On the convergency of a Steffensen-type method

Abstract

Let \(X_{1},X_{2}\) be two Banach spaces, \(f:X_{1}\rightarrow X_{2}\) a nonlinear mapping. We study the convergence of the Steffensen method for solving \(f\left( x\right) =0\): \[x_{n+1}=x_n-[x_{n},g(x_{n});f]^{-1}f(x_n), \quad n=1,…\] Under some simple Holder type conditions on the divided differences of order one of \(f\), of the form \[\|[y, u; f] − [x, y; f]\| ≤ l_1 \|x − u\| ^p + l_2 \|x − y\|^p + l_3 \|y − u\|^p\] we give some error estimations and we determine the convergence order.

Authors

Ion Păvăloiu

Keywords

Steffensen type method; Holder conditions on divided differences; nonlinear equations in Banach spaces; iterative methods

References

[1] Argyros, I.K., The secant method and fixed points of nonlinear operators, Mh. Math. 106, 85–94 (1988).

[2] Pavaloiu, I., Sur la methode de Steffensen pour la resolution des equations operationnelles non lineaires, Revue Roumaine des Mathematiques pures et appliquees, 1, XIII, 149–158 (1968).

[3] Pavaloiu, I., Introduction in the Theory of Approximation of Equations Solutions, Dacia Ed., Cluj-Napoca 1976 (in Romanian).

[4] Pavaloiu, I., Remarks on the secant method for the solution of nonlinear operatorial equations, Research Seminars, Seminar on Mathematical Analysis, Preprint no. 7, (1991), pp. 127 132.

[5] Ul’m, S., Ob obobscenie metod Steffensen dlea resenia nelineinih operatornih urnavnenii, Journal Vicisl., mat. i mat.-fiz. 4, 6 (1964).

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About this paper

Cite this paper as:

I. Păvăloiu, On the convergency of a Steffensen-type method, Research Seminars, Seminar of Mathematical Analysis, Preprint no. 7 (1991), pp. 121-126.

Journal

Seminar on mathematical analysis,
Preprint

Publisher Name

“Babes-Bolyai” University,
Faculty of Mathematics,
Research Seminars

DOI

Not available yet.

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