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{TEXT 223 19 "\334bung 06 - 17.05.05" }} {PARA 216 "" 0 "" }{SECT 1 {PARA 217 "" 0 "" {TEXT 224 10 "Funktionen" }}{PARA 216 "" 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 46 "Le ngth := proc(ll) return nops(ll); end proc:" }{MPLTEXT 1 225 49 "\nLa st := proc(ll) return op(-1, ll); end proc:" }{MPLTEXT 1 225 53 "\n Append := proc(ll, x) return [op(ll), x]; end proc:" }{MPLTEXT 1 225 53 "\nPrepend := proc(ll, x) return [x, op(ll)]; end proc:" }{MPLTEXT 1 225 51 "\nMember := proc(ll, p) return op(p, ll); end proc:" } {MPLTEXT 1 225 23 "\nTake := proc(ll, n)" }{MPLTEXT 1 225 51 "\n \+ if n > 0 then return [ op(1..n, ll) ]" }{MPLTEXT 1 225 54 " \n elif n < 0 then return [ op(n..-1, ll) ]" }{MPLTEXT 1 225 23 "\n end if; " }{MPLTEXT 1 225 21 "\n end proc:" }{MPLTEXT 1 225 20 "\nTotal := proc(ll)" }{MPLTEXT 1 225 22 "\n local i;" }{MPLTEXT 1 225 55 "\n return su m(Member(ll,i),i=1..Length(ll));" }{MPLTEXT 1 225 21 "\n end proc:" }{MPLTEXT 1 225 23 "\nCount := proc(ll, x)" }{MPLTEXT 1 225 24 "\n local i,r;" }{MPLTEXT 1 225 21 "\n r := 0;" }{MPLTEXT 1 225 48 "\n for i from 1 by 1 to Length(ll ) do" }{MPLTEXT 1 225 58 "\n if Member(ll,i) = x then r := r+1 end if;" }{MPLTEXT 1 225 21 "\n end do;" } {MPLTEXT 1 225 23 "\n return r;" }{MPLTEXT 1 225 22 "\n \+ end proc:" }{MPLTEXT 1 225 24 "\nTimes := proc(l1, l2)" } {MPLTEXT 1 225 25 "\n local i, r;" }{MPLTEXT 1 225 22 "\n \+ r := [];" }{MPLTEXT 1 225 45 "\n if Length(l1) = Length(l2) then" }{MPLTEXT 1 225 45 "\n for i from 1 \+ to Length(l1) do" }{MPLTEXT 1 225 62 "\n r := Append(r , Member(l1, i)*Member(l2, i));" }{MPLTEXT 1 225 24 "\n \+ end do: " }{MPLTEXT 1 225 21 "\n end if;" }{MPLTEXT 1 225 23 "\n return r;" }{MPLTEXT 1 225 21 "\n end pro c:" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 22 "with(plots, listplot):" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 20 "with(plots,display):" }}}{EXCHG {PARA 216 "> " 0 "" }}{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" }{SECT 0 {PARA 217 "" 0 "" {TEXT 224 9 "Aufgabe 1" }}{PARA 216 "" 0 "" }{SECT 0 {PARA 218 "" 0 " " {TEXT 226 3 "(a)" }}{PARA 216 "" 0 "" {TEXT 227 6 "DGL: " } {XPPEDIT 18 0 "y[t] = 1+(t-y)^2;" "6#/&%\"yG6#%\"tG,&\"\"\"F)*$,&F'F)F %!\"\"\"\"#F)" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 16 " , 2 <= t <= 3, " }{XPPEDIT 18 0 "y(2) \+ = 1;" "6#/-%\"yG6#\"\"#\"\"\"" }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 14 " L\366sung: " } {XPPEDIT 18 0 "y(t) = t-1/(t-1);" "6#/-%\"yG6#%\"tG,&F'\"\"\"*&F)F),&F 'F)F)!\"\"F,F," }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" }{PARA 216 "" 0 " " {TEXT 227 18 "Max Fehler < 0.02 " }}{PARA 216 "" 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 15 "# Exakte L\366sung" }{MPLTEXT 1 225 2 "\n#" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 49 "dsolve(\{diff (y(t),t)=1+(t-y(t))^2, y(2)=1\},y(t));" }}{PARA 219 "" 1 "" {XPPMATH 20 "6#/-I\"yG6\"6#I\"tGF&*&,(!\"\"\"\"\"*$F(\"\"#F,F(F+F,,&F(F,F+F,F+" }{TEXT 228 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 21 "ye \+ := t -> t-1/(t-1):" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 " > " 0 "" {MPLTEXT 1 225 25 "f := (t,y) -> 1+(t-y)^2: " }}}{EXCHG {PARA 220 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 7 "a : = 2:" }{MPLTEXT 1 225 8 "\nb := 3:" }{MPLTEXT 1 225 9 "\nn := 10:" } {MPLTEXT 1 225 14 "\nh := (b-a)/n:" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 14 "appr := [1.0]:" }{MPLTEXT 1 225 23 "\nfor i from 0 \+ to n-1 do" }{MPLTEXT 1 225 59 "\n appr := Append(appr, Last(appr)+h*f (a+i*h, Last(appr)));" }{MPLTEXT 1 225 8 "\nend do:" }{MPLTEXT 1 225 6 "\nappr;" }}}{EXCHG {PARA 220 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 38 "exct:=evalf( [seq(ye(a+i*h),i=0..n)]);" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 21 "pa := listplot(appr):" } {MPLTEXT 1 225 22 "\npe := listplot(exct):" }{MPLTEXT 1 225 18 "\ndisp lay([pa,pe]);" }}}{EXCHG {PARA 220 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 24 "l := map(abs,appr-exct);" }}}{EXCHG {PARA 216 " > " 0 "" {MPLTEXT 1 225 11 "max(op(l));" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "listplot(l);" }}}{EXCHG {PARA 221 "> " 0 "" }} {EXCHG {PARA 221 "> " 0 "" }}{EXCHG {PARA 221 "> " 0 "" }}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 229 4 "# 3)" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 229 57 "exakt:=dsolve(\{ diff(y(t),t) = 1+(y(t)/t), y(1 )=2\},y(t));" }}{PARA 219 "" 1 "" {XPPMATH 20 "6#>I&exaktG6\"/-I\"yGF% 6#I\"tGF%*&,&-I#lnG6$I*protectedGF0I(_syslibGF%F)\"\"\"\"\"#F2F2F*F2" }{TEXT 228 1 " " }}}{EXCHG {PARA 221 "> " 0 "" }}{EXCHG {PARA 221 "> " 0 "" }}{EXCHG {PARA 221 "> " 0 "" }}{EXCHG {PARA 221 "> " 0 "" }} {EXCHG {PARA 221 "> " 0 "" }}{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" } {SECT 0 {PARA 218 "" 0 "" {TEXT 226 3 "(b)" }}{PARA 216 "" 0 "" {TEXT 227 6 "DGL: " }{XPPEDIT 18 0 "y[t] = cos(2*t)+sin(3*t);" "6#/&%\"yG6# %\"tG,&-%$cosG6#*&\"\"#\"\"\"F'F.F.-%$sinG6#*&\"\"$F.F'F.F." }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 16 " , 0 <= t <= 1, " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"! \"\"\"" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 14 " L\366sung: " }{XPPEDIT 18 0 "y(t) = sin(2*t )/2-cos(3*t)/3+4/3;" "6#/-%\"yG6#%\"tG,(*&-%$sinG6#*&\"\"#\"\"\"F'F/F/ F.!\"\"F/*&-%$cosG6#*&\"\"$F/F'F/F/F6F0F0*&\"\"%F/F6F0F/" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" {TEXT 227 18 "Max Fehler < 0.031" }}{PARA 216 "" 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 37 "ye := t -> sin( 2*t)/2-cos(3*t)/3+4/3:" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 33 "f := (t,y) -> cos(2*t)+sin(3*t): " }} }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 7 "a := 0:" }{MPLTEXT 1 225 8 "\nb := 1:" }{MPLTEXT 1 225 9 "\nn := 20:" }{MPLTEXT 1 225 14 " \nh := (b-a)/n:" }}}{EXCHG {PARA 220 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 14 "appr := [1.0]:" }{MPLTEXT 1 225 23 "\nfor i f rom 0 to n-1 do" }{MPLTEXT 1 225 59 "\n appr := Append(appr, Last(app r)+h*f(a+i*h, Last(appr)));" }{MPLTEXT 1 225 8 "\nend do:" }{MPLTEXT 1 225 13 "\nevalf(appr);" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 38 "exct:=evalf( [seq(ye(a+i*h),i=0..n)]);" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 21 "pa := listplot(appr):" }{MPLTEXT 1 225 22 " \npe := listplot(exct):" }{MPLTEXT 1 225 18 "\ndisplay([pa,pe]);" }}} {EXCHG {PARA 220 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 31 "l := evalf(map(abs,appr-exct));" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 11 "max(op(l));" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "listplot(l);" }}}{EXCHG {PARA 216 "> " 0 "" }} {PARA 216 "" 0 "" }}{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" }{SECT 0 {PARA 217 "" 0 "" {TEXT 224 9 "Aufgabe 2" }}{PARA 216 "" 0 "" }{SECT 0 {PARA 218 "" 0 "" {TEXT 226 3 "(a)" }}{PARA 216 "" 0 "" {TEXT 227 6 "DGL: " }{XPPEDIT 18 0 "y[t] = 1+(t-y)^2;" "6#/&%\"yG6#%\"tG,&\"\"\"F )*$,&F'F)F%!\"\"\"\"#F)" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 16 " , 2 <= t <= 3, " }{XPPEDIT 18 0 "y(2) = 1;" "6#/-%\"yG6#\"\"#\"\"\"" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 14 " L\366sung: " }{XPPEDIT 18 0 "y(t) = t-1/(t-1);" "6#/-%\"yG6#%\"tG,&F'\"\"\"*&F)F), &F'F)F)!\"\"F,F," }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" }{PARA 216 "" 0 " " {XPPEDIT 18 0 "f(t,y) = 1+(t-y)^2;" "6#/-%\"fG6$%\"tG%\"yG,&\"\"\"F* *$,&F'F*F(!\"\"\"\"#F*" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 3 " = " }{XPPEDIT 18 0 "1+t^2-2*t*y+ y^2;" "6#,*\"\"\"F$*$%\"tG\"\"#F$*(F'F$F&F$%\"yGF$!\"\"*$F)F'F$" } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 6 " , " }}{PARA 216 "" 0 "" {XPPEDIT 18 0 "f[t](t,y) = 2* t-2*y-2*t*y[t]+2*y*y[t];" "6#/-&%\"fG6#%\"tG6$F(%\"yG,**&\"\"#\"\"\"F( F.F.*&F-F.F*F.!\"\"*(F-F.F(F.&F*F'F.F0*(F-F.F*F.F2F.F." }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 3 " = " }{XPPEDIT 18 0 "2*(t-y)+2*(y-t)*f(t,y);" "6#,&*&\"\"#\"\"\",&%\"tGF &%\"yG!