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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Digits:=20:\n\n\n" }{TEXT 
-1 42 "Die Differentialgleichung                 " }{XPPEDIT 18 0 "dif
f(y(t),t) = -y(t);" "6#/-%%diffG6$-%\"yG6#%\"tGF*,$-F(6#F*!\"\"" }
{TEXT -1 42 "           mit der Anfangsbedingung       " }{XPPEDIT 18 
0 "y(0) = 1;" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 3 "  :" }{MPLTEXT 1 
0 56 " \n            \node1:=diff(y(t),t)=-y(t):\ninit1:=y(0)=1:\n" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "      Diese Gleichung wird mit dr
ei verschiedenen numerischen Methoden gel\366st: " }{TEXT 286 93 "Gear
-Verfahren mit rationaler Extrapolation, Gear-Verfahren mit   \n      \+
Polynomextrapolation" }{TEXT -1 5 " und " }{TEXT 287 16 "Euler-Verfahr
en." }{MPLTEXT 1 0 1 "\n" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 295 "answer1:=dsolve(\{ode1, init1\}, y(t),type=numeric,method=gea
r[bstoer],stepsize=1,maxord=6,maxpts=10):\nanswer2:=dsolve(\{ode1, ini
t1\}, y(t),type=numeric,method=gear[polyextr],stepsize=1,maxord=6,maxp
ts=10):\nanswer3:=dsolve(\{ode1, init1\}, y(t),type=numeric,method=cla
ssical[foreuler],stepsize=0.1):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
141 "     Graphische Darstellung der numerischen L\366sungen und die g
enaue L\366sung - es ist kaum Unterschied zwischen verschidenen Kurven
 bemerkbar.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 340 "with(plots):\nwit
h(DEtools):\nG:=DEplot(ode1,y(t),t=0..1,[[y(0)=1]],arrows=NONE,thickne
ss=1,linecolor=blue,title=`L\366sungen der Differentialgleichung diff(
y(t),t)=-y(t)`,titlefont=[TIMES,ITALIC,12]):\nF1:=odeplot(answer1,[t,y
(t)],0..1,color=red):\nF2:=odeplot(answer2,[t,y(t)],0..1,color=black):
\nF3:=odeplot(answer3,[t,y(t)],0..1,color=yellow):\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "display(G,F1,F2,F3);\n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 235 "     Erst im kleinen Intervall links von Eins \+
wird deutlich, da\337 die Kurve, die die L\366sung nach Euler-Verfahre
n darstellt, von der genauen L\366sung abweicht, \n     w\344hrend die
 anderen beiden Kurven mit der genauen L\366sung zusammenfallen: " }
{MPLTEXT 1 0 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 608 "G:=DEplot(od
e1,y(t),t=0.9..1,[[y(0)=1]],arrows=NONE,thickness=1,linecolor=blue):\n
T1:=PLOT(TEXT([0.925,0.4],\n'`Exakte L\366sung und `',ALIGNRIGHT,FONT(
TIMES,BOLDITALIC,10),COLOR(HUE,0.7)),\n         TEXT([0.93,0.398],\n'`
die beiden L\366sungen nach Gear `',ALIGNRIGHT,FONT(TIMES,BOLDITALIC,1
0),COLOR(HUE,0.7)),\n         TEXT([0.9,0.39],'`L\366sung mit`',ALIGNR
IGHT,FONT(TIMES,BOLDITALIC,10)),\n         TEXT([0.905,0.388],'`Euler`
',ALIGNRIGHT,FONT(TIMES,BOLDITALIC,10))):\nF1:=odeplot(answer1,[t,y(t)
],0.9..1,color=red):\nF2:=odeplot(answer2,[t,y(t)],0.9..1,color=black)
:\nF3:=odeplot(answer3,[t,y(t)],0.9..1,color=yellow):\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display(T1,G,F1,F2,F3);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 113 "      \n\n         EULER-Differentialgle
ichung:\n                                                             \+
      " }{XPPEDIT 18 0 "diff(x,t) = -y*z;" "6#/-%%diffG6$%\"xG%\"tG,$*
&%\"yG\"\"\"%\"zGF,!\"\"" }{TEXT -1 97 "              x(0) = 0       \+
\n                                                                   \+
" }{XPPEDIT 18 0 "diff(y,t) = -x*z;" "6#/-%%diffG6$%\"yG%\"tG,$*&%\"xG
\"\"\"%\"zGF,!\"\"" }{TEXT -1 90 "              y(0) = 1\n            \+
                                                       " }{XPPEDIT 18 
0 "diff(z,t) = -.5*x*y;" "6#/-%%diffG6$%\"zG%\"tG,$*($\"\"&!\"\"\"\"\"
%\"xGF.%\"yGF.!\"\"" }{TEXT -1 20 "          z(0) = 1\n\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "ode3:=diff(x(t),t)=-y(t)*z(t):\node
4:=diff(y(t),t)=-x(t)*z(t):\node5:=diff(z(t),t)=-0.5*x(t)*y(t):\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "Euler_Dgl:=dsolve(\{ode3,od
e4,ode5,x(0)=0,y(0)=1,z(0)=1\},\{x(t),y(t),z(t)\},type=numeric,method=
gear[bstoer],maxord=6,maxpts=10,errorper=0.1):\n" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 482 "T2:=PLOT(TEXT([1.63,6],'`y(t)`',ALIGNRIGHT,F
ONT(TIMES,BOLDITALIC,10),COLOR(HUE,0.7)),\n         TEXT([1.7,-5.9],'`
x(t)`',ALIGNLEFT,FONT(TIMES,BOLDITALIC,10),COLOR(HUE,1.0)),\n         \+
TEXT([1.72,4.4],'`z(t)`',ALIGNLEFT,FONT(TIMES,BOLDITALIC,10))):\n\nH1:
=odeplot(Euler_Dgl,[t,x(t)],0..1.6,color=red,title=`L\366sungen der EU
LERschen Differentialgleichung`,titlefont=[TIMES,ITALIC,12]):\nH2:=ode
plot(Euler_Dgl,[t,y(t)],0..1.6,color=blue):\nH3:=odeplot(Euler_Dgl,[t,
z(t)],0..1.6,color=green):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "display(T2,H1,H2,H3);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "
          \n\n\n           BESSEL-Differentialgleichung ( f\374r " }
{XPPEDIT 18 0 "J[16];" "6#&%\"JG6#\"#;" }{TEXT -1 70 " ):\n           \+
                                                       " }{XPPEDIT 18 
0 "diff(x,t) = y;" "6#/-%%diffG6$%\"xG%\"tG%\"yG" }{TEXT -1 132 "     \+
                                   x(6) = 1.20195e-6       \n         \+
                                                          " }{XPPEDIT 
18 0 "diff(y,t) = (256/(t^2)-1)*x-y/t;" "6#/-%%diffG6$%\"yG%\"tG,&*&,&
*&\"$c#\"\"\"*$F(\"\"#!\"\"F.\"\"\"F1F.%\"xGF.F.*&F'F.F(F1F1" }{TEXT 
-1 31 "              y(6) = 2.98648e-6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "ode6:=diff(x(t),t)=y(t):\node7:=diff(y(t),t)=(256/t^2
-1)*x(t)-y(t)/t:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "Bess
el_Dgl:=dsolve(\{ode6,ode7,x(6)=1.20195e-6,y(6)=2.98648e-6\},\{x(t),y(
t)\},type=numeric,\\\nmethod=gear[bstoer],maxord=6,maxpts=10,errorper=
0.1):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 367 "T3:=PLOT(TEXT([
7,1.3e-05],'`x(t)`',ALIGNRIGHT,FONT(TIMES,BOLDITALIC,10),COLOR(HUE,1.0
)),\n         TEXT([7,2.55e-05],'`y(t)`',ALIGNRIGHT,FONT(TIMES,BOLDITA
LIC,10),COLOR(HUE,0.7))):\n\nJ1:=odeplot(Bessel_Dgl,[t,x(t)],4.6..7,co
lor=red,title=`L\366sungen der BESSELschen Differentialgleichung`,titl
efont=[TIMES,ITALIC,12]):\nJ2:=odeplot(Bessel_Dgl,[t,y(t)],4.6..7,colo
r=blue):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display(T3,J1
,J2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "     Dieser Graph stammt von vo
n " }{TEXT 256 6 "CLARK " }{TEXT -1 257 "(1966) und repr\344sentiert d
ie Abh\344ngigkeit der Anzahl der ben\366tigten Funktionsauswertungen \+
von der verlangten \n     Fehlerordnung. F\374r alle drei Differential
gleichungen wurde MAXORD auf 6 und MAXPTS auf 10 gesetzt.\n     Die er
ste Differentialgleichung       " }{TEXT 257 7 "y' = -y" }{TEXT -1 8 "
    ,   " }{TEXT 258 0 "" }{TEXT 259 8 "y(0) = 1" }{TEXT 260 0 "" }
{TEXT -1 23 "   wurde an der Stelle " }{TEXT 261 0 "" }{TEXT 262 8 "  \+
t = 20" }{TEXT 263 0 "" }{TEXT -1 57 "   integriert. Gezeigt ist nun d
er absolute Fehler bzgl. " }{XPPEDIT 18 0 "exp(-20);" "6#-%$expG6#,$\"
#?!\"\"" }{TEXT -1 45 ".\n     Die EULER-Dgl. wurde an der Stelle    \+
" }{TEXT 266 0 "" }{TEXT 267 7 "t = 60 " }{TEXT 268 0 "" }{TEXT -1 29 
"  integriert. Gezeichnet ist " }{TEXT 269 44 "quadratischer Mittelwer
t (root mean square) " }{TEXT -1 14 "der Fehler in " }{TEXT 270 0 "" }
{TEXT 271 21 " x(t) , y(t) , z(t).\n" }{TEXT 272 0 "" }{TEXT -1 83 "  \+
   Bei der BESSEL-Dgl. wurden die Fehler aufgenommen bzgl. den gr\366
\337ten Wert von  " }{TEXT 273 0 "" }{TEXT 274 4 "x(t)" }{TEXT 275 0 "
" }{TEXT -1 22 "  , der an der Stelle " }{TEXT 276 0 "" }{TEXT 277 9 "
t = 18.1 " }{TEXT 278 0 "" }{TEXT -1 32 "angenommen wird. Dort ist \n \+
    " }{TEXT 279 0 "" }{TEXT 280 15 "x(18.1) ~ 0.261" }{TEXT 282 0 "" 
}{TEXT -1 58 "... . Der Mittelwert der absoluten Fehlern an den Stelle
n " }{TEXT 283 0 "" }{TEXT 284 30 "t = 6132 , 6134 , 6136 , 6138 " }
{TEXT 285 0 "" }{TEXT -1 118 "ist eingetragen.\n\n     Man kann erkenn
en, da\337 rationale Extrapolation f\374r zwei von diesen Beispiele vo
rteilhafter ist." }{TEXT 281 0 "" }{TEXT -1 0 "" }{TEXT 264 0 "" }
{TEXT 265 0 "" }{TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1788 
"PLOT(\n     POINTS([-0.2,0.69],[-0.05,0.74],[-0.1,0.9],[0.55,0.55],[1
.1,0.45],[2.7,0.28],[3.3,0.2],\n            SYMBOL(CIRCLE)),\n     POI
NTS([0.05,1.1],[0.1,1.2],[0.3,1.05],[1.2,0.93],[1.6,0.86],[2.9,0.88],[
3.55,0.7],\n            SYMBOL(BOX)),\n     CURVES([[-0.25,0.7],[3.3,0
.2]],THICKNESS(2),LINESTYLE(1),COLOR(HUE,0.9)),\n     CURVES([[-0.25,1
.15],[3.9,0.7]],THICKNESS(2),LINESTYLE(1),COLOR(HUE,0.9)),\n     TEXT(
[4.2,0.5],'`y\264 = - y`',ALIGNRIGHT,FONT(TIMES,BOLDITALIC,13),COLOR(H
UE,0.9)),\n    \n     POINTS([0.3,1.78],[1.0,1.75],[1.08,1.67],[1.4,1.
75],[1.6,1.7],[1.78,1.67],[2.9,1.55],\n            [3.7,1.54],[4,1.5],
SYMBOL(CIRCLE)),\n     POINTS([0.9,1.8],[1.2,1.9],[1.25,1.8],[1.5,1.68
],[2.05,1.61],[2.6,1.67],\n            [3.05,1.53],[3.2,1.53],SYMBOL(B
OX)), \n     CURVES([[0.1,1.8],[4,1.5]],THICKNESS(2),LINESTYLE(1),COLO
R(HUE,0.7)),\n     TEXT([4.2,1.5],'`EULER-Dgl.`',ALIGNRIGHT,FONT(TIMES
,BOLDITALIC,11),COLOR(HUE,0.7)),\n     \n     POINTS([0.03,3.45],[1.55
,3.3],[2.6,3.2],[3.0,3.18],[3.8,3.1],SYMBOL(BOX)),\n     POINTS([1.55,
3.15],[2.0,3.1],[2.45,3.09],[2.85,3.06],[4.1,3.0],SYMBOL(CIRCLE)),  \n
     CURVES([[0,3.45],[3.8,3.1]],THICKNESS(2),LINESTYLE(1),COLOR(HUE,1
.0)),\n     CURVES([[1.3,3.15],[4.1,3]],THICKNESS(2),LINESTYLE(1),COLO
R(HUE,1.0)),\n     TEXT([4.2,3.25],'`BESSEL-Dgl.`',ALIGNRIGHT,FONT(TIM
ES,BOLDITALIC,11),COLOR(HUE,1.0)),\n     \n     AXESSTYLE(FRAME),VIEW(
-0.25..4.2,0..4),\n     AXESTICKS([0=`10e-10`,1=`10e-9`,2=`10e-8`,3=`1
0e-7`,4=`10e-6`],\n               [0=`10e2`,1=`10e3`,2=`10e4`,3=`10e5`
,4=`10e6`])\n);\nPLOT(AXESSTYLE(NONE),\n        VIEW(0..10,0..10),\n  \+
      POINTS([0.4,10],SYMBOL(CIRCLE)),\n        TEXT([0.6,10],'` - rat
ionale Extrapolation`',ALIGNRIGHT),\n        POINTS([0.4,9.3],SYMBOL(B
OX)),\n        TEXT([0.6,9.3],'` - Polynomextrapolation`',ALIGNRIGHT))
;\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "20" 0 }{VIEWOPTS 1 1 0 1 1 
1803 }