Local convergence of general Steffensen type methods


  • Ion Păvăloiu Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania




nonlinear scalar equations, Steffensen type method
Abstract views: 241


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 \(2\)).


Download data is not yet available.


Balázs, M., A bilateral approximating method for finding the real roots of real equations, Rev. Anal. Numér. Théor. Approx., 21, no. 2, pp. 111-117, 1992, http://ictp.acad.ro/jnaat/journal/article/view/1992-vol21-no2-art3

Brent, R., Winograd, S. and Walfe, Ph., Optimal iterative processes for root-finding, Numer. Math., 20, no. 5, pp. 327-341, 1973, https://doi.org/10.1007/bf01402555 DOI: https://doi.org/10.1007/BF01402555

Coman, C., Some practical approximation methods for nonlinear equations, Mathematica -- Rev. Anal. Numér. Théor. Approx., 11, nos. 1-2, pp. 41-48, 1982, http://ictp.acad.ro/jnaat/journal/article/view/1982-vol11-nos1-2-art5

Cassulli, V. and Trigiante, D., The convergence order for iterative multipoint procedures, Calcolo, 13, no. 1, pp. 25-44, 1977, https://doi.org/10.1007/bf02576646 DOI: https://doi.org/10.1007/BF02576646

Iancu, C., Păvăloiu, I. and Şerb, I., Méthodes iteratives optimales de type Steffensen obtinues par interpolation inverse, Faculty of Mathematics, "Babeş-Bolyai" University, Seminar on Functional analysis and Numerical Methods, Preprint no. 1, pp. 81-88, 1983.

Kacewicz, B., An integral-interpolation iterative method for the solution of scalar equations, Numer. Math., 26, no. 4, pp. 355-365, 1976, https://doi.org/10.1007/bf01409958 DOI: https://doi.org/10.1007/BF01409958

Ostrowski, A., Solution of Equations in Euclidian and Banach Spaces, Academic Press, New York and London, 1973.

Păvăloiu, I., La résolution des equations par interpolation, Mathematica, 23(46), no. 1, pp. 61-72, 1981.

Păvăloiu, I. and Şerb, I., Sur des méthodes de type intérpolatoire à vitesse de convergence optimale, Rev. Anal. Numér. Théor. Approx., 12, no. 1, pp. 83-88, 1983, http://ictp.acad.ro/jnaat/journal/article/view/1983-vol12-no1-art10

Păvăloiu, I., Optimal efficiency index for iterative methods of interpolatory type, Computer Science Journal of Moldova, 5, no. 1 (13), pp. 20-43, 1997.

Traub, J. F., Iterative Methods for the Solution of Equations, Pretince-Hall, Inc. Englewood Clifs, N.J., 1964.

Turowicz, A. B., Sur les derivées d'ordre superieur d'une fonction inverse, Ann. Polon. Math., 8, pp. 265-269, 1960, https://doi.org/10.4064/ap-8-3-265-269 DOI: https://doi.org/10.4064/ap-8-3-265-269




How to Cite

Păvăloiu, I. (2004). Local convergence of general Steffensen type methods. Rev. Anal. Numér. Théor. Approx., 33(1), 79–86. https://doi.org/10.33993/jnaat331-762