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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "R:=solve(karyht,r);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG6$,(*&\"\"#!\"\"%\"aG\"\"\"F+*(F (F)F*F+%\"bGF+F+*&F(F),**$)F*F(F+F+*(F(F+F1F+F-F+F+*&F1F+)F-F(F+F+*(\" \"%F+F*F+F-F+F)#F+F(F+,(*&F(F)F*F+F+*(F(F)F*F+F-F+F+*&F(F)F/F7F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "R:=subs(a=0.9,b=5/9,[R]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7$^$$\"+++++q!#5$\"+++++5F)^$F' $!+++++5F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "Lis\344ys/muutos" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "yri:=k->a^k*cos(b*k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yriGf*6#%\"kG6\"6$%)operatorG%&arrowGF(*&)%\"aG9$\" \"\"-%$cosG6#*&%\"bGF0F/F0F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(y=yri,lhs(EHY));eval(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$yriG6#,&%\"kG\"\"\"\"\"#F)F)*(%\"aGF),&F)F)%\"bGF) F)-F%6#,&F(F)F)F)F)!\"\"*(F,F)F.F)-F%6#F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)%\"aG,&%\"kG\"\"\"\"\"#F)F)-%$cosG6#*&%\"bGF)F'F)F )F)**F&F),&F)F)F/F)F))F&,&F(F)F)F)F)-F,6#*&F/F)F3F)F)!\"\"**F&F)F/F))F &F(F)-F,6#*&F/F)F(F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "karakt:=simplify(%/a^k,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%'karaktG*&%\"aG\"\"\",2**\"\"#F'F&F'-%$cosG6#*&%\"bGF'%\"kGF'F')-F,6 #F/F*F'F'*&F&F'F+F'!\"\"*,F*F'F&F'-%$sinGF-F'-F8F3F'F2F'F5*(F&F'F+F'F2 F'F5*(F&F'F7F'F9F'F'**F&F'F/F'F+F'F2F'F5**F&F'F/F'F7F'F9F'F'*&F/F'F+F' F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "karakt:=collect(%,c os(k*b));collect(%,sin(k*b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ka raktG,&*(%\"aG\"\"\",,*(\"\"#F(F'F()-%$cosG6#%\"bGF+F(F(F'!\"\"*&F'F(F -F(F1*(F'F(F0F(F-F(F1F0F(F(-F.6#*&F0F(%\"kGF(F(F(*&F'F(,(*(F'F(-%$sinG F5F(-F " 0 "" {MPLTEXT 1 0 56 "\{coeff(karak t,cos(b*k),1)=0,coeff(karakt,sin(b*k),1)=0\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/*&%\"aG\"\"\",,*(\"\"#F'F&F')-%$cosG6#%\"bGF*F'F'F&! \"\"*&F&F'F,F'F0*(F&F'F/F'F,F'F0F/F'F'\"\"!/*&F&F',(*&F&F'-%$sinGF.F'F '**F*F'F&F'F8F'F,F'F0*(F&F'F/F'F8F'F'F'F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Eip\344 tuo aivan yksinkertainen yht\344l\366 ole!" }}}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 2 "2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "with(LinearAlgebra): with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:=<<0,0,0,9>|<1,0,0,-6>|<0,1,0,-8>|<0,0, 1,6>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "X[0]:=<1,0,0,0>; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "n:=20: for i to n do X[ i]:=A.X[i-1] end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq (X[i],i=0..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "b)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "y[0]:=1: y[1]:=0: y[2]:=0: y[3]:=0: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "n:=20: for k from 0 to \+ n do \n y[k+4]:=6*y[k+3]-8*y[k+2]-6*y[k+1]+9*y[k]: end do:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "seq(y[k],k=0..n);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "Eigenvectors(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 2 "3." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "dyht:=y(k+4)-6*y(k+3)+8*y(k+2)+6*y(k+1)-9*y( k)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "karpol:=r^4-6*r^3+ 8*r^2+6*r-9;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "factor(karp ol);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "yri:=k->k*3^k;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "yriy:=subs(y=yri,dyht);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "eval(yriy);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "expand(lhs(%));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Saadaan siis LRT ratlkaisut" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "y1:=k->1: y2:=k->(-1)^k: y3:=k->3^k: y4:=k->k *3^k:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Miten perustellaan LRT" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "Casorati:=<|||>;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "linalg[det](Casorati);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "5" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }