{VERSION 3 0 "IBM INTEL NT" "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 }{CSTYLE " Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "versuc h" -1 256 "Copperplate Gothic Bold" 0 0 255 0 255 1 2 2 2 0 0 2 0 0 0 }{CSTYLE "" 256 257 "Copperplate Gothic Light" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 256 258 "Copperplate Gothic Light" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 20 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 20 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 1 0 1 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits:=20:\n\n" }{TEXT -1 26 "Differentialgleichung " }{XPPEDIT 18 0 "diff(y,t) = -y;" "6#/- %%diffG6$%\"yG%\"tG,$F'!\"\"" }{TEXT -1 38 " und ihre exakte L\366sung an der Stelle " }{XPPEDIT 18 0 "t = 1;" "6#/%\"tG\"\"\"" }{TEXT -1 1 ":" }{MPLTEXT 1 0 116 "\n\node:=diff(y(t),t)=-y(t):\ny.0:=1:\ninit:=y( 0)=y.0:\ndsolve(\{ode,init\}, y(t));\nExakte_lsg:=evalf(subs(t=1,rhs(% )));\n\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 " Die Zahlenfolg e n = 2 , 4 , 6 , 8 , 12 , 16 ... und damit die Folge der Schrittweite n " }{XPPEDIT 18 0 "h[i];" "6#&%\"hG6#%\"iG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "1/2;" "6#*&\"\"\"\"\"\"\"\"#!\"\"" }{TEXT -1 4 " , " } {XPPEDIT 18 0 "1/4;" "6#*&\"\"\"\"\"\"\"\"%!\"\"" }{TEXT -1 4 " , " } {XPPEDIT 18 0 "1/6;" "6#*&\"\"\"\"\"\"\"\"'!\"\"" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "1/8;" "6#*&\"\"\"\"\"\"\"\")!\"\"" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "1/12;" "6#*&\"\"\"\"\"\"\"#7!\"\"" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "1/16;" "6#*&\"\"\"\"\"\"\"#;!\"\"" }{TEXT -1 23 " ... \+ werden definiert:" }{MPLTEXT 1 0 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "n[0]:=2:\nn[1]:=4:\nn[2]:=6:\nfor i from 3 to 5 do\n n[i]:=2 *n[i-2];\nod:\nfor i from 0 to 5 do\n h[i]:=1/n[i];\nod:\n\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 " Die Prozedur zur Berechnung d er Interpolation-Startwerte " }{XPPEDIT 18 0 "R[0]^i;" "6#)&%\"RG6#\" \"!%\"iG" }{TEXT -1 109 ". Das sind die numerischen L\366sungen der Di fferentialgleichung nach der Gragg-Methode an der Stelle " } {XPPEDIT 18 0 "t = 1;" "6#/%\"tG\"\"\"" }{TEXT -1 2 ": " }{MPLTEXT 1 0 1 "\n" }{TEXT -1 55 " \+ " }{XPPEDIT 18 0 "eta[0] := y[0];" "6#>&%$etaG6#\"\"!&%\"yG6#F '" }{TEXT -1 57 " \n \+ " }{XPPEDIT 18 0 "eta[1] := y[0]-h[i]*f(t[0],y[0]);" "6#>&%$etaG6 #\"\"\",&&%\"yG6#\"\"!\"\"\"*&&%\"hG6#%\"iGF--%\"fG6$&%\"tG6#F,&F*6#F, F-!\"\"" }{TEXT -1 55 " \n \+ " }{TEXT 261 6 " f\374r " }{XPPEDIT 262 0 "j = 1;" "6#/%\"jG \"\"\"" }{TEXT 263 19 " , 2 , ... , n-1 :" }{TEXT -1 4 " " } {XPPEDIT 18 0 "eta[j+1] := eta[j-1]-2*h[i]*f(t[j],eta[j]);" "6#>&%$eta G6#,&%\"jG\"\"\"\"\"\"F),&&F%6#,&F(F)\"\"\"!\"\"F)*(\"\"#F)&%\"hG6#%\" iGF)-%\"fG6$&%\"tG6#F(&F%6#F(F)F0" }{TEXT -1 96 "\n \+ \+ " }{XPPEDIT 18 0 "t[j] := t[j-1]+h[i];" "6#>&%\"tG6#%\"jG,&&F%6# ,&F'\"\"\"\"\"\"!\"\"F,&%\"hG6#%\"iGF," }{TEXT -1 56 "\n \+ " }{XPPEDIT 18 0 "y(t,h[i]) : = (eta[n[i]]+eta[n[i]-1]+h[i]*f(t[n[i]],eta[n[i]]))/2;" "6#>-%\"yG6$% \"tG&%\"hG6#%\"iG*&,(&%$etaG6#&%\"nG6#F+\"\"\"&F/6#,&&F26#F+F4\"\"\"! \"\"F4*&&F)6#F+F4-%\"fG6$&F'6#&F26#F+&F/6#&F26#F+F4F4F4\"\"#F;" } {TEXT -1 19 " " }{MPLTEXT 1 0 3 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "Gragg:=proc(i)\nlocal eta, j:\nglo bal h, n:\neta[0]:=y.0:\neta[1]:=y.0-h[i]*y.0:\nfor j from 1 to n[i]-1 do\n eta[j+1]:=eta[j-1]-2*h[i]*eta[j];\nod:\nRETURN(evalf(eta[n[i]]+ eta[n[i]-1]-h[i]*eta[n[i]])/2):\nend:\n\nfor i from 0 to 5 do\n R[i, 0]:=Gragg(i);\nod:\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 " J etzt wird nach dem Dreieck-Schema intrerpoliert nach der \344hnlichen \+ Formel wie " }{TEXT 257 3 "(2)" }{TEXT 259 0 "" }{TEXT 26 0 "" }{TEXT 260 0 "" }{TEXT -1 12 " , aber mit " }{XPPEDIT 18 0 "(h[i]/h[i+m])^2; " "6#*$*&&%\"hG6#%\"iG\"\"\"&F&6#,&F(F)%\"mGF)!