\"\"F&F&*(F%F&,&F)F&F(F*F&-%\"fG6$F(F)F&F&" }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {XPPEDIT 18 0 "T[2](w[i],t[i]) = f(w[i],t[i])+h*f[t ](w[i],t[i])/2;" "6#/-&%\"TG6#\"\"#6$&%\"wG6#%\"iG&%\"tGF,,&-%\"fGF)\" \"\"*(%\"hGF3-&F26#F/F)F3F(!\"\"F3" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {TEXT 227 20 "Max Fehler < 0.0018" }}{PARA 216 "" 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 21 "ye := t -> t-1/(t-1):" }}} {EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 25 "f := (t,y) -> 1+(t-y)^2:" }{MPLTEXT 1 225 39 "\nf1 := (t,y) - > 2*(t-y)+2*(y-t)*f(t,y):" }{MPLTEXT 1 225 38 "\ntt := (t,y) -> f(t,y) +(h/2)*f1(t,y): " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 7 "a := 2:" }{MPLTEXT 1 225 8 "\nb := 3:" }{MPLTEXT 1 225 9 "\nn := 10:" } {MPLTEXT 1 225 14 "\nh := (b-a)/n:" }}}{EXCHG {PARA 220 "> " 0 "" }} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 14 "appr := [1.0]:" } {MPLTEXT 1 225 23 "\nfor i from 0 to n-1 do" }{MPLTEXT 1 225 60 "\n a ppr := Append(appr, Last(appr)+h*tt(a+i*h, Last(appr)));" }{MPLTEXT 1 225 8 "\nend do:" }{MPLTEXT 1 225 6 "\nappr;" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 38 "exct:=evalf( [seq(ye(a+i*h),i=0..n)]);" }}} {EXCHG {PARA 220 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 31 "l := evalf(map(abs,appr-exct));" }{MPLTEXT 1 225 12 "\nmax(op( l));" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "listplot(l);" } }}{EXCHG {PARA 220 "> " 0 "" }}{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" } {SECT 0 {PARA 218 "" 0 "" {TEXT 226 3 "(b)" }}{PARA 216 "" 0 "" {TEXT 227 6 "DGL: " }{XPPEDIT 18 0 "y[t] = cos(2*t)+sin(3*t);" "6#/&%\"yG6# %\"tG,&-%$cosG6#*&\"\"#\"\"\"F'F.F.-%$sinG6#*&\"\"$F.F'F.F." }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 16 " , 0 <= t <= 1, " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"! \"\"\"" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 14 " L\366sung: " }{XPPEDIT 18 0 "y(t) = sin(2*t )/2-cos(3*t)/3+4/3;" "6#/-%\"yG6#%\"tG,(*&-%$sinG6#*&\"\"#\"\"\"F'F/F/ F.!\"\"F/*&-%$cosG6#*&\"\"$F/F'F/F/F6F0F0*&\"\"%F/F6F0F/" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" {XPPEDIT 18 0 "f(t) = cos(2*t)+sin(3*t);" "6#/ -%\"fG6#%\"tG,&-%$cosG6#*&\"\"#\"\"\"F'F.F.-%$sinG6#*&\"\"$F.F'F.F." } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 5 " , " }{XPPEDIT 18 0 "f[t] = -2*sin(2*t)+3*cos(3*t);" "6 #/&%\"fG6#%\"tG,&*&\"\"#\"\"\"-%$sinG6#*&F*F+F'F+F+!\"\"*&\"\"$F+-%$co sG6#*&F2F+F'F+F+F+" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" {XPPEDIT 18 0 "T[2](w[i],t[i]) = f(w[i],t[i])+h*f[t](w[i],t[i])/2" "6#/-&%\"TG6#\"\"#6$&%\"wG6#%\"iG&% \"tGF,,&-%\"fGF)\"\"\"*(%\"hGF3-&F26#F/F)F3F(!\"\"F3" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 " " }{PARA 216 "" 0 "" {TEXT 227 19 "Max Fehler < 0.0033" }}{PARA 216 "" 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 37 "ye := t -> sin(2* t)/2-cos(3*t)/3+4/3:" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 33 "f := (t,y) -> cos(2*t)+sin(3*t):" } {MPLTEXT 1 225 38 "\nf1 := (t,y) -> 3*cos(3*t)-2*sin(2*t):" }{MPLTEXT 1 225 39 "\ntt := (t,y) -> f(t,y)+(h/2)*f1(t,y): " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 7 "a := 0:" }{MPLTEXT 1 225 8 "\nb := 1:" }{MPLTEXT 1 225 9 "\nn := 20:" }{MPLTEXT 1 225 14 "\nh := (b-a)/n:" } }}{EXCHG {PARA 220 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 14 "appr := [1.0]:" }{MPLTEXT 1 225 23 "\nfor i from 0 to n-1 do" }{MPLTEXT 1 225 60 "\n appr := Append(appr, Last(appr)+h*tt(a+i*h, La st(appr)));" }{MPLTEXT 1 225 8 "\nend do:" }{MPLTEXT 1 225 13 "\nevalf (appr);" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 38 "exct:=evalf( [seq(ye(a+i*h),i=0..n)]);" }}}{EXCHG {PARA 220 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 31 "l := evalf(map(abs,appr-exct)); " }{MPLTEXT 1 225 12 "\nmax(op(l));" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "listplot(l);" }}}{EXCHG {PARA 220 "> " 0 "" }} {PARA 216 "" 0 "" }}{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" }{SECT 1 {PARA 217 "" 0 "" {TEXT 224 9 "Aufgabe 3" }}{PARA 216 "" 0 "" }{SECT 0 {PARA 218 "" 0 "" {TEXT 226 3 "(a)" }}{PARA 216 "" 0 "" {TEXT 227 6 "DGL: " }{XPPEDIT 18 0 "y[t] = 1+(t-y)^2;" "6#/&%\"yG6#%\"tG,&\"\"\"F )*$,&F'F)F%!