On some Aitken-Steffensen-Halley-type method for approximating the roots of scalar equations

We extend the Aitken-Steffensen method to the Halley transformation. Under some rather simple assumptions we obtain error bounds for each iteration step; moreover, the convergence order of the iterates is 3, i.e. higher than for the Aitken-Steffensen case.

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

nonlinear equations in R; Aitken-Steffensen-Halley method.

PDF-LaTeX file on the journal website.

I. Păvăloiu, On some Aitken-Steffensen-Halley-type method for approximating the roots of scalar equations, Rev. Anal. Numér. Théor. Approx., 30 (2001) no. 2, pp. 207-212.

[1] Argyros, I.K. On the convergence of some projection methods with perturbations, J. Comp. Appl. Math. 36, (1991), 255–258.

[2] Argyros, I.K. On an application of the Zincenko method to the approximation of implicit functions, Z.A.A. 10, 3, (1991), 391– 396.

[3] Argyros, I.K. and Szidarovszky, F. The Theory and Application of Iteration Methods, C.R.C. Press, Inc., Boca Raton, Florida, 1993.

[4] Catinas, E. On some iterative methods for solving nonlinear equations, Revue d’analyse Numerique et de theorie de l’approximation, 23, 1, (1994), 47–53.

[5] Kantorovich, L.V. The method of successive approximation for functional equations, Acta Math. 71 (1939), 63–97.

[6] Pavaloiu, I. Sur une generalisation de la methode de Steffensen, Revue d’analyse Numerique et de theorie de l’approximation, 21, 1, (1992), 59–65.

[7] Pavaloiu, I. Bilateral approximations for the solutions of scalar equations, Revue d’analyse numerique et de theorie de l’approximation, 23, 1, (1994), 95–100.