Das Newton-Verfahren (und das Paket "Student")
with(plots):
with(Student[Calculus1]):
Das Paket "Student[Calculus1]" wird geladen. Es enth\303\244lt eine Reihe von Tutorials und Routinen zum Nacharbeiten und selbst\303\244ndigen Erforschen von Standardthemen der Differential- und Integralrechnung. Zun\303\244chst wird eine Funktion LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0YyL0Y2USdub3JtYWxGJw== definiert:
f:=x->(x-1)*(x-3)*(x-4);
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigsJkYkIiIiRishIiJGKywmRiRGKyIiJEYsRissJkYkRisiIiVGLEYrRiVGJUYl
Der Mittelwertsatz (Englisch: Mean value theorem) kann mit Hilfe des Kommandos "MeanValueTheorem" aus dem "Student" Paket illustriert werden:
MeanValueTheorem(f(x),x=1..5);
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Man erh\303\244lt eine Zeichnung des Graphen der Funktion LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0YyL0Y2USdub3JtYWxGJw== im angegebenen Intervall [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 ], die Sehne, die den Punkt ( 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 ) mit dem Punkt ( 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 ) verbindet, sowie Tangenten an Punkte des Graphen, die parallel zu der Sehne sind (der Mittelwertsatz garantiert, dass wenigstens ein Punkt im Intervall existiert, in dem die Tangente parallel zu der Sehne ist).
Wir betrachten jetzt den Graphen der Funktion LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0YyL0Y2USdub3JtYWxGJw== sowie die Graphen der ersten und der zweiten Ableitung: Da LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0YyL0Y2USdub3JtYWxGJw== eine ganzrationale Funktion (Polynom) vom Grad 3 ist, ist die Ableitung vom Grad 2, hat also als Graphen eine Parabel. Die zweite Ableitung hat Grad 1 (ist eine lineare Funktion), ihr Graph ist also eine Gerade. Man sieht durch Betrachten der Graphen: In dem Teilintervall, in dem die zweite Ableitung negativ ist, f\303\244llt die erste Ableitung monoton, die Funktion ist in diesem Teilintervall konkav. In dem Teilintervall, in dem die zweite Ableitung positiv ist, w\303\244chst die erste Ableitung monoton, die Funktion ist in diesem Teilintervall konvex. Randpunkt dieser beiden Teilintervalle ist der Wendepunkt (Nullstelle der zweiten Ableitung mit Extremwert der ersten Ableitung), in dem die erste Ableitung ihr Minimum hat. Die beiden Nullstellen der ersten Ableitung liefern bei dieser Funktion lokale Extrema (ein Minimum und ein Maximum).
plot(f(x),x=0..5);
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plot(D(f)(x),x=0..5);
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plot(D(D(f))(x),x=0..5);
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Um das Newton-Verfahren zu illustrieren zeichnen wir jetzt den Graphen der Funktion im Intervall [ LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1JI21uR0YkNiVRJDMuNUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEiLEYnRjVGOC8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYyNiVRIjZGJ0Y1RjgvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjVGOEYrRldGNUY4 ], in dem die Funktion konvex ist, gemeinsam mit der Sekante durch die Punkte ( 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 ) und ( 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 ) sowie der Tangente an den Graphen im Punkt (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 ). Man sieht, dass die Sehne oberhalb des Graphen liegt, die Tangente unterhalb, wie dies bei konvexen Funktionen stets der Fall ist. Danach wird das Gleiche f\303\274r den Graphen der Funktion LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1JI21vR0YkNi5RKiZ1bWludXMwO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSy1GLDYmUSJmRicvJSdpdGFsaWNHUSV0cnVlRidGNS9GOVEnaXRhbGljRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjVGOEYrRlZGNUY4 gemacht, die konkav ist, bei der also umgekehrt die Sehne unterhalb, die Tangente oberhalb des Graphen verl\303\244uft.
plot({f(x),((f(6)-f(3.5))/2.5)*(x-3.5),D(f)(6)*(x-6)+f(6)},x=2..6.5);
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plot({-f(x),-((f(6)-f(3.5))/2.5)*(x-3.5),-D(f)(6)*(x-6)-f(6)},x=2..6.5);
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Nun sehen wir uns an, wie das "Student"-Paket das Newton-Verfahren illustriert: Man sieht wieder den Graphen, als Startpunkt wird hier 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 gew\303\244hlt. Man sieht jeweils die Tangente in den Punkten mit Abszisse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JlEibkYnRjVGOEY7LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y4L0Y8USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRitGQ0Y4RkY= und den Schnittpunkt LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNihGKy1GIzYoLUYsNiZRIm5GJ0Y1RjhGOy1JI21vR0YkNi5RIitGJ0Y4L0Y8USdub3JtYWxGJy8lJmZlbmNlR0Y6LyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRmVuLUkjbW5HRiQ2JVEiMUYnRjhGSS8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOEZJRitGXG9GOEZJLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Zcb0Y4Rkk= dieser Tangente mit der LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0YyL0Y2USdub3JtYWxGJw== -Achse f\303\274r die ersten 5 Schritte des Verfahrens. Die Nullstelle 4 wird damit schon recht gut approximiert.
NewtonsMethod(f(x),7,output=plot);
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Anschlie\303\237end ein Beispiel daf\303\274r, wie man das gleich Verfahren in MAPLE von Hand programmieren und die entsprechenden Graphen zeichnen kann:
x[0]:=7.0;
JCIjcSEiIg==
for n from 1 to 7 do x[n]:=x[n-1]-f(x[n-1])/D(f)(x[n-1]) end do;
JCIrbm1tbWMhIio=
JCIrRWUjZSNbISIq
JCIra0x4SVYhIio=
JCIrXCFvYDMlISIq
JCIrSFU/M1MhIio=
JCIreiIpMytTISIq
JCIrNSsrK1MhIio=
funktion:=plot(f(x),x=2..8,color=blue):
tangenten:=plot({D(f)(x[0])*x+f(x[0])-D(f)(x[0])*x[0],D(f)(x[1])*x+f(x[1])-D(f)(x[1])*x[1]},x=2..8):
display(funktion);
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display(funktion,tangenten);
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