\"\"\"\"#F)" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 16 " , 2 <= t <= 3, " }{XPPEDIT 18 0 "y(2) = 1;" "6#/-%\"yG6#\"\"#\"\"\"" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 14 " L\366sung: " }{XPPEDIT 18 0 "y(t) = t-1/(t-1);" "6#/-%\"yG6#%\"tG,&F'\"\"\"*&F)F), &F'F)F)!\"\"F,F," }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" }{PARA 216 "" 0 " " {TEXT 227 13 "Max Fehler < " }{XPPEDIT 18 0 "4.5*10^(-7);" "6#*&-%&F loatG6$\"#X!\"\"\"\"\")\"#5,$\"\"(F(F)" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 21 "ye := t -> t-1/(t- 1):" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 25 "f := (t,y) -> 1+(t-y)^2: " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 7 "a := 2:" }{MPLTEXT 1 225 8 "\nb := 3:" } {MPLTEXT 1 225 9 "\nn := 10:" }{MPLTEXT 1 225 14 "\nh := (b-a)/n:" }}} {EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "appr := [1]:" }{MPLTEXT 1 225 23 "\nfor i from 0 to n-1 do" } {MPLTEXT 1 225 40 "\n k1 := evalf( h*f(a+i*h, Last(appr)));" } {MPLTEXT 1 225 51 "\n k2 := evalf( h*f(a+(i+1/2)*h, Last(appr)+k1/2)) ;" }{MPLTEXT 1 225 51 "\n k3 := evalf( h*f(a+(i+1/2)*h, Last(appr)+k2 /2));" }{MPLTEXT 1 225 47 "\n k4 := evalf( h*f(a+(i+1)*h, Last(appr)+ k3));" }{MPLTEXT 1 225 60 "\n appr := Append(appr, Last(appr)+(1/6)*( k1+2*k2+2*k3+k4));" }{MPLTEXT 1 225 8 "\nend do:" }{MPLTEXT 1 225 13 " \nevalf(appr);" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 38 "exct: =evalf( [seq(ye(a+i*h),i=0..n)]);" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 21 "pa := listplot(appr):" }{MPLTEXT 1 225 22 "\npe := \+ listplot(exct):" }{MPLTEXT 1 225 18 "\ndisplay([pa,pe]);" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 24 "l \+ := map(abs,appr-exct);" }{MPLTEXT 1 225 12 "\nmax(op(l));" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "listplot(l);" }}}{EXCHG {PARA 216 "> " 0 "" }}{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" }{SECT 0 {PARA 218 "" 0 "" {TEXT 226 3 "(b)" }}{PARA 216 "" 0 "" {TEXT 227 6 "DGL: " }{XPPEDIT 18 0 "y[t] = cos(2*t)+sin(3*t);" "6#/&%\"yG6#%\"tG,&-%$cosG 6#*&\"\"#\"\"\"F'F.F.-%$sinG6#*&\"\"$F.F'F.F." }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 16 " , 0 \+ <= t <= 1, " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 14 " L\366sung: " }{XPPEDIT 18 0 "y(t) = sin(2*t)/2-cos (3*t)/3+4/3;" "6#/-%\"yG6#%\"tG,(*&-%$sinG6#*&\"\"#\"\"\"F'F/F/F.!\"\" F/*&-%$cosG6#*&\"\"$F/F'F/F/F6F0F0*&\"\"%F/F6F0F/" }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }} {PARA 216 "" 0 "" {TEXT 227 13 "Max Fehler < " }{XPPEDIT 18 0 "1.3*10^ (-7);" "6#*&-%&FloatG6$\"#8!\"\"\"\"\")\"#5,$\"\"(F(F)" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 37 "ye := t -> sin(2*t)/2-cos(3*t)/3+4/3:" }}}{EXCHG {PARA 216 "> " 0 "" }} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 33 "f := (t,y) -> cos(2*t)+s in(3*t): " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 7 "a := 0:" } {MPLTEXT 1 225 8 "\nb := 1:" }{MPLTEXT 1 225 9 "\nn := 20:" }{MPLTEXT 1 225 14 "\nh := (b-a)/n:" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 14 "appr := [1.0]:" }{MPLTEXT 1 225 23 "\nfor i from 0 to n-1 do" }{MPLTEXT 1 225 40 "\n k1 := evalf( h*f(a+i*h, Last(appr)));" }{MPLTEXT 1 225 51 "\n k2 := evalf( h*f(a+ (i+1/2)*h, Last(appr)+k1/2));" }{MPLTEXT 1 225 51 "\n k3 := evalf( h* f(a+(i+1/2)*h, Last(appr)+k2/2));" }{MPLTEXT 1 225 47 "\n k4 := evalf ( h*f(a+(i+1)*h, Last(appr)+k3));" }{MPLTEXT 1 225 60 "\n appr := App end(appr, Last(appr)+(1/6)*(k1+2*k2+2*k3+k4));" }{MPLTEXT 1 225 8 "\ne nd do:" }{MPLTEXT 1 225 6 "\nappr;" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 38 "exct:=evalf( [seq(ye(a+i*h),i=0..n)]);" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 21 "pa := listplot(appr):" } {MPLTEXT 1 225 22 "\npe := listplot(exct):" }{MPLTEXT 1 225 18 "\ndisp lay([pa,pe]);" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 31 "l := evalf(map(abs,appr-exct));" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 11 "max(op(l));" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "listplot(l);" }}}{EXCHG {PARA 216 "> \+ " 0 "" }}{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" } {SECT 1 {PARA 217 "" 0 "" {TEXT 224 9 "Aufgabe 4" }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {TEXT 227 6 "DGL's:" }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "u1[t] = 4*u1-u2+u3-(t+2)^2;" "6#/&%#u1G6#%\"tG, **&\"\"%\"\"\"F%F+F+%#u2G!\"\"%#u3GF+*$,&F'F+\"\"#F+F1F-" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 5 " , " }{XPPEDIT 18 0 "u1(0) = 3;" "6#/-%#u1G6#\"\"!\"\"$" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 2 " " }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "u2[t] \+ = -u1+3*u2-2*u3+2*t^2+t+15;" "6#/&%#u2G6#%\"tG,.%#u1G!\"\"*&\"\"$\"\" \"F%F-F-*&\"\"#F-%#u3GF-F**&F/F-*$F'F/F-F-F'F-\"#:F-" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 5 " , \+ " }{XPPEDIT 18 0 "u2(0) = -7;" "6#/-%#u2G6#\"\"!,$\"\"(!\"\"" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 2 " " }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "u3[t] \+ = u1-2*u2+3*u3-3*t^2+t-10;" "6#/&%#u3G6#%\"tG,.%#u1G\"\"\"*&\"\"#F*%#u 2GF*!\"\"*&\"\"$F*F%F*F**&F0F**$F'F,F*F.F'F*\"#5F." }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 5 " , " }{XPPEDIT 18 0 "u3(0) = -2;" "6#/-%#u3G6#\"\"!,$\"\"#!\"\"" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 2 " " }}{PARA 216 "" 0 "" {TEXT 227 25 " 0 <= t <= 1 , h = 0.1 " }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {TEXT 227 14 "Exakte L\366sun g:" }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "u1(t) = 2*exp (3*t)+exp(6*t)+t;" "6#/-%#u1G6#%\"tG,(*&\"\"#\"\"\"-%$expG6#*&\"\"$F+F 'F+F+F+-F-6#*&\"\"'F+F'F+F+F'F+" }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 2 " " }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "u2(t) = -5-2*exp(t)+exp(3*t)-exp(6*t );" "6#/-%#u2G6#%\"tG,*\"\"&!\"\"*&\"\"#\"\"\"-%$expGF&F-F*-F/6#*&\"\" $F-F'F-F--F/6#*&\"\"'F-F'F-F*" }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 2 " " }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "u3(t) = -2*exp(t)-exp(3*t)+exp(6*t)+ t^2;" "6#/-%#u3G6#%\"tG,**&\"\"#\"\"\"-%$expGF&F+!\"\"-F-6#*&\"\"$F+F' F+F.-F-6#*&\"\"'F+F'F+F+*$F'F*F+" }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 2 " " }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {TEXT 227 41 "Obere Schranken f\374r den relativen Fehler:" }}{PARA 216 "" 0 "" {TEXT 227 22 " u1: Fehler < 0.37 %" }} {PARA 216 "" 0 "" {TEXT 227 22 " u2 : Fehler < 0.41 %" }}{PARA 216 "" 0 "" {TEXT 227 22 " u3: Fehler < 0.73 %" }}{PARA 216 "" 0 "" } {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 15 "# Exakte L\366sung" } {MPLTEXT 1 225 2 "\n#" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 51 "dsolve( \{diff(u1(t),t)=4*u1(t)-u2(t)+u3(t)-(t+2)^2," }{MPLTEXT 1 225 51 "\n diff(u2(t),t)=-u1(t)+3*u2(t)-2*u3(t)+2*t^2+t+15," } {MPLTEXT 1 225 50 "\n diff(u3(t),t)=u1(t)-2*u2(t)+3*u3(t)-3*t^2+t-10, " }{MPLTEXT 1 225 32 "\n u1(0)=3, u2(0)=-7, u3(0)=-2\}," }{MPLTEXT 1 225 25 "\n \{u1(t),u2(t),u3(t)\} );" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 34 "u1e := t -> 2*exp(3*t)+exp(6*t)+t:" }{MPLTEXT 1 225 43 "\nu2e := t -> -5-2*exp(t)+exp(3*t)-exp(6*t):" }{MPLTEXT 1 225 45 "\nu3e := t -> -2*exp(t)-exp(3*t)+exp(6*t)+t^2:" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 15 "# Approx imation" }{MPLTEXT 1 225 2 "\n#" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 42 "f1 := (t,u1,u2,u3) -> 4*u1-u2+u3-(t+2)^2: " } {MPLTEXT 1 225 48 "\nf2 := (t,u1,u2,u3) -> -u1+3*u2-2*u3+2*t^2+t+15:" }{MPLTEXT 1 225 47 "\nf3 := (t,u1,u2,u3) -> u1-2*u2+3*u3-3*t^2+t-10:" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 7 "a := 0:" }{MPLTEXT 1 225 8 "\nb := 1:" }{MPLTEXT 1 225 9 "\nn : = 10:" }{MPLTEXT 1 225 14 "\nh := (b-a)/n:" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 11 "au1 := [3]:" } {MPLTEXT 1 225 13 "\nau2 := [-7]:" }{MPLTEXT 1 225 13 "\nau3 := [-2]:" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 22 "for i from 0 to n-1 \+ do" }{MPLTEXT 1 225 1 "\n" }{MPLTEXT 1 225 63 "\n k11 := evalf( h*f1( a+i*h, Last(au1), Last(au2), Last(au3)));" }{MPLTEXT 1 225 63 "\n k12 := evalf( h*f2(a+i*h, Last(au1), Last(au2), Last(au3)));" }{MPLTEXT 1 225 63 "\n k13 := evalf( h*f3(a+i*h, Last(au1), Last(au2), Last(au3 )));" }{MPLTEXT 1 225 1 "\n" }{MPLTEXT 1 225 35 "\n k21 := evalf( h*f 1( a+(i+1/2)*h," }{MPLTEXT 1 225 65 "\n Last(au1)+k11/2, La st(au2)+k12/2, Last(au3)+k13/2));" }{MPLTEXT 1 225 35 "\n k22 := eval f( h*f2( a+(i+1/2)*h," }{MPLTEXT 1 225 65 "\n Last(au1)+k11 /2, Last(au2)+k12/2, Last(au3)+k13/2));" }{MPLTEXT 1 225 35 "\n k23 : = evalf( h*f3( a+(i+1/2)*h," }{MPLTEXT 1 225 65 "\n Last(au 1)+k11/2, Last(au2)+k12/2, Last(au3)+k13/2));" }{MPLTEXT 1 225 1 "\n" }{MPLTEXT 1 225 35 "\n k31 := evalf( h*f1( a+(i+1/2)*h," }{MPLTEXT 1 225 65 "\n Last(au1)+k21/2, Last(au2)+k22/2, Last(au3)+k23/ 2));" }{MPLTEXT 1 225 35 "\n k32 := evalf( h*f2( a+(i+1/2)*h," } {MPLTEXT 1 225 65 "\n Last(au1)+k21/2, Last(au2)+k22/2, Las t(au3)+k23/2));" }{MPLTEXT 1 225 35 "\n k33 := evalf( h*f3( a+(i+1/2) *h," }{MPLTEXT 1 225 65 "\n Last(au1)+k21/2, Last(au2)+k22/ 2, Last(au3)+k23/2));" }{MPLTEXT 1 225 3 "\n " }{MPLTEXT 1 225 33 "\n k41 := evalf( h*f1( a+(i+1)*h," }{MPLTEXT 1 225 59 "\n La st(au1)+k31, Last(au2)+k32, Last(au3)+k33));" }{MPLTEXT 1 225 33 "\n \+ k42 := evalf( h*f2( a+(i+1)*h," }{MPLTEXT 1 225 59 "\n Last (au1)+k31, Last(au2)+k32, Last(au3)+k33));" }{MPLTEXT 1 225 33 "\n k4 3 := evalf( h*f3( a+(i+1)*h," }{MPLTEXT 1 225 59 "\n Last(a u1)+k31, Last(au2)+k32, Last(au3)+k33));" }{MPLTEXT 1 225 1 "\n" } {MPLTEXT 1 225 61 "\n au1 := Append(au1, Last(au1)+(1/6)*(k11+2*k21+2 *k31+k41));" }{MPLTEXT 1 225 61 "\n au2 := Append(au2, Last(au2)+(1/6 )*(k12+2*k22+2*k32+k42));" }{MPLTEXT 1 225 61 "\n au3 := Append(au3, \+ Last(au3)+(1/6)*(k13+2*k23+2*k33+k43));" }{MPLTEXT 1 225 1 "\n" } {MPLTEXT 1 225 8 "\nend do:" }{MPLTEXT 1 225 5 "\nau1;" }{MPLTEXT 1 225 5 "\nau2;" }{MPLTEXT 1 225 5 "\nau3;" }}}{EXCHG {PARA 216 "> " 0 " " }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "# Vergleiche" } {MPLTEXT 1 225 2 "\n#" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 40 "eu1 := evalf( [seq(u1e(a+i*h),i=0..n)]);" }{MPLTEXT 1 225 41 "\neu 2 := evalf( [seq(u2e(a+i*h),i=0..n)]);" }{MPLTEXT 1 225 41 "\neu3 := e valf( [seq(u3e(a+i*h),i=0..n)]);" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 21 "pa1 := listplot(au1):" }{MPLTEXT 1 225 22 "\npe1 := listplot(eu1):" }{MPLTEXT 1 225 20 "\ndisplay([pa1,pe1]);" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "re le1 := []:" }{MPLTEXT 1 225 13 "\nrele2 := []:" }{MPLTEXT 1 225 13 "\n rele3 := []:" }{MPLTEXT 1 225 23 "\nfor i from 1 to n+1 do" }{MPLTEXT 1 225 76 "\n rele1 := Append(rele1, abs((Member(au1,i)-Member(eu1,i)) /Member(eu1,i)));" }{MPLTEXT 1 225 76 "\n rele2 := Append(rele2, abs( (Member(au2,i)-Member(eu2,i))/Member(eu2,i)));" }{MPLTEXT 1 225 76 "\n rele3 := Append(rele3, abs((Member(au3,i)-Member(eu3,i))/Member(eu3, i)));" }{MPLTEXT 1 225 8 "\nend do:" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 51 "[ max(op(rele1)), max(op(rele2)), max(op(rele3)) ]; " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 59 "display([listplot(r ele1),listplot(rele2),listplot(rele3)]);" }}}{EXCHG {PARA 216 "> " 0 " " }}{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" }{SECT 1 {PARA 217 "" 0 "" {TEXT 224 9 "Aufgabe 5" }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {TEXT 227 4 "DGL:" }}{PARA 216 "" 0 "" {TEXT 227 3 " " }{XPPEDIT 18 0 "y[t tt]+2*y[tt]-y[t]-2*y = exp(t);" "6#/,*&%\"yG6#%$tttG\"\"\"*&\"\"#F)&F& 6#%#ttGF)F)&F&6#%\"tG!