February 2015
Intermediate to advanced
392 pages
17h 52m
German
Content preview from Mathematik für Wirtschaftswissenschaftler, 2nd Edition
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8.5. Margin al analyse
243
Beispiel 8.20 (Numerische Lösung einer kubischen Gleichung)
Das Polynom dritten Grades f(x) = x
3
+ x+2 ist wegen
f'(x)
—
3x
2
+
1
> 0 eine steigende Funktion, hat ξ = -1
als einzige Nullstelle und verläuft wegen f"(x) = 6x für
positive χ konvex und für negative χ konkav. Wird ein
positiver Startpunkt
XQ
gewählt, so fällt die Näherungs-
folge so lange, bis x
n
erstmals links von der Nullstelle -2
liegt oder diese (zufällig) genau erreicht hat. Anschlie-
ßend konvergiert die Folge steigend gegen die Nullstelle.
Neben der geometrisch einsichtigen Konvergenz lässt sich hier auch leicht die quantitative Konver-
genzaussage
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