## Abstract

We study the local convergence of some Aitken–Steffensen–Hermite type methods of order three. We obtain that under some reasonable conditions on the monotony and convexity of the nonlinear function, the iterations offer bilateral approximations for the solution, which can be efficiently used as a posteriori estimations.

## Authors

Ion **Păvăloiu**

(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Emil **Cătinaş**

(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

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## Cite this paper as

I. Păvăloiu, E. Cătinaş, *Bilateral approximations for some Aitken-Steffensen-Hermite type methods of order three*, Appl. Math. Comput., **217** (2011) 12, pp. 5838-5846

doi: 10.1016/j.amc.2010.12.067

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## References

[1] S. Amat, S. Busquier, *On a Steffensen’s type method and its behavior for semismooth equations*, Appl. Math. Comput. 177 (2006) 819–823.

[2] S. Amat, S. Busquier, *A two-step Steffensen’s method under modified convergence conditions*, J. Math. Anal. Appl. 324 (2006) 1084–1092.

[3] S. Amat, S. Busquier, V. Candela, *A class of quasi-Newton generalized Steffensen methods on Banach spaces*, J. Comput. Appl. Math. 149 (2002) 397– 406.

[4] I.K. Argyros, *A new convergence theorem for the Steffensen method in Banach space and applications*, Rev. Anal. Numér. Théor. Approx. 29 (2) (2000) 119–128.

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

[6] E. Catinas, *Methods of Newton and Newton-Krylov Type*, Risoprint, Cluj-Napoca, 2007.

[7] M. Frontini, *Hermite interpolation and a new iterative method for the computation of the roots of non-linear equations*, Calcolo 40 (2003) 109–119.

[8] M. Grau, *An improvement to the computing of nonlinear equation solution*, Numer. Algorithms 34 (2003) 1–12.

[9] P. Jain, *Steffensen type methods for solving non-linear equations*, Appl. Math. Comput. 194 (2007) 527–533.

[10] M.A. Ostrovski, *Solution of Equations and Systems of Equations*, Academic Press, New York, 1982.

[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, N. Pop, *Interpolation and Applications*, Risoprint, Cluj-Napoca, Romania, 2005 (in Romanian).

[13] I. Pavaloiu, *Bilateral approximations of solutions of equations by order three Steffensen-type methods*, Studia Univ. Babes-Bolyai, Mathematica LI (3) (2006) 87–94.

[14] I. Pavaloiu, *Optimal efficiency index of a class of Hermite iterative methods with two steps*, Rev. Anal. Numér. Théor. Approx. 29 (1) (2009) 83–89.

[15] I. Pavaloiu, E. Catinas, *On a Steffensen type method, in: SYNASC 2007, Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing*, IEEE Computer Society, Timisoara, Romania 26–29 September 2007.

[16] I. Pavaloiu, E. Catinas, *On a Steffensen–Hermite method of order three*, Appl. Math. Comput. 215 (2009) 2663–2672.

[17] J.R. Sharma, *A composite third order Newton–Steffensen method for solving nonlinear equations*, Appl. Math. Comput. 169 (2005) 342–346.

[18] B.A. Turowicz, *Sur le derivées d’ordre superieur d’une fonction inverse*, Ann. Polon. Math. 8 (1960) 265–269.

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