{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 2 0 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 0 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 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 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 "" 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 "" 0 259 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 "" 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 "" 0 261 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 "" 0 262 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 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 30 "5. \334bungsblatt Comput eralgebra" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 10 "Aufgabe 1:" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "elsym := proc( n, j)\nlocal P, X, z, i;\nP := expand(mul(X - z[i], i=1..n));\nsort((- 1)^j*coeff(P, X, n-j))\nend;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "for j to 5 do\npsi[j] := elsym(5, j)\nod;" }}}{EXCHG {PARA 258 " " 0 "" {TEXT -1 10 "Aufgabe 2:" }}{PARA 261 "" 0 "" {TEXT -1 3 "a) " } {TEXT 256 74 "Der i-te Koeffizient ist bis aufs Vorzeichen gleich dem \+ f\374hrenden mal der " }{TEXT -1 5 "(n-i)" }{TEXT 260 108 "-ten elemen tarsymmetrischen Funktion, ausgewerter an den Nullstellen. Falls diese alle ganzzahlig sind, ist " }{XPPEDIT 18 0 "a[i];" "6#&%\"aG6#%\"iG" }{TEXT -1 1 " " }{TEXT 257 38 "somit ein ganzzahliges Vielfaches von \+ " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 0 "" }{TEXT 258 2 ".\n" }{XPPEDIT 18 0 "a[0]/a[n];" "6#*&&%\"aG6#\"\"!\"\"\"&F%6#%\"nG !\"\"" }{TEXT -1 1 " " }{TEXT 259 4 "ist " }{XPPEDIT 18 0 "(-1)^m*psi( z[1],`...`,z[n]) = (-1)^n*product(z[i],i = 1 .. n);" "6#/*&),$\"\"\"! \"\"%\"mGF'-%$psiG6%&%\"zG6#F'%$...G&F.6#%\"nGF'*&),$F'F(F3F'-%(produc tG6$&F.6#%\"iG/F<;F'F3F'" }{TEXT -1 1 " " }{TEXT 261 37 " und somit of fensichtlich durch alle " }{XPPEDIT 18 0 "z[i];" "6#&%\"zG6#%\"iG" } {TEXT -1 1 " " }{TEXT 262 7 "teilbar" }{TEXT -1 5 ".\n\nb)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "f := X^9 - 3*X^8 - 275*X^7 + 829*X^6 + 1129 9*X^5 - 34997*X^4 - 11025*X^3 + 78271*X^2 - 44100;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 338 "probiere := proc(x)\n# testet ob x Nullste lle von f ist; falls ja, wird f/(X-x)\n# zurueckgegeben, sonst f selbs t. Nullstellen werden zusammen\n# mit ihrer Vielfachheit ausgedruckt. \nglobal f;\nlocal i;\ni := 0;\nwhile subs(X=x, f) = 0 do\nf := normal (f/(X-x)); i := i+1\nod;\nif i > 0 then\nprintf(\"%10d ist %d-fache Nu llstelle\\n\", x, i)\nfi;\nf\nend;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ifactor(subs(X=0, f));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "probiere(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "probiere(3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "probi ere(-3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "probiere(5);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "probiere(-5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "probiere(15);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "probiere(-15);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "probiere(7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "probiere(-7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "prob iere(1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "probiere(-1);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 38 "Damit sind alle Nullstellen g efunden.\n" }}{PARA 260 "" 0 "" {TEXT -1 10 "Aufgabe 3:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1194 "with(plots): wit h(plottools):\n\ndemoRiemann := proc(f, a, b)\nlocal i, j, n, m, M, x1 , x2, m0, M0, delta, L,\nUSumme, OSumme, unten, oben, Abstand;\noben \+ := maximize(f, x=a..b); \nunten := minimize(f, x=a..b);\n# in Maple V .x hat dies die Syntax maximize(f, x, a..b) usw.\nAbstand := (oben - u nten)*0.05;\nj := 0;\nfor n in [5, 10, 25, 50, 100] do\nj := j+1; L[j] := []; USumme[j] := 0; OSumme[j] := 0;\ndelta := (b-a)/n;\nfor i to n do\nx1 := evalf(a + (i-1)*delta);\nx2 := evalf(a + i*delta);\nm := mi nimize(f, x=x1..x2);\nM := maximize(f, x=x1..x2);\nUSumme[j] := USumme [j] + delta*m;\nOSumme[j] := OSumme[j] + delta*M;\nif evalf(m) >= 0 th en m0 := 0; M0 := m\nelif evalf(M) >= 0 then m0 := 0; M0 := 0\nelse m0 := M; M0 := 0 fi;\nL[j] := [op(L[j]),\nrectangle([x1, m0], [x2, m], s tyle=patch, color=green),\nrectangle([x1, M0], [x2, M], style=patch, c olor=red)];\nod\nod;\ndisplay([seq(\ndisplay([op(L[j]),\nplot(f, x=a.. b, color=blue, thickness=5),\ntextplot([(a+b)/2, unten-Abstand, sprint f(\"Untersumme = %5.2f\", evalf(USumme[j]))],\nfont=[HELVETICA, 24], c olor=green),\ntextplot([(a+b)/2, oben+Abstand, sprintf(\"Obersumme = % 5.2f\", evalf(OSumme[j]))],\nfont=[HELVETICA, 24], color=red)]), j=1.. 5)],\ninsequence=true)\nend;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "demoRiemann(42*x^3-319*x^2+668*x-231, 0, 5);" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 10 "Aufgabe 4:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 737 "Felder := []:\nEinheit := [\"Tausend\", \"Millionen\", \"Milliard en\",\n\"Billionen\", \"Billiarden\", \"Trillionen\"]:\nfor i to 8 do \nfor j to 8 do\nKoerner := 2^(8*(j-1)+i-1);\nif (i+j) mod 2 = 1 then \+ Farbe := coral\nelse Farbe := yellow fi;\nif Koerner < 1000 then\nFeld er := [op(Felder),\nrectangle([i,-j],[i+1,-j-1], color=Farbe),\ntextpl ot([i+1/2, -j-1/2, sprintf(\"%d\", Koerner)],\nfont=[HELVETICA, 18])] \nelse\nGruppen := floor(evalf(log[1000](Koerner)));\nFaktor := Koerne r/10^(3*Gruppen);\nFelder := [op(Felder),\nrectangle([i,-j],[i+1,-j-1] , color=Farbe),\ntextplot([[i+1/2, -j-1/3, sprintf(\"%5.3f\", Faktor)] ,\n[i+1/2, -j-2/3, Einheit[Gruppen]]],\nfont=[HELVETICA, 12])]\nfi\nod \nod:\nAnz_weiss := add(2^(2*i-2), i=1..32);\nAnz_schwarz := add(2^(2* i-1), i=1..32);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "display ([op(Felder),\ntextplot([4.5, -9.5, sprintf(\"%d K\366rner auf wei\337 en Feldern\", Anz_weiss)],\nfont=[HELVETICA, 12], color=yellow),\ntext plot([4.5, -10, sprintf(\"%d K\366rner auf schwarzen Feldern\", Anz_sc hwarz)],\nfont=[HELVETICA, 12], color=coral)],\nscaling=constrained, a xes=none);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }