## 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

## 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

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## About this paper

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[1] S. Amat, J. Blanda, S. Busquier, *A Steffensen type method with modified functions*, Riv. Mat. Univ. Parma 7 (2007) 125–133.

[2] S. Amat, S. Busquier, J.A. Ezquerro, M.A. Hernández-Verón, *A Steffensen type method of two steps in banach spaces with applications*, J. Comput. Appl. Math. 291 (2016) 317–331.

[3] I.K. Argyros, *A new convergence theorem for the Steffensen method in Banach space and applications*, Rev. Anal. Numér. Théor. Approx. 29 (1) (2000) 119–127.

[4] I.K. Argyros, S. Hilout, *An improved local convergence analysis for Newton–Steffensen-type method*, J. Appl. Math. Comp. 32 (1) (2010) 111–118.

[5] E. Catinas, *On some Steffensen-type iterative method for a class of nonlinear equations*, Rev. Anal. Numér. Théor. Approx. 24 (1-2) (1995) 37–43.

[6] C. Chun, *Certain improvements of Chebyshev–Halley methods with accelerated fourth-order convergence*, Appl. Math. Comput. 189 (1) (2007) 597–601.

[7] A. Cordero, J.L. Hueso, E. Martinez, J.R. Torregrosa, *Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolations*, Math. Comput. Model. 57 (7-8) (2013) 1950–1956.

[8] A. Cordero, J.R. Torregrosa, M.P. Vassileva, *A family of modified Ostrowski’s methods with optimal eighth order of convergence*, Appl. Math. Lett. 24 (12) (2011) 2082–2086.

[9] J. Kou, X. Wang, *Some improvements of Ostrowski’s method*, Appl. Math. Lett. 23 (1) (2010) 92–96.

[10] L. Liu, X. Wang, *Eighth-order methods with high efficiency index for solving nonlinear equations*, Appl. Math. Comput. 215 (9) (2010) 3449–3454.

[11] I. Pavaloiu, *Approximation of the root of equations by Aitken–Steffensen-type monotonic sequences*, Calcolo 32 (1-2) (1995) 69–82.

[12] I. Pavaloiu , E. Catinas , Bilateral approximation for some Aitken–Steffensen–Hermite type method of order three, Appl. Math. Comput. 217 (12) (2011) 5838–5846.

[13] I. Pavaloiu, E. Catinas , On an Aitken–Newton type method, Numer. Algorithm 62 (2) (2013) 253–260.

[14] M.S. Petkovic, On a general class of multipoint root-finding methods of high computational efficiency, SIAM J. Numer. Anal. 47 (6) (2010) 4402–4414.

[15] J.R. Sharma, A composite third order Newton–Steffensen method for solving nonlinear equations, Appl. Math. Comput. 169 (1) (2005) 242–246.

[16] F. Soleymani, S.K. Vanani, Optimal Steffensen-type method with eighth order of convergence, Comput. Math. Appl. 62 (12) (2011) 4619–4626.