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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Contributor : Jean-Michel Muller <>
Submitted on : Wednesday, April 12, 2006 - 11:02:17 AM
Last modification on : Wednesday, November 20, 2019 - 7:36:24 AM
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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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