{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 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 }{PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 257 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 54 "Math 262a, Fall 1999, G lenn Tesler\nHomework 9\n12/10/99" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 9 "Problem 2" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Here's how I generated the sequence:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "rec := proc(n)\n option remember;\n (3*n^2+2*n-1)*rec(n-2)+(4*n-5)*rec(n-1);\nend:\nrec(0 ):=1: rec(1):=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "terms : = ['rec(k)'$k=0..16];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&termsG73\" \"\"\"\"#\"#@\"$6#\"%wM\"&k)p\"(g5u\"\")?EA^\"+g,TV<\",ghfjt'\".SmqtQ \"H\"0+W[&p3&R\"\"1+/NUkoCt\"3+'p\")y)))y%=%\"5+c9U-\\0r%e#\"7+kS*=q'R **>;<\"9+?J\\YCFPj2>7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Normally you would find the start of the sequence from some empirical or brute force means." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 414 "guessr ec := proc(terms,I,J)\n local i,j,k,template,grec,eqs,sols,N,frec; \n\n k := nops(terms)-1;\n template := sum(sum(a[i,j]*n^i*f(n+ j),i=0..I),j=0..J);\n\n frec := unapply(terms[n+1],n); # make this a function\n grec := subs(f=frec,template);\n \n eqs := \{ s eq(eval(subs(n=N,grec)),N=0..k-J) \};\n sols := solve(eqs);\n \+ if sols=NULL then RETURN(FAIL) fi;\n\n subs(sols,template) = 0;\ne nd: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "guessrec(terms,0,0) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "guessrec(terms,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " guessrec(terms,2,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.*&&%\"aG6$ \"\"!\"\"#\"\"\"-%\"fG6#%\"nGF+!#:*(F&F+F/F+F,F+!#9*(F&F+F/F*F,F+!\"$* &F&F+-F-6#,&F/F+F+F+F+F4*(F&F+F/F+F6F+!\"%*&F&F+-F-6#,&F/F+F*F+F+F+F) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "simplify(\"/a[0,2]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.-%\"fG6#%\"nG!#:*&F(\"\"\"F%F+!#9* &F(\"\"#F%F+!\"$-F&6#,&F(F+F+F+F/*&F(F+F0F+!\"%-F&6#,&F(F+F.F+F+\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "collect(\",\{f(n),f(n+1 ),f(n+2)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%\"fG6#,&%\"nG\"\" \"\"\"#F*F**&,(F)!#9*$F)F+!\"$!#:F*F*-F&6#F)F*F**&,&F)!\"%F0F*F*-F&6#, &F)F*F*F*F*F*\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Using too f ew terms gives spurious solutions:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "guessrec(terms[1..4],2,2);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/,4*&,&&%\"aG6$\"\"!\"\"\"!\"#&F(6$F* \"\"#!#@F+-%\"fG6#%\"nGF+F+*(&F(6$F+F*F+F4F+F1F+F+*(,0F'#!# " 0 "" {MPLTEXT 1 0 26 "guessrec(terms[1..9],2,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,4*&,&&%\"aG6$\"\"!\"\"#!#:&F(6$F+F+#!1?*o,(**yFU\" .JoIl]$>\"\"\"-%\"fG6#%\"nGF2F2*(,&F'!#9F-#\"1k#=0z1,&fF1F2F6F2F3F2F2* (,&F'!\"$F-#\"2G'[/mFwzCF1F2F6F+F3F2F2*&,&F'F>F-#\"1gW3&)\\*Q6#F1F2-F4 6#,&F6F2F2F2F2F2*(,&F'!\"%F-#\"0k#o$*yS$)[F1F2F6F2FEF2F2*(F-F2F6F+FEF2 #\"1j+5Z\"er>%F1*&F'F2-F46#,&F6F2F+F2F2F2*(F-F2F6F2FQF2#!0#\\As(e-I*F1 *(F-F2F6F+FQF2F2F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Using too \+ high order/degree gives multiples of the desired recurrence with lots \+ of a[i,j]'s still free:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "bigrec : = guessrec(terms,3,3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'bigrecG/, <*&,&&%\"aG6$\"\"!\"\"#!#:&F*6$F,\"\"$!$0\"\"\"\"-%\"fG6#%\"nGF3F3*(,* &F*6$F3F1\"$g\"F)\"\"'F/\"$U\"&F*6$F3F3\"\"&F3F7F3F4F3F3*(,*F)#\"#ZF1F /\"$.#F:#\"$[%F1F?#\"#9F1F3F7F-F4F3F3*(,*F:\"#KF)\"\"%F/\"#[F?F3F3F7F1 F4F3F3*&,&F)!\"$F/!#`F3-F56#,&F7F3F3F3F3F3*(F?F3F7F3FTF3F3*(,*F:#\"#oF 1F)#\"#;F1F/\"#hF?#FNF1F3F7F-FTF3F3*(F:F3F7F1FTF3FR*&F)F3-F56#,&F7F3F- F3F3F3*(,*F:#FSF1F)#!\"%F1F/!#?F?#!\"\"F1F3F7F3F\\oF3F3*(F:F3F7F-F\\oF 3Fco*&F/F3-F56#,&F7F3F1F3F3F3*(F:F3F7F3FioF3F3F," }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 336 "The distinction between the last two things is th at in bigrec, the number of equations k-J+1 greatly exceeds the number of unknowns, (I+1)(J+1), yet is still solvable; but with too few term s, the number of equations is smaller than the number of unknowns, and of course there's a solution, but it's meaningless due to insufficien t data." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "In \+ bigrec, the coefficient of each a[i,j] is a recurrence:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "as := indets(bigrec,indexed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#asG<&&%\"aG6$\"\"!\"\"#&F'6$F)\"\"$&F'6$\"\"\"F -&F'6$F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "map(collect,[ coeffs(expand(lhs(bigrec)),as)],\{f(n),f(n+1),f(n+2),f(n+3)\},factor); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&,(*&,&\"\"\"F'%\"nG#!\"%\"\"$F'- %\"fG6#,&F(F'\"\"#F'F'F'**,&F(F+\"\"&F'F',&F(F'F+F'F',&F(\"\"%!\"$F'F' -F-6#F(F'#F'F+*(F5F',&F(F6F+F'F'-F-6#,&F(F'F'F'F'F:,**&F(F'F,F'!#?**F2 F',&F(\"#;!\"(F'F'F4F'F8F'F'*&,&!#`F'*$F(F0\"#hF'F=F'F'-F-6#F4F',**(F( F',&F(\"#7\"#`F'F'F,F'#!\"\"F+**F(F'F4F'F2F'F8F'#\"#KF+*(F(F0,&!#oF'F( \"\"*F'F=F'FS*&F(F'FLF'F',(FAFSFUF:*(F(F'F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 }