\"\"*&F+F)F&F)F2-%$expGF0" }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 21 " , \+ 0 <= t <= 3 , " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 4 " , " }{XPPEDIT 18 0 "y[t](0) = 2;" "6#/-&%\"yG6#%\"tG6# \"\"!\"\"#" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 4 " , " }{XPPEDIT 18 0 "y[tt](0) = 0;" "6#/-&%\" yG6#%#ttG6#\"\"!F*" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 12 " , h = 0.1" }}{PARA 216 "" 0 "" } {PARA 216 "" 0 "" {TEXT 227 14 "Exakte L\366sung:" }}{PARA 216 "" 0 "" {TEXT 227 3 " " }{XPPEDIT 18 0 "y(t) = 1/6*t*exp(t)+43/36*exp(t)-4/ 9*exp(-2*t)+1/4*exp(-t);" "6#/-%\"yG6#%\"tG,***\"\"\"F*\"\"'!\"\"F'F*- %$expGF&F*F**(\"#VF*\"#OF,F-F*F**(\"\"%F*\"\"*F,-F.6#,$*&\"\"#F*F'F*F, F*F,*(F*F*F3F,-F.6#,$F'F,F*F*" }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 2 " " }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {TEXT 227 9 "Vorgehen:" }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "u1(t) = y(t);" "6#/-%#u1G6#%\"tG-%\"yGF&" } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 5 " , " }{XPPEDIT 18 0 "u2(t) = y[t](t);" "6#/-%#u2G6#%\"t G-&%\"yGF&F&" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 5 " , " }{XPPEDIT 18 0 "u3(t) = y[tt](t); " "6#/-%#u3G6#%\"tG-&%\"yG6#%#ttGF&" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 30 " Daraus folgt das D GL-System:" }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "u1[t] = u2;" "6#/&%#u1G6#%\"tG%#u2G" }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 5 " , " }{XPPEDIT 18 0 " u2[t] = u3;" "6#/&%#u2G6#%\"tG%#u3G" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 5 " , " }{XPPEDIT 18 0 "u3[t] = y[ttt];" "6#/&%#u3G6#%\"tG&%\"yG6#%$tttG" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 3 " = " }{XPPEDIT 18 0 "exp(t)-2*y[tt]+y[t]+2*y = exp(t)-2*u3+u2+2*u1;" "6#/,* -%$expG6#%\"tG\"\"\"*&\"\"#F)&%\"yG6#%#ttGF)!\"\"&F-F'F)*&F+F)F-F)F),* F%F)*&F+F)%#u3GF)F0%#u2GF)*&F+F)%#u1GF)F)" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }}{PARA 216 "" 0 "" } {PARA 216 "" 0 "" {TEXT 227 41 "Obere Schranken f\374r den relativen F ehler:" }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "y(t);" "6 #-%\"yG6#%\"tG" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 14 " : Fehler < " }{XPPEDIT 18 0 "2.5*10^ (-4);" "6#*&-%&FloatG6$\"#D!\"\"\"\"\")\"#5,$\"\"%F(F)" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 2 " % " }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "y[t](t);" "6#-& %\"yG6#%\"tGF&" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " } {TEXT 227 1 " " }{TEXT 227 13 " : Fehler < " }{XPPEDIT 18 0 "2.7*10^( -4);" "6#*&-%&FloatG6$\"#F!\"\"\"\"\")\"#5,$\"\"%F(F)" }{TEXT 227 1 " \+ " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 2 " %" }}{PARA 216 "" 0 "" {TEXT 227 2 " " }{XPPEDIT 18 0 "y[tt](t);" "6#-& %\"yG6#%#ttG6#%\"tG" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " \+ " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 11 ": Fehler < " } {XPPEDIT 18 0 "9.0*10^(-4);" "6#*&-%&FloatG6$\"#!*!\"\"\"\"\")\"#5,$\" \"%F(F)" }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 1 " " }{TEXT 227 2 " %" }}{PARA 216 "" 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 15 "# Exakte L\366sung" }{MPLTEXT 1 225 2 "\n#" } }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 76 "dsolve( \{diff(y(t),t$ 3) + 2*diff(y(t),t$2) - diff(y(t),t) - 2*y(t) = exp(t)," }{MPLTEXT 1 225 48 "\n y(0)=1,D(y)(0)=2,D(D(y))(0)=0\}, y(t) );" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 64 "u1e := t -> 1/6*t*exp(t)+43/36* exp(t)-4/9*exp(-2*t)+1/4*exp(-t):" }{MPLTEXT 1 225 65 "\nu2e := t -> 4 9/36*exp(t)+1/6*t*exp(t)+8/9*exp(-2*t)-1/4*exp(-t):" }{MPLTEXT 1 225 66 "\nu3e := t -> 55/36*exp(t)+1/6*t*exp(t)-16/9*exp(-2*t)+1/4*exp(-t) :" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 15 "# Approximation" }{MPLTEXT 1 225 2 "\n#" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 26 "f1 := (t,u1,u2,u3) -> u2: " } {MPLTEXT 1 225 26 "\nf2 := (t,u1,u2,u3) -> u3:" }{MPLTEXT 1 225 43 "\n f3 := (t,u1,u2,u3) -> exp(t)-2*u3+u2+2*u1:" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 7 "a := 0:" }{MPLTEXT 1 225 8 "\nb := 3:" }{MPLTEXT 1 225 9 "\nn := 30:" }{MPLTEXT 1 225 14 "\nh := (b-a)/n:" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> \+ " 0 "" {MPLTEXT 1 225 11 "au1 := [1]:" }{MPLTEXT 1 225 12 "\nau2 := [2 ]:" }{MPLTEXT 1 225 12 "\nau3 := [0]:" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 22 "for i from 0 to n-1 do" }{MPLTEXT 1 225 1 "\n" } {MPLTEXT 1 225 63 "\n k11 := evalf( h*f1(a+i*h, Last(au1), Last(au2), Last(au3)));" }{MPLTEXT 1 225 63 "\n k12 := evalf( h*f2(a+i*h, Last( au1), Last(au2), Last(au3)));" }{MPLTEXT 1 225 63 "\n k13 := evalf( h *f3(a+i*h, Last(au1), Last(au2), Last(au3)));" }{MPLTEXT 1 225 1 "\n" }{MPLTEXT 1 225 35 "\n k21 := evalf( h*f1( a+(i+1/2)*h," }{MPLTEXT 1 225 65 "\n Last(au1)+k11/2, Last(au2)+k12/2, Last(au3)+k13/ 2));" }{MPLTEXT 1 225 35 "\n k22 := evalf( h*f2( a+(i+1/2)*h," } {MPLTEXT 1 225 65 "\n Last(au1)+k11/2, Last(au2)+k12/2, Las t(au3)+k13/2));" }{MPLTEXT 1 225 35 "\n k23 := evalf( h*f3( a+(i+1/2) *h," }{MPLTEXT 1 225 65 "\n Last(au1)+k11/2, Last(au2)+k12/ 2, Last(au3)+k13/2));" }{MPLTEXT 1 225 1 "\n" }{MPLTEXT 1 225 35 "\n \+ k31 := evalf( h*f1( a+(i+1/2)*h," }{MPLTEXT 1 225 65 "\n La st(au1)+k21/2, Last(au2)+k22/2, Last(au3)+k23/2));" }{MPLTEXT 1 225 35 "\n k32 := evalf( h*f2( a+(i+1/2)*h," }{MPLTEXT 1 225 65 "\n \+ Last(au1)+k21/2, Last(au2)+k22/2, Last(au3)+k23/2));" }{MPLTEXT 1 225 35 "\n k33 := evalf( h*f3( a+(i+1/2)*h," }{MPLTEXT 1 225 65 "\n Last(au1)+k21/2, Last(au2)+k22/2, Last(au3)+k23/2));" } {MPLTEXT 1 225 3 "\n " }{MPLTEXT 1 225 33 "\n k41 := evalf( h*f1( a+ (i+1)*h," }{MPLTEXT 1 225 59 "\n Last(au1)+k31, Last(au2)+k 32, Last(au3)+k33));" }{MPLTEXT 1 225 33 "\n k42 := evalf( h*f2( a+(i +1)*h," }{MPLTEXT 1 225 59 "\n Last(au1)+k31, Last(au2)+k32 , Last(au3)+k33));" }{MPLTEXT 1 225 33 "\n k43 := evalf( h*f3( a+(i+1 )*h," }{MPLTEXT 1 225 59 "\n Last(au1)+k31, Last(au2)+k32, \+ Last(au3)+k33));" }{MPLTEXT 1 225 1 "\n" }{MPLTEXT 1 225 61 "\n au1 : = Append(au1, Last(au1)+(1/6)*(k11+2*k21+2*k31+k41));" }{MPLTEXT 1 225 61 "\n au2 := Append(au2, Last(au2)+(1/6)*(k12+2*k22+2*k32+k42)); " }{MPLTEXT 1 225 61 "\n au3 := Append(au3, Last(au3)+(1/6)*(k13+2*k2 3+2*k33+k43));" }{MPLTEXT 1 225 1 "\n" }{MPLTEXT 1 225 8 "\nend do:" } {MPLTEXT 1 225 5 "\nau1;" }{MPLTEXT 1 225 5 "\nau2;" }{MPLTEXT 1 225 5 "\nau3;" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "# Vergleiche" }{MPLTEXT 1 225 2 "\n#" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 40 "eu1 := evalf( [seq(u1e(a+i*h),i =0..n)]);" }{MPLTEXT 1 225 41 "\neu2 := evalf( [seq(u2e(a+i*h),i=0..n) ]);" }{MPLTEXT 1 225 41 "\neu3 := evalf( [seq(u3e(a+i*h),i=0..n)]);" } }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 21 "pa1 := listplot(au1):" }{MPLTEXT 1 225 22 "\npe1 := listplot(eu1):" }{MPLTEXT 1 225 20 "\ndi splay([pa1,pe1]);" }}}{EXCHG {PARA 216 "> " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 12 "rele1 := []:" }{MPLTEXT 1 225 13 "\nrele2 : = []:" }{MPLTEXT 1 225 13 "\nrele3 := []:" }{MPLTEXT 1 225 23 "\nfor i from 2 to n+1 do" }{MPLTEXT 1 225 76 "\n rele1 := Append(rele1, abs( (Member(au1,i)-Member(eu1,i))/Member(eu1,i)));" }{MPLTEXT 1 225 76 "\n rele2 := Append(rele2, abs((Member(au2,i)-Member(eu2,i))/Member(eu2, i)));" }{MPLTEXT 1 225 76 "\n rele3 := Append(rele3, abs((Member(au3, i)-Member(eu3,i))/Member(eu3,i)));" }{MPLTEXT 1 225 8 "\nend do:" }}} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 51 "[ max(op(rele1)), max(op (rele2)), max(op(rele3)) ];" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 225 59 "display([listplot(rele1),listplot(rele2),listplot(rele3)]);" } }}{EXCHG {PARA 216 "> " 0 "" }}{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" } {PARA 216 "" 0 "" }{PARA 222 "" 0 "" }{PARA 223 "" 0 "" }{PARA 224 "" 0 "" }{PARA 225 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }