{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "MS Sans Serif" 1 8 128 0 128 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 "Roter Text" -1 256 "Tahoma" 0 0 255 0 0 1 2 1 1 0 0 2 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helv etica" 1 9 128 0 128 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 8 0 128 128 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Seite numbruch" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 1 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 260 1 {CSTYLE "" -1 -1 "Time s" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 3" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 12 "Aufgabe (4 9)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Di ese Aufgabe behandelt ein etwas umfangreicheres Beispiel zu Satz 8 in \+ \247 11 und dem daran anschlie\337enden Verfahren." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 168 "Es handelt sich aber im mer noch um ein \"sch\366n konstruiertes\" Beispiel, bei dem die L\366 sungen exakt angegeben werden k\366nnen, ohne auf N\344herungsverfahre n zur\374ckzugreifen. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Die Aufgabe ist soweit vorbereitet, da\337 man sich auf das nach Satz 8 beschriebene Verfahren konzentrieren kann. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Dies ist \+ dann Ihnen \374berlassen!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(Groebner):" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Bevor Sie beginnen m\374ssen Sie die folgenden (nicht op timalen) Prozeduren" }}{PARA 0 "" 0 "" {TEXT -1 11 "aktivieren:" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 " Prozeduren" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 953 "Kbasis := proc(PList,VList,torder)\n local B, Bs,C,G,i,v,t,l,m,pB,r;\n if is_finite(PList,VList) then\n \+ G := gbasis(PList,torder);\n B := [1];\n for v in VLis t do\n m := degree(univpoly(v,G,VList),v);\n C := B;\n for t in C do\n for l to m-1 do\n t := t*v;\n if normalf(t,G,to rder) = t then\n B := [op(B),t]\n \+ fi\n od\n od\n od;\n \+ Bs:=B;\n r:=nops(B);\n pB := 0;\n for i to \+ r do pB := pB+B[i] end do;\n i := 0; Bs := []; while pB <> 0 \+ do\n l := Groebner:-leadmon(pB,torder);\n i := i +1;\n Bs := [l[2], op(Bs)];\n pB := pB-l[2]\n \+ end do;\n B:=Bs;\n RETURN(B)\n else\n E RROR(`Das Ideal ist nicht null-dimensional, und es existiert keine end liche Basis.`)\n fi\n end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1471 "holMatrix := proc(f,KbasisList,GBList,VList,torder)\n loc al KK,GG,r,ff,fff,XX,i,k,l,Ku,pKu;\n global B,Mf;\n \+ Ku := KbasisList;\n r := nops(Ku);\n \+ XX := VList;\n GG := GBList;\n # \n # zuerst wird \"Ku\" sortiert gem\344\337 \"torder\" : \n #\n pKu := 0;\n for i t o r do pKu := pKu+Ku[i] od;\n i := 0;\n \+ KK := [];\n while pKu <> 0 do l := leadmon(pKu,tor der); i := i+1; KK := [l[2],op(KK)]; pKu := pKu-l[2] od;\n \+ B:=KK;\n # print(`Die nach der vorgegebenen Monomord nung sortierte K-Basis ist : B = `,B);\n #\n \+ # jetzt wird die Matrix \"Mf\" bestimmt :\n #\n \+ ff := f;\n Mf := matrix(r,r,0);\n \+ for i to r do\n fff := ff*KK[i];\n \+ R||i := normalf(fff,GG,torder);\n wh ile R||i <> 0 do\n l := leadmon(R||i,torder); \n for k to r do if KK[k] = l[2] then Mf[k,i] := l[1]; k := r fi od;\n R||i := R||i-l[1]*l[ 2]\n od\n od;\n #p rint(`Die Matrix` ,` M`[f],` der Multiplikation mit f` = f,` in `,'K '*XX*`/ I`,\n # ` bezueglich dieser K-Basis ist` );\n evalm(Mf);\n end:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 557 "zdimradical := proc(PList,VList)\n l ocal p,pred,v,RList;\n if is_finite(PList,VList) the n\n RList := PList;\n for \+ v in VList do\n p := univpoly(v,PList,VList) ;\n pred := simplify(p/gcd(p,diff(p,v)));\n \+ RList := [op(RList),pred]\n od ;\n RETURN(RList)\n else ERROR (`Das vorgegebene Ideal ist nicht nulldimensional !!!`)\n \+ fi\n end:" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 67 "Das Ideal f\374r diese Aufgabe sei er zeugt von folgender Polynomliste " }{XPPEDIT 18 0 "GI;" "6#%#GIG" } {TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 295 "X:=[x[1],x[2],x[3 ],x[4]];\ng1:=x[3]^2+2*x[1]-4*x[3]-x[4]+4;\ng2:=x[2]^2-2*x[2]+1;\ng3:= -6*x[3]-x[4]^2+x[4]+4+2*x[4]*x[3]+2*x[1];\ng4:= 4*x[4]*x[1]-x[4]^2+4*x [3]-3*x[4]-6*x[1];\ng5:=4*x[3]*x[1]-x[4]^2-4*x[3]+5*x[4]-6*x[1];\ng6:= x[1]^2-x[1]+x[3]-x[4];\ng7:=-18*x[1]+2*x[4]^3-21*x[4]^2+79*x[4]-48-12* x[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "GI:= ['g1','g2','g 3','g4','g5','g6','g7'];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 33 "Die Standardmonombasis BI f\374r I :" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "BI:=Kbasis(GI,X,tdeg(op(X)));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "B erechnung des Radikalideals VI, der reduzierten Gr\366bnerbasis GVI un d der zugeh\366rigen Standardmonombasis BVI :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "GVI:=map(sort,gbasis(zdimradical(GI,X),tdeg(op(X))),t deg(op(X)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "BVI:=Kbasis (GVI,X,tdeg(op(X)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Am Ergebn is liest man ab, da\337 I kein Radikalideal ist. Warum ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Wir arbeiten mit \+ " }{XPPEDIT 18 0 "VI;" "6#%#VIG" }{TEXT -1 12 " bzw. mit " } {XPPEDIT 18 0 "BVI;" "6#%$BVIG" }{TEXT -1 9 " weiter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Wahl einer Linearfor m " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 62 " mit zuf\344lligen gan zzahligen Koeffizienten im Bereich [-a,a] :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "a:=10000:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "c:=ran d((-a)...a):f:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "for i to 4 do \+ f:=f+c()*x[i];od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f:=sor t(f,[x[4],x[3],x[2],x[1]],tdeg):'f'=f;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Berechnung von " } {XPPEDIT 18 0 "M[f];" "6#&%\"MG6#%\"fG" }{TEXT -1 29 " und der Transpo nierten von " }{XPPEDIT 18 0 "M[f];" "6#&%\"MG6#%\"fG" }{TEXT -1 2 " \+ :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "tMf:=transpose(holMatrix(f,BVI ,GVI,X,tdeg(op(X))));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "Berechnung einer Eigenvektorbasis f\374r " }{XPPEDIT 18 0 "tMf;" "6#%$tMfG" }{TEXT -1 2 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvects(tMf);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "Jetzt sind Sie dran ! " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Bestimmen Sie die L\366 sungen des durch " }{XPPEDIT 18 0 "GI;" "6#%#GIG" }{TEXT -1 91 " vorg elegten algebraischen Gleichungssystems mit dem nach Satz 8 beschriebe nen Verfahren ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }