290296600 5871080791805e-9*a[20,14]^2*a[12,10]+.25381188368176047296176829602950 0391868417043103721006856188207062597130345e-12*a[20,14]*a[19,14]-.866 8497333114188537957637177284721317458486541289950308188305792491910354 82e-10*a[20,14]*a[12,10]-.13725086575181377038853986472947725756432173 9618200620637468500034511123194e-10*a[20,15]*a[20,14]+.215744687060198 752326868547699487583212486612653378714939932096516349775591e-8*a[19,1 4]*a[12,11]+.127260270314044161884445167702744695210391449707268672858 423117535169829553e-9*a[20,14]*a[12,10]*a[19,14]-.69722835992316674823 5076727014078329968765050376892466592251610632025045442e-8*a[19,14]*a[ 12,11]*a[20,14]-.24412019529803358490523152163148409433044801282135282 7653628785175077592438e-5*a[20,15]^2*a[12,11]+.75455906757420539086799 1116690058544809196935579422460536996517903597718913e-9*a[20,15]*a[12, 10]-.46465577713394957727177613548841775431029562659474246491974697705 8210184665e-7*a[20,15]*a[12,11]+.2477377372644993385219569582372836381 48812656094720922088553863608702675448e-10*a[19,14]^2*a[12,10]-.904097 335185079582121129142079424239555929076183372049711146694883956677423e -8*a[20,14]^2*a[12,11]-.1357295374414595975571694038816866944851085388 34073981717984519476962833461e-8*a[19,14]^2*a[12,11]-.3550653918923897 30876997545639285812847779081130932308282367213643871731429e-10*a[19,1 4]*a[12,10]-.718477895694484230824815816478385767344748887694780900481 113793640619156086e-6*a[20,14]*a[12,10]^2*a[20,15]+.528932970358941897 693069546863716290573396829998274664967142600387906079595e-8*a[20,14]* a[12,11]+.403636840256386823821772577228122427676442822233366368609664 506827842760441e-13+.4116923622429044042039697119684615477480922464443 29712318028398723178002928e-7*a[20,15]^2*a[12,10]+.3139591556354149470 38487609500894057828342527623486162399284250463368358378e-4*a[19,14]*a [12,11]*a[20,15]*a[12,10]+.8448687619043091527220823623454704795636160 89770946725095840961927728157496e-4*a[20,14]*a[12,10]*a[20,15]*a[12,11 ]-.4838282850362395171019725088680878007562898893812565368612478192671 11166833e-5*a[20,14]*a[12,10]*a[19,14]*a[12,11]:" }}}{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 61 "sm L := expand(subs(e23,add(add(a[i,j]^2,j=1..i-1),i=2..25))):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "smL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1335 "smL := -21906006 0.627322774625516087304177891607696405491932284256405059907969304070*a [20,15]*a[19,14]-1223973.492747013292331264298586003962299731974038100 03975627342958540778689*a[20,14]+7522848.87859251405392465293774614064 577434028588720160005475490582028242669*a[20,15]-403598.64327730263902 7344383375092057289309467739958679700664246666965733634*a[19,14]+60.67 8686638275749584142881521036012421202071731996243985604869241892651809 2*a[12,11]-.7620734896360026227362765430776701011949657048216672284104 45141487134611645*a[12,10]+39139091.9493762045438702712852715619231770 669373268138240981183080195146460*a[20,14]^2+3235408190.47088911535921 888906469309858033642765492946972902704278467545504*a[20,15]^2+4223977 .96425497314495899635443219364227404498237858553875919241966623487908* a[19,14]^2+272.7431956219662438232380307315175223315853971507317575594 19428299068108004*a[12,10]^2-38855.44737361499299369084210175370516084 00481690293184988255595614924314465*a[12,11]*a[12,10]+1423814.87347729 562434461867733850533286809683678127069433629111050071715174*a[12,11]^ 2+25713773.46608603458442515984128409116641105182842698310634034634292 14367664*a[20,14]*a[19,14]-667056446.356489559914100257025577311120081 405552838737884213418042492123025*a[20,15]*a[20,14]+19941.609788434914 4981122291141357122777675803753885981013238252718758274790:" }}}{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 25 "smP := sm+smL*3*10^(-20):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 82 "Again we try to minimize the weighted sum with respect to the pair of parameters " }{XPPEDIT 18 0 "a[12,10];" "6#&%\"aG6$\"#7\"#5" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[12, 11];" "6# &%\"aG6$\"#7\"#6" }{TEXT -1 32 " and the triple of parameters " } {XPPEDIT 18 0 "a[19,14];" "6#&%\"aG6$\"#>\"#9" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[20, 14];" "6#&%\"aG6$\"#?\"#9" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "a[20,15];" "6#&%\"aG6$\"#?\"#:" }{TEXT -1 39 " alternat ely using the Maple procedure " }{TEXT 0 8 "minimize" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 106 "This process initially minimzes both q uantities. Eventually the balance is lost, so we stop at this stage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 833 "Digits := 50:\neA := \{a[19,14]=0.,a[20,14]=0.,a[20,15]=0.\}:\n for ct to 1000 do\n minimize(eval(smP,eA),a[12,10]=-.5e-1..0,a[12,11 ]=0..0.2e-3,'location'):\n eB,ss := op(1,op(op(2,[%]))),sqrt(op(2,op (op(2,[%])))):\n minimize(eval(smP,eB),a[19,14]=-1.5..1.5,a[20,14]=- 0.5..0.5,a[20,15]=-.2..0.2,'location'):\n eA,ss := op(1,op(op(2,[%]) )),sqrt(op(2,op(op(2,[%])))):\n if `mod`(ct,50)=0 then\n print( ct);\n print(a[12,10]=evalf[30](subs(eB,a[12,10])),a[12,11]=evalf [30](subs(eB,a[12,11])));\n print(a[19,14]=evalf[30](subs(eA,a[19 ,14])),a[20,14]=evalf[30](subs(eA,a[20,14])),\n a[20,15 ]=evalf[30](subs(eA,a[20,15])));\n eAB := `union`(eA,eB):\n \+ print(`principal error norm`=evalf[30](sqrt(eval(sm,eAB))));\n pr int(`norm of linking coefficients`=evalf[30](sqrt(eval(smL,eAB))));\n \+ end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"#7\"# 5$!?nf!#J/&F%6$F'\"#6$\"?r()fF=vSd$y7F'eZ5!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"#>\"#9$\"?!4_F2PQL'>7n`%oQ'!#I/&F %6$\"#?F($!?(e]&H%)f3)=q\"[Bcf6F+/&F%6$F/\"#:$\"?qjDPu,6a,5Y)\\g8\"!#J " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"?tb,B%> 8Qa;/42&eV!#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%=norm~of~linking~co efficientsG$\"?v(*\\!*)ePj(*=9sKM.#!#F" }}{PARA 261 "" 0 "" {TEXT -1 3 " : " }}{PARA 260 "" 0 "" {TEXT -1 3 " : " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"#7\"#5$!?2x.o%\\G4)Qs.VP;K!#J/&F%6$F'\"#6$\" ?32;5G5Jjpnh._==!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"aG6$\"#>\" #9$\"??wyxgc#*HX\"=U$pil!#I/&F%6$\"#?F($!?E\\$y%pHC\"o'e_HC!)**!#J/&F% 6$F/\"#:$\"?2;CgQqrZ#Q38G[D\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5 principal~error~normG$\"?-&yz#fb#*f@R%z;98%!#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%=norm~of~linking~coefficientsG$\"?/86k[&*ziy'*G=FL>!# F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"aG6$\"#7\"#5$!?#zG\"#9$\"?\"4U.$)y8uHC(Q3qyl!#I/&F%6$\"#?F($ !?^(**GJ)>u*o_\"R%p-$**!#J/&F%6$F/\"#:$\"?]*z()[:?UKl^@7$e7F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%5principal~error~normG$\"?/:B&)[c(\\ ncT\\/<8%!#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%=norm~of~linking~coe fficientsG$\"?j7]VgbQdav4K6I>!#F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "[a[12,10]=-.321637430372388 092849468037707e-1,a[12,11]=.181852036167696331102810160708e-3,\n a[19 ,14]=.656269342181452992566077787620,a[20,14]=-.9980242952586681242969 47834926e-1,\n a[20,15]=.125482813083824771703860241607e-1]:\nconvert( %,rational,5);\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/&% \"aG6$\"#7\"#5#!#6\"$U$/&F&6$F(\"#6#\"\"\"\"%*\\&/&F&6$\"#>\"#9#\"#@\" #K/&F&6$\"#?F8#!#]\"$,&/&F&6$F?\"#:#\"#8\"%O5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/&%\"aG6$\"#7\"#5$!+pUP;K!#6/&F&6$F(\"#6$\"+dC^==!#8/ &F&6$\"#>\"#9$\"+++]il!#5/&F&6$\"#?F8$!+?*R+)**F,/&F&6$F?\"#:$\"+bi#[D \"F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 314 "Digits := 30:\nc_2 := 39/88: c_4 := 101/152: c_9 := \+ 1/5: c_10 := 1/3:\nb_2 := -11/100: b_3 := -17/100: b_6 := -19/100: b_7 := -21/100:\nb_9 := -221/958: b_10 := -27/100: b_11 := -29/100:\na12_ 10 := -11/342: a12_11 := 1/5499:\na19_14 := 21/32: a20_14 := -50/501: \+ a20_15 := 13/1036:\ncalc_RKcoeffs('give_characteristics'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeff sGF&$\"+]%)><)*!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% " 0 "" {MPLTEXT 1 0 312 "Dig its := 30:\nc_2 := 39/88: c_4 := 101/152: c_9 := 1/5: c_10 := 1/3:\nb_ 2 := -11/100: b_3 := -17/100: b_6 := -19/100: b_7 := -21/100:\nb_9 := \+ -13/56: b_10 := -27/100: b_11 := -29/100:\na12_10 := -11/342: a12_11 : = 1/5499:\na19_14 := 21/32: a20_14 := -50/501: a20_15 := 13/1036:\ncal c_RKcoeffs('give_characteristics'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+1n'Qt*!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%AA%!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We inspect the stability region and the imaginary ax is inclusion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 472 "Digits := \+ 30:\nA := subs(e23,Matrix([seq([seq(a[i,j],j=1..i-1),seq(0,j=i..25)],i =1..25)])):\nB := subs(e23,Matrix([[seq(b[i],i=1..25)]])):\nid := Matr ix([seq([1],i=1..25)]):\nadd(z^j/j!,j=0..12)+add((B.A^(j-1).id)[1,1]*z ^j,j=13..25):\nmap(convert,%,rational,24): R := unapply(%,z):\npts := \+ []: z0 := 0:\nfor ct from 0 to 220 do\n zz := newton(R(z)=exp(ct*Pi/ 20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend \+ do:\nplot(pts,color=COLOR(RGB,0,.35,.35),style=point);" }}{PARA 13 "" 1 "" {GLPLOT2D 383 404 404 {PLOTDATA 2 "6'-%'CURVESG6#7ix7$$\"\"!F)F(7 $$\"?@])o+EGPO,7V(Goz!#]$\"?[9_05.i'*[zEjzq:!#I7$$\"?&>t@-c\"3*o.6ntVH \"!#X$\"?!))oVF$R*>!)*e`EfTJF07$$\"?LHG*pxi?$pH%3utr$!#V$\"?B_U:(QvpO' Q!)*)Q7ZF07$$\"?LuZK12!)*fc>&>eI?!#T$\"?-1!)G'Qg\"*e[sI&=$G'F07$$\"?%* 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