Choosing Starting Values for Newton-Raphson Computation of Reciprocals, Square-Roots and Square-Root Reciprocals

Peter Kornerup 1, 2 Jean-Michel Muller
1 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : We aim at finding the best possible seed values when computing reciprocals, square-roots and square-root reciprocals in a given interval using Newton-Raph- son iterations. 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).
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Peter Kornerup, Jean-Michel Muller. Choosing Starting Values for Newton-Raphson Computation of Reciprocals, Square-Roots and Square-Root Reciprocals. [Research Report] RR-4687, LIP RR-2002-48, INRIA,LIP. 2003. ⟨inria-00071899⟩

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