{VERSION 5 0 "HP RISC UNIX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 2 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 262 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE " " -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 31 "11. \334bungsblatt Compu teralgebra" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "restart; with(plots): with(plottools):" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 10 "Aufgabe 1:" }}{PARA 258 "" 0 "" {TEXT -1 2 "a)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Periode := proc(a, p, x0)\nlocal x , n;\nx := x0;\nfor n by 1 do\nx := a*x mod p;\nif x = x0 then break f i\nod;\nn\nend: " }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 2 "b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Periode(3, 31, 15);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Periode(13, 31, 15);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 2 "c)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "zeichnung : = proc(a)\nlocal x, i;\nx[0] := 15:\nfor i to 30 do x[i] := a*x[i-1] m od 31 od:\ndisplay([seq(disk([x[i], x[i+1]],0.2,color=blue), i=0..29)] ,\nscaling=constrained);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "zeichnung(3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ze ichnung(13);" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 116 "d) Zumindest f \374r zweidimensionale Simulationen ist der mit a=13 besser, denn er f \374llt die Ebene deutlich besser aus." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 10 "Aufgabe 2:" }}{PARA 263 "" 0 "" {TEXT -1 3 "a) " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "xnplus1 := (2^16+3)*xn;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "xnplus2 := (2^16+3)*xnplus1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "xnplus2 - 6*xnplus1 + 9*xn;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "% mod 2^31;" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 37 "Also verschwindet diese Summe modulo " }{XPPEDIT 18 0 "2^31;" " 6#*$\"\"#\"#J" }{TEXT -1 52 ".\nb) Nat\374rlich liegen sie alle auf E benen der Form " }{XPPEDIT 18 0 "z-6*y+9*x = k*2^31;" "6#/,(%\"zG\"\" \"*&\"\"'F&%\"yGF&!\"\"*&\"\"*F&%\"xGF&F&*&%\"kGF&*$\"\"#\"#JF&" } {TEXT -1 11 " mit k aus " }{TEXT 257 0 "" }{TEXT 258 1 "Z" }{TEXT 259 0 "" }{TEXT -1 0 "" }{TEXT 256 4 " . \n" }{TEXT -1 24 "Da x,y,z zwisch en 0 und " }{XPPEDIT 18 0 "2^31-1;" "6#,&*$\"\"#\"#J\"\"\"F'!\"\"" } {TEXT -1 86 " liegen, mu\337 k Werte zwischen -5 und 9 annehmen, d.h. \+ es gibt nur 15 solche Ebenen.\nc)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "Zufallszahl := rand():\nRANDU := proc()\nglobal Zufallszahl;\nZuf allszahl := (2^16+3)*Zufallszahl mod 2^31;\nend:\nPunkte := [seq([RAND U(), RANDU(), RANDU()], i=1..10000)]:\nspacecurve(Punkte, style=point, orientation=[62,45], color=red);" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 10 "Aufgabe 3:" }}{PARA 266 "" 0 "" {TEXT -1 2 "a)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "n := 0: x := rand() :\nfor i to 100000 do\ny := rand();\nif igcd(x, y) = 1 then n := n+1 f i;\nx := y\nod: n;" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 2 "b)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Theorie := evalf(6/Pi^2);\nPraxis \+ := evalf(n/100000);\nProzentuale_Abweichung := 100*(Praxis - Theorie) /Theorie;" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 2 "c)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p := evalf(sqrt(600000/n));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 47 "Prozentuale_Abweichung := 100*evalf((p-Pi)/P i);" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 10 "Aufgabe 4:" }{MPLTEXT 1 0 0 "" }}{PARA 270 "" 0 "" {TEXT -1 2 "a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "Summe := 0:\nzuf := rand(37):\nfor i to 10000 do\nKa pital := 100;\nEinsatz := 1;\n\nfor j while Kapital > Einsatz do\nif ( zuf() mod 2) = 1\nthen Kapital := Kapital + Einsatz; break\nelse Kapit al := Kapital - Einsatz; Einsatz := 2*Einsatz; fi;\nod:\nSumme := Summ e + (Kapital-100);\nod:\nGewinn := evalf(Summe/10000, 5);" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 17 "b) Abgesehen vom " }{TEXT 260 6 "break- " }{TEXT -1 31 "Kommando ist das derselbe Code:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "Summe := 0:\nzuf := rand(37):\nfor i to 10000 do\nKa pital := 100;\nEinsatz := 1;\n\nfor j while Kapital > Einsatz do\nif ( zuf() mod 2) = 1\nthen Kapital := Kapital + Einsatz; # break\nelse Kap ital := Kapital - Einsatz; Einsatz := 2*Einsatz; fi;\nod:\nSumme := Su mme + (Kapital-100);\nod:\nGewinn := evalf(Summe/10000, 5);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 38 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }