Abstract
The problem of obtaining optimal starting values for the calculation of square root using Newton-Raphson's Method is considered. This paper presents the best starting values theory in order to optimize the maximum absolute error after a given number of iterations. Two different methods are shown, and a third, which can be considered as a mixture of the previous two, is briefly discussed. The approach combines analytical and numerical methodologies, which gives more interesting results on the main characteristics of the behavior of the absolute error for different initializations. A comparison table between the traditional optimal relative error results and the absolute error ones is provided.
Zusammenfassung
Wir betrachten das Problem, optimale Startwerte für die Berechnung von Quadratwurzeln bei Verwendung der Newton-Raphson Methode zu erhalten.
In dieser Arbeit wird die Theorie zur Wahl der besten Startwerte in Hinblick auf die Optimirung des maximalen absoluten Fehlers nach einer vorgegebenen Anzahl von Iterationen dargestellt. Es werden zwei verschiedene Methoden gezeigt und eine dritte, die als Mischung dieser beiden Methoden angesehen werden kann, betrachtet. Eine Vergleichstabelle zwischen den herkömmlichen Ergebnissen bezüglich des optimalen relativen Fehlers und des absoluten Fehlers wird angegeben.
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Montuschi, P., Mezzalama, M. Optimal absolute error starting values for Newton-Raphson calculation of square root. Computing 46, 67–86 (1991). https://doi.org/10.1007/BF02239012
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DOI: https://doi.org/10.1007/BF02239012