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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 52 "Numerik fuer Anfangswert
aufgaben - 5. Uebungsblatt \n" }{TEXT 256 0 "" }{TEXT -1 53 "Christine
 Schweinem, mit Ergaenzungen von Lars Gruene" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Aufgabe 1" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart; with(linalg): with(plots):
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 
7 "Gegeben" }{TEXT -1 40 ": 3 RK-Verfahren, Butcher-Tableaus s. u." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Verfahren
 " }{TEXT 257 3 "(1)" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 76 "A1:=matrix(2,2,[[0,0],[2/3,0]]); c1:=vector([0,2/3]);
 b1:=vector([1/4,3/4]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 10 "Verfahren " }{TEXT 258 3 "(2)" }{TEXT -1 
1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "A2:=matrix(3,3,[[0,
0,0],[2/3,0,0],[0,2/3,0]]);\nc2:=vector([0,2/3,2/3]); b2:=vector([1/4,
3/8,3/8]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "Verfahren " }{TEXT 259 3 "(3)" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "A3:=matrix(3,3,[[5/36,2/9-sqrt(15)
/15,5/36-sqrt(15)/30],\n                [5/36+sqrt(15)/24,2/9,5/36-sqr
t(15)/24],\n                [5/36+sqrt(15)/30,2/9+sqrt(15)/15,5/36]]);
\nc3:=vector([1/2-sqrt(15)/10,1/2,1/2+sqrt(15)/10]); b3:=vector([5/18,
4/9,5/18]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 55 "Explizites und implizites Eulerverfahren zum Vergleich:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "A.eulx:=matrix(1,1,[0])
; b.eulx:=vector(1,[1]);\nA.eulm:=matrix(1,1,[1]); b.eulm:=vector(1,[1
]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
261 7 "Gesucht" }{TEXT -1 44 ": Die A-Stabilitaetsgebiete der 3 Verfah
ren." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "D
afuer muessen wir die Funktion " }{XPPEDIT 18 0 "R(z);" "6#-%\"RG6#%\"
zG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1+z*b(Id-z*A)^(-1)*Eins;" "6#,&
\"\"\"\"\"\"*(%\"zGF%)-%\"bG6#,&%#IdGF%*&F'F%%\"AGF%!\"\",$\"\"\"F0F%%
%EinsGF%F%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "z = x+iy;" "6#/%\"zG,&%\"
xG\"\"\"%#iyGF'" }{TEXT -1 5 " aus " }{XPPEDIT 18 0 "C;" "6#%\"CG" }
{TEXT -1 9 ", kennen." }}{PARA 0 "" 0 "" {TEXT -1 12 "Hierbei ist " }
{XPPEDIT 18 0 "Id;" "6#%#IdG" }{TEXT -1 34 " eine passende Einheitsmat
rix und " }{XPPEDIT 18 0 "Eins;" "6#%%EinsG" }{TEXT -1 26 " ein passen
der Einsvektor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 15 "Definition von " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 
1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Id:=n->matrix(n,n,(
i,j)->if i=j then 1 else 0 fi):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "
Eins:=n->vector(n,[1$n]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "R:=(i
,z)->simplify(1+z*multiply(b.i,inverse(matadd(Id(rowdim(A.i)),scalarmu
l(A.i,z),1,-1)),Eins(rowdim(A.i)))):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 96 "'R(1,z)'=R(1,z); 'R(2,z)'=R(2,z); 'R(3,z)'=R(3,z); 'R
(eulx,z)'=R(eulx,z); 'R(eulm,z)'=R(eulm,z);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 9 "z:=x+I*y;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 62 "Jetzt muessen wir herausfinden, wie die
 A-Stabilitaetsgebiete " }{XPPEDIT 18 0 "S[R];" "6#&%\"SG6#%\"RG" }
{TEXT -1 4 " = \{" }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 5 " aus " }
{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "abs(R(z)
) < 1;" "6#2-%$absG6#-%\"RG6#%\"zG\"\"\"" }{TEXT -1 11 "\} aussehen." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Leider \+
habe ich keine Lust, die Gleichungen " }{XPPEDIT 18 0 "betrag.i < 1;" 
"6#2(%'betragG%\"iG\"\"\"" }{TEXT -1 60 " wirklich zu loesen, aber zum
indest die Raender der Gebiete " }{XPPEDIT 18 0 "S[R];" "6#&%\"SG6#%\"
RG" }{TEXT -1 14 " kann ich mit " }{TEXT 266 14 "implicitplot()" }
{TEXT -1 10 " zeichnen." }}{PARA 0 "" 0 "" {TEXT 265 0 "" }{TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 15 "Fuer Verfahren " }{TEXT 262 3 "(1)" 
}{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "betrag1:=
factor(evalc(abs(R(1,z))));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "solve(betrag1=1,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 
"implicitplot(betrag1<1,x=-10..10,y=-10..10,color=cyan,thickness=3,num
points=10000,scaling=constrained,title=\"Verfahren (1)\");" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 16 " Fuer Verfahren " }{TEXT 263 3 "(2)" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "betrag2:=evalc(abs(R(2,z)));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(betrag2=1,y);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 81 "S
ieht gruselig aus, beschreibt aber offenbar den Rand des A-Stabilitaet
sbereiches" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 125 "implicitplot(betrag2<1,x=-10..10,y=-10..10,color=c
yan,thickness=3,numpoints=10000,scaling=constrained,title=\"Verfahren \+
(2)\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 15 "Fuer Verfahren " }{TEXT 264 3 "(3)" }
{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "betrag3:=s
implify(evalc(abs(R(3,z))));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "solve(betrag3=1, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
125 "implicitplot(betrag3<1,x=-20..20,y=-20..20,color=cyan,thickness=5
,numpoints=10000,scaling=constrained,title=\"Verfahren (3)\");" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 32 "Fuer die beiden Euler-Verfahren:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 152 "implicitplot(evalc(abs(R(eulx,z)))<1,x=-10..10,
y=-10..10,color=navy,thickness=3,numpoints=10000,scaling=constrained,t
itle=\"Explizites Euler-Verfahren\");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 151 "implicitplot(evalc(abs(R(eulm,z)))<1,x=-10..10,y=-10
..10,color=navy,thickness=3,numpoints=9500,scaling=constrained,title=
\"Implizites Euler-Verfahren\");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Aufgabe 2" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "z:=evaln(z):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 7 "Gegeben" 
}{TEXT -1 40 ": 2 RK-Verfahren, Butcher-Tableaus s. u." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Verfahren " }{TEXT 
267 3 "(a)" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
80 "Aa:=matrix(2,2,[[0,0],[1/2,1/2]]); ca:=vector(2,[0,1]); ba:=vector
(2,[1/2,1/2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 10 "Verfahren " }{TEXT 269 3 "(b)" }{TEXT -1 1 ":" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "Ab:=matrix(2,2,[[(2-sqrt(2)
)/2,0],[sqrt(2)/2,(2-sqrt(2))/2]]); cb:=vector(2,[(2-sqrt(2))/2,1]); b
b:=vector(2,[sqrt(2)/2,(2-sqrt(2))/2]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 5 "Frage" }{TEXT -1 37 ": Si
nd die beiden Verfahren A-stabil?" }}{PARA 0 "" 0 "" {TEXT -1 103 "   \+
        (Diese Frage soll entschieden werden, ohne die A-Stabilitaetsg
ebiete explizit zu bestimmen.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 271 22 "Satz aus der Vorlesung" }{TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "abs(R(
z)) < 1;" "6#2-%$absG6#-%\"RG6#%\"zG\"\"\"" }{TEXT -1 8 "  f. a. " }
{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 5 " aus " }{XPPEDIT 18 0 "C_min
us;" "6#%(C_minusG" }{TEXT -1 4 " <=>" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 9 "         " }{TEXT 273 1 "i" }{TEXT 
274 1 ")" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "Re(z[pol]);" "6#-%#ReG6#&%
\"zG6#%$polG" }{TEXT -1 24 " > 0 fuer jeden Pol von " }{XPPEDIT 18 0 "
R;" "6#%\"RG" }}{PARA 0 "" 0 "" {TEXT -1 36 "                         \+
        und" }}{PARA 0 "" 0 "" {TEXT -1 9 "         " }{TEXT 272 2 "ii
" }{TEXT 275 1 ")" }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(R(I*t)) <= 1;" 
"6#1-%$absG6#-%\"RG6#*&%\"IG\"\"\"%\"tGF,\"\"\"" }{TEXT -1 7 " f. a. \+
" }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 5 " aus " }{XPPEDIT 18 0 "R;
" "6#%\"RG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 42 "Wir wissen weiter aus der Vorlesung, dass " }
{XPPEDIT 18 0 "R(z) = P(z)/Q(z);" "6#/-%\"RG6#%\"zG*&-%\"PG6#F'\"\"\"-
%\"QG6#F'!\"\"" }{TEXT -1 8 ", wobei " }{XPPEDIT 18 0 "P;" "6#%\"PG" }
{TEXT -1 5 " und " }{XPPEDIT 18 0 "Q;" "6#%\"QG" }{TEXT -1 22 " Polyno
me vom Grad <= " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 36 " (Anzahl \+
der Verfahrens-Stufen) und " }{XPPEDIT 18 0 "P(0)-Q(0) = 1;" "6#/,&-%
\"PG6#\"\"!\"\"\"-%\"QG6#F(!\"\"\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "P(z);" "6#-%\"PG
6#%\"zG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "det(Id-z*A)+z*b*Adjungierte
(Id-z*A)*Eins;" "6#,&-%$detG6#,&%#IdG\"\"\"*&%\"zGF)%\"AGF)!\"\"F)**F+
F)%\"bGF)-%,AdjungierteG6#,&F(F)*&F+F)F,F)F-F)%%EinsGF)F)" }{TEXT -1 
0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "Q(Z);" "6#-%\"QG6#%\"ZG" }
{TEXT -1 3 " = " }{XPPEDIT 18 0 "det(Id-z*A);" "6#-%$detG6#,&%#IdG\"\"
\"*&%\"zGF(%\"AGF(!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "P:=(i,z)->  det( matad
d(Id(rowdim(A.i)),scalarmul(A.i,z),1,-1) )\n                          \+
         +\nz*multiply( b.i,adjoint(matadd(Id(rowdim(A.i)),scalarmul(A
.i,z),1,-1)),Eins(rowdim(A.i)) ):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "Q:=(i,z)->det(matadd(Id(rowdim(A.i)),scalarmul(A.i,z)
,1,-1)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 282 69 "Pole sind Nullstellen des Nenners Q(z), aber nicht des Z
aehlers P(z):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 74 "Polmenge_a:=solve(\{Q(a,pol_a)=0,P(a,pol_a)<>0\}
,pol_a); assign(Polmenge_a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 74 "Polmenge_b:=solve(\{Q(b,pol_b)=0,P(b,pol_b)<>0\},pol_b); assign(
Polmenge_b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 80 "Beide Pole liegen auf der reellen Achse, die Vorzeichen
 der Realteile sind also:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "VZa=signum(pol_a); VZb=signum(pol_b);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 6 "D. h. " }{TEXT 276 1 "i" 
}{TEXT 277 1 ")" }{TEXT 285 39 " gilt, und die Verfahren koennen damit
 " }{TEXT -1 1 "b" }{TEXT 286 19 "eide A-stabil sein." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 
259 "" 0 "" {TEXT -1 72 "Jetzt ist noch der Wert fuer R mit rein imagi
naerem Argument zu testen. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "R:=(i, z)-> P(i,z)/Q(i,z);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assume(x, real);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(abs(R(a, I*x)));" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 8 "
Also ok." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "assume(x, real);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "simplify(abs(R(b, I*x)));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "solve(%=1, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 279 97 "Wegen Stetigkeit reicht es also n
achzuweisen, dass |R(b, I*x)| fuer x<0 und x>0 kleiner als 1 ist" }}
{PARA 0 "" 0 "" {TEXT 283 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "evalf(abs(R(b, I*1)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "evalf(abs(R(b, -I*1)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 280 18 "Also ebenfalls ok!" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}}}{MARK "2 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }