{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {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 "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 256 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 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 "" -1 262 "" 0 1 0 0 42 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 "" 0 1 115 109 105 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 7 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 2 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 20 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 107 169 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 44 0 84 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 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 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 47 "Musterloesung zur Aufgabe 5 aus der 17. Sitzung" }{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Hintergrund der Aufgabe:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Die Klasse der numerischen Verfahren vom " }{TEXT 279 1 "s" } {TEXT -1 87 "-stufigen Runge-Kutta-Typ zur Loesung eines gewoehnlichen Differentialgleichungssystems" }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "dy/dt = f(t,y) " "/*&%#dyG\"\"\"%#dtG!\"\"-%\"fG6$%\"tG %\"yG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "mit gegebener S tartbedingung " }{XPPEDIT 18 0 "y(t[0])=y[0]" "/-%\"yG6#&%\"tG6#\"\"!& F$6#F)" }{TEXT -1 34 " ist die folgende Approximation " }{XPPEDIT 18 0 "y[1]" "&%\"yG6#\"\"\"" }{TEXT -1 8 " fuer " }{XPPEDIT 18 0 "y( t[0]+h)" "-%\"yG6#,&&%\"tG6#\"\"!\"\"\"%\"hGF*" }{TEXT -1 2 " :" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y[1]:=y[0]+h*sum(b[j]* k[j],j=1..s)" ">&%\"yG6#\"\"\",&&F$6#\"\"!\"\"\"*&%\"hGF+-%$sumG6$*&&% \"bG6#%\"jGF+&%\"kG6#F5F+/F5;\"\"\"%\"sGF+F+" }{TEXT -1 2 " ," }} {PARA 0 "" 0 "" {TEXT -1 26 "wobei die Zwischenstufen " }{XPPEDIT 18 0 "k[j]" "&%\"kG6#%\"jG" }{TEXT -1 22 " definiert sind durch" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "k[i] = f*``(t[0]+c[i]* h, y[0]+h*sum(a[i,j]*k[j],j=1..s))*` `,` `*i=1.. s*` .`" "6$/&%\"kG6#% \"iG*(%\"fG\"\"\"-%!G6$,&&%\"tG6#\"\"!F**&&%\"cG6#F'F*%\"hGF*F*,&&%\"y G6#F2F**&F7F*-%$sumG6$*&&%\"aG6$F'%\"jGF*&F%6#FDF*/FD;\"\"\"%\"sGF*F*F *%\"~GF*/*&FKF*F'F*;\"\"\"*&FJF*%#~.GF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "Die unbekannten Parameter " }{XPPEDIT 18 0 "a[i,j], \+ b[j], c[i] " "6%&%\"aG6$%\"iG%\"jG&%\"bG6#F'&%\"cG6#F&" }{TEXT -1 29 " sind so zu bestimmen, dass " }{XPPEDIT 18 0 "y[1]-y(t[0]+h) = O(h^(p +1))" "/,&&%\"yG6#\"\"\"\"\"\"-F%6#,&&%\"tG6#\"\"!F(%\"hGF(!\"\"-%\"OG 6#)F0,&%\"pGF(\"\"\"F(" }{TEXT -1 10 " gilt, wo " }{XPPEDIT 18 0 "p" " I\"pG6\"" }{TEXT -1 79 " die Ordnung des Verfahrens ist. Durch Reihene ntwicklung nach der Schrittweite " }{XPPEDIT 18 0 "h" "I\"hG6\"" } {TEXT -1 80 " stellt man fest, dass Ordnung 4 erreicht wird, wenn die \+ folgenden 8 Gleichungen" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eq1:=sum (b[i1],i1=1..s)-1" }{TEXT -1 5 " (=0)" }{MPLTEXT 1 0 36 "; \neq2:=sum( b[i1]*c[i1],i1=1..s)-1/2" }{TEXT -1 5 " (=0)" }{MPLTEXT 1 0 38 "; \neq 3:=sum(b[i1]*c[i1]^2,i1=1..s)-1/3" }{TEXT -1 5 " (=0)" }{MPLTEXT 1 0 38 "; \neq4:=sum(b[i1]*c[i1]^3,i1=1..s)-1/4" }{TEXT -1 5 " (=0)" } {MPLTEXT 1 0 57 "; eq5:=sum(b[i1]*sum(a[i1,i2]*c[i2],i2=1..s),i1=1..s) -1/6" }{TEXT -1 5 " (=0)" }{MPLTEXT 1 0 64 "; \neq6:=sum(b[i1]*c[i1]*s um(a[i1,i2]*c[i2],i2=1..s),i1=1..s)-1/8" }{TEXT -1 5 " (=0)" } {MPLTEXT 1 0 60 "; eq7:=sum(b[i1]*sum(a[i1,i2]*c[i2]^2,i2=1..s),i1=1.. s)-1/12" }{TEXT -1 5 " (=0)" }{MPLTEXT 1 0 80 "; eq8:=sum(b[i1]*sum(a[ i1,i2]*sum(a[i2,i3]*c[i3],i3=1..s),i2=1..s),i1=1..s)-1/24" }{TEXT -1 5 " (=0)" }{MPLTEXT 1 0 1 ";" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "e rfuellt sind. Zusaetzlich muss " }{XPPEDIT 18 0 "c[i]=sum(a[i,j],j=1. .s)" "/&%\"cG6#%\"iG-%$sumG6$&%\"aG6$F&%\"jG/F-;\"\"\"%\"sG" }{TEXT -1 2 " " }{XPPEDIT 18 0 "``(i=1..s) " "-%!G6#/%\"iG;\"\"\"%\"sG" } {TEXT -1 10 " gelten. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Setzt man die Matrix " }{XPPEDIT 18 0 "a[i,j]" "&% \"aG6$%\"iG%\"jG" }{TEXT -1 55 " als untere Dreiecksmatrix voraus, so \+ lassen sich die " }{TEXT 278 1 "s" }{TEXT -1 17 " Zwischenstufen " } {XPPEDIT 18 0 "k[1],`..`,k[s]" "6%&%\"kG6#\"\"\"%#..G&F$6#%\"sG" } {TEXT -1 55 " leicht rekursiv berechen : das Verfahren heisst dann " }{TEXT 277 8 "explizit" }{TEXT -1 42 ". Betrachte 4-stufige explizite \+ Verfahren:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "s:=4: a:=matrix(s,s): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "for i to s do\n for j from i to s do\n a[i,j]:=0;\n od;\nod;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "p rint(a);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Die Gleichungen " } {XPPEDIT 18 0 "c[i]=sum(a[i,j],j=1..s)" "/&%\"cG6#%\"iG-%$sumG6$&%\"aG 6$F&%\"jG/F-;\"\"\"%\"sG" }{TEXT -1 78 " lassen sich durch eine geei gnete Parametrisierung trivialerweise erfuellen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c[1]:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for i t o 4 do a[i,1]:=c[i]-sum(a[i,'j'],'j'=2..s):od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(a);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Die \+ 10 Parameter des Verfahrens sind damit " }{XPPEDIT 18 0 "b[1],b[2],b[ 3],b[4],c[2],c[3],c[4],a[3,2],a[4,2],a[4,3] " "6,&%\"bG6#\"\"\"&F$6#\" \"#&F$6#\"\"$&F$6#\"\"%&%\"cG6#\"\"#&F16#\"\"$&F16#\"\"%&%\"aG6$\"\"$ \"\"#&F;6$\"\"%\"\"#&F;6$\"\"%\"\"$" }{TEXT -1 54 " , die fuer Ordnung 4 zu erfuellenden Gleichungen sind" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "for i to 8 do eq.i; od; " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 10 "Aufgabe 5:" }{TEXT -1 105 " Bestimme alle 4-stufigen expliziten R unge-Kutta-Verfahren der Ordnung 4. Das heisst im Klartext: finde " } {TEXT 257 4 "alle" }{TEXT -1 28 " Loesungen der 8 Gleichungen" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eq1:=b[1]+b[2] +b[3] +b [4] - 1 ;" }{MPLTEXT 0 21 5 "(= 0)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eq2:= b[2]*c[2] +b[3]*c[3] +b[4]*c[4] -1/2;" } {MPLTEXT 0 21 5 "(= 0)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eq3:= \+ b[2]*c[2]^2+b[3]*c[3]^2+b[4]*c[4]^2-1/3;" }{MPLTEXT 0 21 5 "(= 0)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eq4:= b[2]*c[2]^3+b[3]*c[3]^3+b [4]*c[4]^3-1/4;" }{MPLTEXT 0 21 5 "(= 0)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eq5:=b[3]*a[3,2]*c[2]+b[4]*a[4,2]*c[2]+b[4]*a[4,3]*c[ 3]-1/6;" }{MPLTEXT 0 21 5 "(= 0)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "eq6:=b[3]*a[3,2]*c[3]*c[2]+b[4]*a[4,2]*c[4]*c[2]+b[4]*a[4,3]*c[4]*c[3 ]-1/8;" }{MPLTEXT 0 21 5 "(= 0)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " eq7:=b[3]*a[3,2]*c[2]^2+b[4]*a[4,2]*c[2]^2+b[4]*a[4,3]*c[3]^2-1/12;" } {MPLTEXT 0 21 5 "(= 0)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eq8:=b[4] *a[4,3]*a[3,2]*c[2]-1/24;" }{MPLTEXT 0 21 5 "(= 0)" }}{PARA 0 "" 0 "" {TEXT -1 23 "in den 10 Unbekannten " }{XPPEDIT 18 0 "b[1],b[2],b[3],b [4],c[2],c[3],c[4],a[3,2],a[4,2],a[4,3] " "6,&%\"bG6#\"\"\"&F$6#\"\"#& F$6#\"\"$&F$6#\"\"%&%\"cG6#\"\"#&F16#\"\"$&F16#\"\"%&%\"aG6$\"\"$\"\"# &F;6$\"\"%\"\"#&F;6$\"\"%\"\"$" }{TEXT -1 3 " !" }{MPLTEXT 1 0 0 "" } }{PARA 0 "" 0 "" {TEXT -1 36 "Das zu loesende Gleichungssystem ist" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqs:=\{seq(eq.i,i=1..8)\};" }} {PARA 0 "" 0 "" {TEXT -1 20 "mit den Unbekannten:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 70 "unbekannte:=[a[3,2],a[4,2],a[4,3],b[1],b[2],b[3],b[ 4],c[2],c[3],c[4]];" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Anleitung :" }}{EXCHG {PARA 0 "" 0 "" {TEXT 260 10 "Anleitung:" }{TEXT -1 73 " \+ Das Gleichungssystem ist zu komplex, um direkt mit den Hilfsmittel des " }{TEXT 265 7 "grobner" }{TEXT -1 195 "-Pakets geloest zu werden. Zu naechst muss die Anzahl der Unbekannten reduziert werden. Die Tatsache , dass die Gleichungen linear in einigen der Unbekannten sind, ist hie rbei hilfreich. Betrachte " }{TEXT 262 4 " eq1" }{TEXT -1 1 "," } {TEXT 259 12 "eq2,eq3,eq4 " }{TEXT -1 46 ". Dies ist ein lineares Glei chungssystem fuer " }{XPPEDIT 18 0 "b[1],b[2],b[3],b[4]" "6&&%\"bG6#\" \"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 15 " mit der von " } {XPPEDIT 18 0 "c[1]:=0,c[2],c[3],c[4]" ">&%\"cG6#\"\"\"6&\"\"!&F$6#\" \"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 62 " erzeugten Vandermonde-Matrix. \+ Es ist eindeutig in Termen von " }{XPPEDIT 18 0 "c[2], c[3], c[4]" "6% &%\"cG6#\"\"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 16 " loesbar, wenn " } {XPPEDIT 18 0 "c[1]:=0,c[2],c[3],c[4]" ">&%\"cG6#\"\"\"6&\"\"!&F$6#\" \"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 103 " paarweise verschieden sind. \+ Untersuche zunaechst systematisch alle Faelle, in denen mindestens 2 d er " }{XPPEDIT 18 0 "c[i]" "&%\"cG6#%\"iG" }{TEXT -1 31 " uebereinstim men (Hilfsmittel: " }{TEXT 263 8 "solvable" }{TEXT -1 9 " aus dem " } {TEXT 264 7 "grobner" }{TEXT -1 71 "-Paket). Zeige, dass Loesungen nur in den folgenden Faellen existieren:" }}{PARA 0 "" 0 "" {TEXT -1 9 "F all 1) " }{XPPEDIT 18 0 "c[3]=c[1] *`= `*0" "/&%\"cG6#\"\"$*(&F$6#\" \"\"\"\"\"%#=~GF+\"\"!F+" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 9 "Fall 2) " }{XPPEDIT 18 0 "c[3] =c[2] *`<>`*0" "/&%\"cG6#\"\"$*( &F$6#\"\"#\"\"\"%#<>GF+\"\"!F+" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 9 "Fall 3) " }{XPPEDIT 18 0 "c[4]=c[2]*`<>`*0" "/&%\"cG6#\" \"%*(&F$6#\"\"#\"\"\"%#<>GF+\"\"!F+" }{TEXT -1 2 " ," }}{PARA 0 "" 0 " " {TEXT -1 9 "Fall 4) " }{XPPEDIT 18 0 "c[1]=0,c[2], c[3], c[4]" "6&/ &%\"cG6#\"\"\"\"\"!&F%6#\"\"#&F%6#\"\"$&F%6#\"\"%" }{TEXT -1 29 " sin d paarweise verschieden." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 68 "In den Faellen 1,2,3 kann mit den Hilfsmitteln des Groebner-Pakets " }{TEXT 258 7 "grobner" }{TEXT -1 90 " jeweils lei cht eine 1-parametrige Schar von Loesungen gefunden werden (verwende d ie in " }{TEXT 269 11 "unbekannte " }{TEXT -1 38 " angegebene Ordnung der Unbekannten). