# On a robust Aitken-Newton method based on the Hermite polynomial

## Abstract

We introduce an Aitken–Newton iterative method for nonlinear equations, which is obtained by using the Hermite inverse interpolation polynomial of degree 2, with two nodes given by the Newton method. The local convergence of these iterates is shown to be 8, and the efficiency index is $$\sqrt[5]{8}\approx 1.51$$, which is not optimal in the sense of Kung and Traub. However, we show that under supplementary conditions (sometimes easy to verify) the inner and outer iterates converge monotonically to the solution. This aspect allows an improved control of the iteration stopping (avoiding divisions by zero) and offer an alternative way to the estimation of radius of attraction balls in ensuring the convergence of the iterates. Numerical examples show that this method may become competitive and in certain circumstances even more robust than certain optimal methods of same convergence order.

## Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

## Keywords

Nonlinear equations in  R; Newton-type iterative methods; inverse interpolation; (Sided) convergence domains; monotone convergence; divided differences.

## Paper coordinates

I. Păvăloiu, E. Cătinaş, On a robust Aitken-Newton method based on the Hermite polynomial, Appl. Math. Comput., 287-288 (2016), pp. 224-231.
doi: 10.1016/j.amc.2016.03.036

## PDF

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Elsevier

0096-3003