Berechnung der Quadratwurzel
Bearbeitungszeit: ca. 10 Minuten für erste Lösung; ca. 15 Minuten für beide Lösungen.
Implementieren Sie eine Methode sqrt ( double x, double epsilon) , die die Quadratwurzel von x mit einer Genauigkeit von epsilon berechnet. Verwenden Sie dazu das Newton-Raphson-Verfahren: y n + 1 = 1 2 ( y n + x y n ) y_{n+1} = \frac{1}{2}\left(y_n + \frac{x}{y_n}\right) y n + 1 = 2 1 ( y n + y n x ) . Implementieren Sie die Methode einmal rekursiv und einmal iterativ.
Der Abbruch soll erfolgen, wenn ∣ y n + 1 − y n ∣ < ϵ |y_{n+1} - y_n| < \epsilon ∣ y n + 1 − y n ∣ < ϵ .
Hinweis
Für die rekursive Variante kann es sinnvoll sein neben der Hauptmethode eine zweite Hilfsmethode zu implementieren.
Beispiel
Enter number to compute SQRT for: 9
Enter epsilon: 0.0001
9.0 = > 5.0
.. .
3.00009155413138 = > 3.000000001396984
The SQRT of 9.0 is 3.000000001396984 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