Choosing Starting Values for certain Newton-Raphson Iterations

Abstract : We aim at finding the best possible seed values when computing $a^{\frac1p}$ using the Newton-Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function $f(a)$ in the interval $[a_{\min},a_{\max}]$, by building the sequence $x_n$ defined by the Newton-Raphson iteration, the natural choice consists in choosing $x_0$ equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between $x_0$ and $f(a)$. And yet, if we perform $n$ iterations, what matters is to minimize the maximum possible distance between $x_n$ and $f(a)$. In several examples, the value of the best starting point varies rather significantly with the number of iterations.
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Theoretical Computer Science, Elsevier, 2006, 351 (1), pp.101-110. 〈10.1016/j.tcs.2005.09.056〉
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Contributeur : Jean-Michel Muller <>
Soumis le : mercredi 12 avril 2006 - 11:02:17
Dernière modification le : jeudi 17 mai 2018 - 12:52:03
Document(s) archivé(s) le : samedi 3 avril 2010 - 21:18:27

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Jean-Michel Muller, Peter Kornerup. Choosing Starting Values for certain Newton-Raphson Iterations. Theoretical Computer Science, Elsevier, 2006, 351 (1), pp.101-110. 〈10.1016/j.tcs.2005.09.056〉. 〈ensl-00000009〉

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