\"\"\"\"#" }{TEXT -1 24 " im Nenner anstatt von " }{XPPEDIT 18 0 "h[i]/h[i+m];" "6#*&&%\"h G6#%\"iG\"\"\"&F%6#,&F'F(%\"mGF(!\"\"" }{TEXT -1 50 " :\n\n \+ " }{XPPEDIT 18 0 "R[m-1]^(i+1)+(R[m -1]^(i+1)-R[m-1]^i)/((h[i]/h[i+m]-1)^2);" "6#,&)&%\"RG6#,&%\"mG\"\"\" \"\"\"!\"\",&%\"iGF*\"\"\"F*F**&,&)&F&6#,&F)F*\"\"\"F,,&F.F*\"\"\"F*F* )&F&6#,&F)F*\"\"\"F,F.F,F**$,&*&&%\"hG6#F.F*&FB6#,&F.F*F)F*F,F*\"\"\"F ,\"\"#F,F*" }{TEXT -1 16 " " }{TEXT 258 4 "(2a)" } {TEXT -1 30 " " }{MPLTEXT 1 0 1 "\n" } {TEXT -1 0 "" }{MPLTEXT 1 0 2 "\n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "for m from 1 to 5 do \n for i from 0 to 5-m do\n R[i,m]:=R[i+1, m-1]+(R[i+1,m-1] -R[i,m-1])/((h[i]/h[i+m])^2-1);\n od;\nod:\n\n\n\n\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 795 "printf(`Tabelle 5: Di e Werte R[m]^i f\374r die Polynom-Extrapolation unter Verwendung von G raggschen Formeln:\\n`);\nprintf(`------------------------------------ ---------------------------------------------------------------------- ---------\\n`);\nprintf(` i h[i] m | 0 \+ 1 2 3 4 5 \+ \\n`);\nprintf(`----------------------------------------------------- --------------------------------------------------------------\\n`);\n for i from 0 to 5 do\n printf(` %d 1/%2d | `,i,n[i ]);\n for m from 0 to 5-i do\n printf(`%11.9f `, R[i,m]);\n o d;\n printf(`\\n`);\nod;\n\nprintf(`--------------------------------- ---------------------------------------------------------------------- ------------\\n`);\n\n\n\n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 813 "\n\n\n\n\n\nprintf(`Tabelle 6: Die Fehler in R[m]^i f\374r P olynom-Extrapolation unter Verwendung von Graggschen Formeln:\\n`);\np rintf(`--------------------------------------------------------------- ----------------------------------------------------\\n`);\nprintf(` \+ i h[i] m | 0 1 2 \+ 3 4 5 \\n`);\nprintf(`------- ---------------------------------------------------------------------- --------------------------------------\\n`);\nfor i from 0 to 5 do\n \+ printf(` %d 1/%2d | `,i,n[i]);\n for m from 0 to 5 -i do\n printf(`%11.9f `, R[i,m]-Exakte_lsg);\n od;\n printf( `\\n`);\nod;\nprintf(`------------------------------------------------ -------------------------------------------------------------------\\n `);\n\n\n\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 " " } {MPLTEXT 1 0 5 "\n\n\n\n\n" }{TEXT -1 5 " " }{TEXT 264 0 "" } {TEXT 265 16 "Arbeitsaufwand:\n" }{MPLTEXT 1 0 1 "\n" }{TEXT -1 130 " \+ Anzahl der Funktionsauswerungen bei der Extrapolation mit Euler (T abelle 3) und bei der Extrapolation mit Gragg (Tabelle 6):\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "for i from 0 to 5 do\n Anzahl_Euler[i,0] :=2^i;\nod:\n\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 291 "for m from 1 to 5 d o \n for i from 0 to 5-m do\n Anzahl_Euler[i,m]:=sum('Anzahl_Euler[j ,0]','j'=i..i+m)-m;\n od;\nod:\n\nfor i from 0 to 5 do\n Anzahl_Grag g[i,0]:=n[i]+1;\nod:\n\nfor m from 1 to 5 do \n for i from 0 to 5-m d o\n Anzahl_Gragg[i,m]:=sum('Anzahl_Gragg[j,0]','j'=i..i+m)-m;\n od; \nod:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 780 "printf(`Tabel le 7: Die Anzahl der Funktionsauswertungen f\374r\\n`); \nprintf(` \+ Polynom-Extrapolation nach Euler-Methode / nach Graggschen Form eln :\\n`);\nprintf(`----------------------------------------------- ------------------------------------------------\\n`);\nprintf(` \+ i m | 0 1 2 3 4 \+ 5 \\n`);\nprintf(`------------------------------------ -----------------------------------------------------------\\n`);\nfor i from 0 to 5 do\n printf(` %d | `,i);\n for m fro m 0 to 5-i do\n printf(`%2d/%2d `, Anzahl_Euler[i,m],Anzahl_G ragg[i,m] );\n od;\n printf(`\\n`);\nod;\nprintf(`----------------- ---------------------------------------------------------------------- --------\\n`);\n\n\n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{MARK "11" 0 }{VIEWOPTS 1 1 0 1 1 1803 }