## Abstract

We study the local convergence of a generalized Steffensen method. We show that this method substantially improves the convergence order of the classical Steffensen method. The convergence order of our method is greater or equal to the number of the controlled nodes used in the Lagrange-type inverse interpolation, which, in its turn, is substantially higher than the convergence orders of the Lagrange type inverse interpolation with uncontrolled nodes (since their convergence order is at most (2)).

## Author

## Keywords

nonlinear equations in R; Steffensen method.

## References

[1] Balazs, M.,

*A bilateral approximating method for finding the real roots of real equations*, Rev. Anal. Num ́er. Theor. Approx., 21 , no. 2, pp. 111–117, 1992.[2] Brent, R., Winograd, S. and Walfe, Ph.,

*Optimal iterative processes for root-finding*, Numer. Math., 20 , no. 5, pp. 327–341, 1973.[3] Coman, C.,

*Some practical approximation methods for nonlinear equations*, Mathematica – Rev. Anal. Num ́er. Theor. Approx., 11, nos. 1–2, pp. 41–48, 1982.[4] Cassulli, V. and Trigiante, D.,

*The convergence order for iterative multipoint procedures*, Calcolo, 13, no. 1, pp. 25–44, 1977.[5] Iancu, C., Pavaloiu, I. and Serb, I.,

*Methodes iteratives optimales de type Steffensen obtinues par interpolation inverse*, Faculty of Mathematics, “Babes-Bolyai” University, Seminar on Functional analysis and Numerical Methods, Preprint no.1, pp. 81–88, 1983.[6] Kacewicz, B.,

*An integral-interpolation iterative method for the solution of scalar equations*, Numer. Math., 26 , no. 4, pp. 355–365, 1976.[7] Ostrowski, A.,

*Solution of Equations in Euclidian and Banach Spaces*, Academic Press, New York and London, 1973. [8] Pavaloiu, I.,

*La resolution des equations par interpolatio*n, Mathematica, 23(46), no. 1, pp. 61–72, 1981.[9] Pavaloiu, I. and Serb, I.,

*Sur des methodes de type interpolatoire `a vitesse de convergence optimale*, Rev. Anal. Numer. Theor. Approx., 12 , no. 1, pp. 83–88, 1983.[10] Pavaloiu, I.,

*Optimal efficiency index for iterative methods of interpolatory type*, Computer Science Journal of Moldova, 5 , no. 1 (13), pp. 20–43, 1997.[11] Traub, J. F.,

*Iterative Methods for the Solution of Equations*, Pretince-Hall, Inc. Englewood Clifs, N.J., 1964.[12] Turowicz, A. B.,

*Sur les deriv ́ees d’ordre superieur d’une fonction inverse*, Ann. Polon. Math., 8, pp. 265–269, 1960Scanned paper.

PDF-LaTeX version of the paper (soon).

## About this paper

##### Cite this paper as:

I. Păvăloiu, *Local convergence of general Steffensen type methods*, Rev. Anal. Numér. Théor. Approx., **33** (2004) 1, pp. 79-86.

##### Publisher Name

##### Article on the journal website

##### Print ISSN

1222-9024

##### Online ISSN

2457-8126