{VERSION 3 0 "SGI MIPS UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "with(MS):with(linalg ):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Consider the poly-exponenti al function " }{XPPEDIT 18 0 "s(x) = exp(2*x)*x^3+exp(3*x);" "6#/-%\"s G6#%\"xG,&*&-%\$expG6#*&\"\"#\"\"\"F'F/F/*\$F'\"\"\$F/F/-F+6#*&\"\"\$F/F'F /F/" }{TEXT -1 122 ". Notice that deg^P(s(x)) = 5 and deg^d(s(x)) = 3 . Hence to multisection by 7 at 4, we need only look at the values fo r " }{XPPEDIT 18 0 "b[4];" "6#&%\"bG6#\"\"%" }{TEXT -1 1 "," } {XPPEDIT 18 0 "b[11];" "6#&%\"bG6#\"#6" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[18];" "6#&%\"bG6#\"#=" }{TEXT -1 5 "..., " }{XPPEDIT 18 0 "b[74]; " "6#&%\"bG6#\"#u" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "s := exp(2*x)*x^3 + exp(3*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&*&-%\$expG6#,\$%\"xG\"\"#\"\"\")F+\"\"\$\"\"\"F--F (6#,\$F+F/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "`pe/metric/P `(s,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "`pe/metric/d`(s,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "for i from 4 to 74 by 7 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " b[i] := eval(diff(s,x\$i),x=0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"%\"\$H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#6\"'(eI%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#=\"*&%\"bG 6#\"#D\"-VY+q^!*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#K\"1hH >nu**o=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#R\"4\"4=j!oJBj0 %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#Y\"7p0!Q'\\!yERP'))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#`\";nclogKn'>RS\$Q>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#g\">TA2A!\\#>m%\\'y=\"RU" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#n\"AnuRY\$)4#>'[X+Xo%4F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"#u\"E\"GU3([(p=jVi?G#of bF?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Set the matrix " } {XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 10 " equal to " }{XPPEDIT 18 0 "matrix([[b[11], b[18], b[25], b[32], b[39]], [b[18], b[25], b[32], b[ 39], b[46]], [b[25], b[32], b[39], b[46], b[53]], [b[32], b[39], b[46] , b[53], b[60]], [b[39], b[46], b[53], b[60], b[67]]]);" "6#-%'matrixG 6#7'7'&%\"bG6#\"#6&F)6#\"#=&F)6#\"#D&F)6#\"#K&F)6#\"#R7'&F)6#\"#=&F)6# \"#D&F)6#\"#K&F)6#\"#R&F)6#\"#Y7'&F)6#\"#D&F)6#\"#K&F)6#\"#R&F)6#\"#Y& F)6#\"#`7'&F)6#\"#K&F)6#\"#R&F)6#\"#Y&F)6#\"#`&F)6#\"#g7'&F)6#\"#R&F)6 #\"#Y&F)6#\"#`&F)6#\"#g&F)6#\"#n" }{TEXT -1 1 "." }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "C := matrix (5,5,[seq(seq(b[4+7*(i+j-1)],i=1..5),j=1..5)]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Set the vector " }{XPPEDIT 18 0 "v;" "6#%\"vG" } {TEXT -1 10 " equal to " }{XPPEDIT 18 0 "vector([b[44], b[51], b[58], \+ b[65], b[72]]);" "6#-%'vectorG6#7'&%\"bG6#\"#W&F(6#\"#^&F(6#\"#e&F(6# \"#l&F(6#\"#s" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "v := vector(5, [seq(b[4+7*i+35],i=1..5)]):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 10 "Now solve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "linsolve(C,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6# 7'\"-sAMoqe!,_6K9'=\"*c%zLA!([!=7\"%*p#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "This gives a linear recurrence relation of " }{XPPEDIT 18 0 "d[i] = d[i-35]*587068342272-18614321152*b[i-28]+223379456*b[i-21 ]-1218048*b[i-14]+2699*b[i-7];" "6#/&%\"dG6#%\"iG,,*&&F%6#,&F'\"\"\"\" #N!\"\"F-\"-sAMoqeF-F-*&\",_6K9'=F-&%\"bG6#,&F'F-\"#GF/F-F/*&\"*c%zLAF -&F46#,&F'F-\"#@F/F-F-*&\"([!=7F-&F46#,&F'F-\"#9F/F-F/*&\"%*p#F-&F46#, &F'F-\"\"(F/F-F-" }{TEXT -1 75 ". This could have also been done by t he Maple function `pe/ms/linalg/sym`." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "`pe/ms/linalg/sym`(s,f, x,7,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&/-%\"fG6#%\"xG,,-F%6#,&F'\"\"\"!#NF,\"-sAMoqe-F%6#,&F'F ,!#GF,!,_6K9'=-F%6#,&F'F,!#@F,\"*c%zLA-F%6#,&F'F,!#9F,!([!=7-F%6#,&F'F ,!\"(F,\"%*p#F%F'7E/-F%6#\"\"!FG/-F%6#F,FG/-F%6#\"\"#\"\"*/-F%6#\"\"\$F G/-F%6#\"\"%FG/-F%6#\"\"&FG/-F%6#\"\"'FG/-F%6#\"\"(FG/-F%6#\"\")FG/-F% 6#FO\"&R>&/-F%6#\"#5FG/-F%6#\"#6FG/-F%6#\"#7FG/-F%6#\"#8FG/-F%6#\"#9FG /-F%6#\"#:FG/-F%6#\"#;\")T=dq/-F%6#\"#FG/-F%6 #\"#?FG/-F%6#\"#@FG/-F%6#\"#AFG/-F%6#\"#B\"-.uM&G0\"/-F%6#\"#CFG/-F%6# \"#DFG/-F%6#\"#EFG/-F%6#\"#FFG/-F%6#\"#GFG/-F%6#\"#HFG/-F%6#\"#I\"0H([ fng\"4#/-F%6#\"#JFG/-F%6#\"#KFG/-F%6#\"#LFG/-F%6#\"#MFG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Which is the same result." }}}}{MARK "1 0 2" 38 }{VIEWOPTS 1 1 0 1 1 1803 }