{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 9 "with(MS):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 303 "This example will demonstrate how the me thods of multisectioning can be applied to functions with symbolic par ameters for parameters of the exponentials of rational poly-exponentia l functions. Define the \{\\em Bernoulli polynomials\}'' to be the \+ coefficients of the exponential generating function of " }{XPPEDIT 18 0 "x*exp(t*x)/(exp(x)-1);" "6#*(%\"xG\"\"\"-%$expG6#*&%\"tGF%F$F%F% ,&-F'6#F$F%\"\"\"!\"\"F/" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "x;" "6#% \"xG" }{TEXT -1 0 "" }{TEXT -1 163 ". The denominator and numerator o f this function have very complicated lacunary recurrence relations, e ven when multisectioning by a small value such as 3 (at 0)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "top := x* exp(t*x):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "bot := exp(x)-1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "botlrr := bottom/ms(bot, f, x, 3) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'botlrrG6&/-%\"fG6#%\"xG-F(6#,& F*\"\"\"!\"'F.F(F*7(/-F(6#\"\"!F4/-F(6#F.F4/-F(6#\"\"#F4/-F(6#\"\"$\" \"'/-F(6#\"\"%F4/-F(6#\"\"&F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "toplrr := collect([top/ms/linalg/sym(top,bot, f, x, 3, 0)],f); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'toplrrG7&/-%\"fG6#%\"xG,2*&,H*$)%\"tG\"#9\"\"\"!%_r*$)F0\"#;F2F3*$)F0\"#6F2\"%K>*$)F0\"#=F2!%*f$*$)F0 \"#?F2!$S)*$)F0\"\"'F2!\"\"*$)F0\"#F2F:*$)F0\"#BF2FY*$)F0\"#5F2FB*$)F0\"\"*F2FR*$)F0FYF2 F>\"\"\"-F(6#,&F*F\\p!#CF\\pF\\pF\\p*(,HF;\"\"%FG!#UF4\"$;#FK!$A(F.\"% kFV\"%?zF[o!%SfFq!%WbFgo\"%?8/-F(6# FUFhw/-F(6#F1Fhw/-F(6#FM,4\"#IF\\pF[r!%l8Fdq\"&I+$F7!&!Q;FC\"&!4!*F[o! &X]%FdoFh[lFq!&!4!*Fjo\"%IF/-F(6#F6Fhw/-F(6#FIFhw/-F(6#F=,8F0FgtF.!&? n$Fgq\"&![CF[r!%WtF7!'oFAFK\"%'*[FS\"&!o&)FV\"'G,qF[o!(SFJ\"Fq!'%Q6\" Fgo\"'g^()/-F(6#FoFhw/-F(6#FAFhw/-F(6#FQ,<\"#UF\\pF.!'gR\")F[r!%!*RFd q\"'!\\.#F4F_^lF7!)![\"e5F;\"%!)zFC\"(?zi\"FG!&?=(F[o!(qk#Fdo\"(_(fxF q!'_n(*Fjo\"(S2H&/-F(6#FfnFhw/-F(6#FcoFhw" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Now, if " }{XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 78 " this will reduce to the situation of looking at the normal Ber noulli numbers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(t=0 ,[toplrr]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7#7&/-%\"fG6#%\"xG,.-F' 6#,&F)\"\"\"!#=F.!\"\"-F'6#,&F)F.!#:F.\"\"#-F'6#,&F)F.!#7F.F.-F'6#,&F) F.!\"*F.!\"%-F'6#,&F)F.!\"'F.F.-F'6#,&F)F.!\"$F.F5F'F)7:/-F'6#\"\"!FK/ -F'6#F.FK/-F'6#F5FK/-F'6#\"\"\$\"\"'/-F'6#\"\"%FK/-F'6#\"\"&FK/-F'6#FVF K/-F'6#\"\"(FK/-F'6#\"\")FK/-F'6#\"\"*\"#=/-F'6#\"#5FK/-F'6#\"#6FK/-F' 6#\"#7FK/-F'6#\"#8FK/-F'6#\"#9FK/-F'6#\"#:\"#I/-F'6#\"#;FK/-F'6#\"#FK/-F'6#\"#?FK/-F'6#\"#@\"#U/-F'6#\"#AFK/-F'6#\"# BFK" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 179 "This example is interesting because it demonstrat es how large and complicated the results get when done symbolically, b ut still shows that feasibility of doing these calculations." }}}} {MARK "6 0 2" 78 }{VIEWOPTS 1 1 0 1 1 1803 }