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 24 "Fall 4: Die Gleichungen " }{TEXT 261 12 "eq5,eq6,eq7 " }{TEXT -1 7 "stellen" }{TEXT 266 9 " lineare " }{TEXT -1 11 "Gleichung en" }{TEXT 268 1 " " }{TEXT -1 6 "fuer " }{XPPEDIT 18 0 "a[3,2], a[4, 2], a[4,3]" "6%&%\"aG6$\"\"$\"\"#&F$6$\"\"%\"\"#&F$6$\"\"%\"\"$" } {TEXT -1 79 " dar, welche hiermit eliminiert werden koennen (zeige, d ass notwendigerweise " }{XPPEDIT 18 0 "b[3]<>0" "0&%\"bG6#\"\"$\"\"! " }{TEXT -1 1 " " }{XPPEDIT 18 0 "``<>b[4]" "0%!G&%\"bG6#\"\"%" } {TEXT -1 31 " gelten muss). Dann eliminiere " }{XPPEDIT 18 0 "b[1],b[2 ], b[3], b[4]" "6&&%\"bG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" } {TEXT -1 34 " mittels des Vandermonde-Systems " }{TEXT 267 15 "eq1,eq 2,eq3,eq4" }{TEXT -1 25 " . Die letzte Gleichung " }{TEXT 270 3 "eq8 " }{TEXT -1 37 " liefert dann eine Bedingung an die " }{XPPEDIT 18 0 "c[i]" "&%\"cG6#%\"iG" }{TEXT -1 2 " ." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Zeige, das s notwendigerweise " }{XPPEDIT 18 0 "b[3]<>0" "0&%\"bG6#\"\"$\"\"!" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``<>b[4]" "0%!G&%\"bG6#\"\"%" }{TEXT -1 13 " gelten muss:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grobner[sol vable](subs(b[3]=0,eqs));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "grobne r[solvable](subs(b[4]=0,eqs));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Untersuche die Faelle " }{XPPEDIT 18 0 "c[i]=c[j]" "/&%\"cG6#%\"iG&F $6#%\"jG" }{TEXT -1 36 " , die zu Loesungen fuehren koennen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c[1]:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "for i to 4 do\n for j from i+1 to 4 do\n grobner[solvable]( subs(c[j]=c[i],eqs));\n print(`loesbar fuer `,c[j]=c[i],` :`*\");\n od;\nod;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Fall 1: " } {XPPEDIT 18 0 "c[3] = c[1]*``(`=0`)" "/&%\"cG6#\"\"$*&&F$6#\"\"\"\"\" \"-%!G6#%#=0GF+" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "grobner[g solve](subs(c[3]=0,eqs),subs(c[3]=NULL,unbekannte));" }}{PARA 0 "" 0 " " {TEXT -1 15 "Mit beliebigem " }{XPPEDIT 18 0 "b[3]<>0" "0&%\"bG6#\" \"$\"\"!" }{TEXT -1 54 " ergibt sich hieraus eine 1-parametrige Loesun gsschar:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "solve(\{op(op(\"))\},su bs(c[3]=NULL,b[3]=NULL,\{op(unbekannte)\}));" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 276 5 "Test:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "normal(subs (c[3]=0,\",eqs));" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Fall 2: " }{XPPEDIT 18 0 "c[3] = c[2]" "/&%\"cG6#\"\"$&F$6#\"\"#" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "grobner[gsolve](subs(c[3]=c[2],eqs) ,subs(c[3]=NULL,unbekannte));" }}{PARA 0 "" 0 "" {TEXT -1 15 "Mit beli ebigem " }{XPPEDIT 18 0 "b[3]<>0" "0&%\"bG6#\"\"$\"\"!" }{TEXT -1 54 " ergibt sich hieraus eine 1-parametrige Loesungsschar:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "solve(\{op(op(\"))\},subs(c[3]=c[2],b[3]=NULL, \{op(unbekannte)\}));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 5 "Test:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "normal(subs(c[3]=c[2],\",eqs));" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Fall 3: " }{XPPEDIT 18 0 "c[4] = c[2]" "/&%\"cG6#\"\"%&F$6#\"\"#" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "grobner[gsolve](subs(c[4]=c[2],eqs),subs(c[4]=NULL,un bekannte));" }}{PARA 0 "" 0 "" {TEXT -1 15 "Mit beliebigem " } {XPPEDIT 18 0 "b[4]<>0" "0&%\"bG6#\"\"%\"\"!" }{TEXT -1 54 " ergibt si ch hieraus eine 1-parametrige Loesungsschar:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "solve(\{op(op(\"))\},subs(c[4]=c[2],b[4]=NULL,\{op(un bekannte)\}));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 5 "Test:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "normal(subs(c[4]=c[2],\",eqs));" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Fall 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Die Gleichungen " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "e q5;eq6;eq7;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "aufgefasst als lin eare Gleichungen fuer " }{XPPEDIT 18 0 "a[3,2],a[4,2],a[4,3] " "6%&% \"aG6$\"\"$\"\"#&F$6$\"\"%\"\"#&F$6$\"\"%\"\"$" }{TEXT -1 46 " , sind \+ im Fall 4 eindeutig loesbar (beachte " }{XPPEDIT 18 0 "b[3]<>0" "0&% \"bG6#\"\"$\"\"!" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``<>b[4]" "0%!G&%\"b G6#\"\"%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[1]=0,c[2],c[3],c[4]" "6&/ &%\"cG6#\"\"\"\"\"!&F%6#\"\"#&F%6#\"\"$&F%6#\"\"%" }{TEXT -1 65 " paar weise verschieden ). Die Determinante der Koeffizienten ist:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "matrix([\n [coeff(eq5,a[3,2]),coeff(eq 5,a[4,2]),coeff(eq5,a[4,3])],\n [coeff(eq6,a[3,2]),coeff(eq6,a[4,2]) ,coeff(eq6,a[4,3])],\n [coeff(eq7,a[3,2]),coeff(eq7,a[4,2]),coeff(eq 7,a[4,3])] \n]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(linalg[d et](\"));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Loese" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "a_eqs:=factor(solve(\{eq5,eq6,eq7\},\{a[3,2],a [4,2],a[4,3]\}));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Das Vandermo nde-System " }{TEXT 271 15 "eq1,eq2,eq3,eq4" }{TEXT -1 8 " fuer " } {XPPEDIT 18 0 "b[1],b[2],b[3],b[4] " "6&&%\"bG6#\"\"\"&F$6#\"\"#&F$6# \"\"$&F$6#\"\"%" }{TEXT -1 34 " ist in Fall 4 eindeutig loesbar:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "b_eqs:=factor(solve(\{eq1,eq2,eq3,e q4\},\{b[1],b[2],b[3],b[4]\}));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Einsetzen von " }{XPPEDIT 18 0 "a[3,2],a[4,2],a[4,3],b[1],b[2],b[3 ],b[4]" "6)&%\"aG6$\"\"$\"\"#&F$6$\"\"%\"\"#&F$6$\"\"%\"\"$&%\"bG6#\" \"\"&F16#\"\"#&F16#\"\"$&F16#\"\"%" }{TEXT -1 27 " in das Gleichungss ystem " }{TEXT 272 3 "eqs" }{TEXT -1 3 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "factor(subs(a_eqs,b_eqs,eqs));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Da " }{XPPEDIT 18 0 "c[2]<>0" "0&%\"cG6#\"\"#\"\"!" }{TEXT -1 21 " gelten muss, folgt " }{XPPEDIT 18 0 "c[4]=1" "/&%\"cG6 #\"\"%\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 66 "Somit er gibt sich die folgende 2-parametrige Schar von Loesungen :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "factor(subs(c[4]=1,b_eqs));" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "factor(subs(c[4]=1,\",a_eqs));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 5 "Test:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "normal(subs(\",\"\",c[4]=1,eqs));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "4" 0 }{VIEWOPTS 1 1 0 3 2 1804 }