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
(Tiberiu Popoviciu Institute of Numerical Analysis)
Keywords
Steffensen type method; Holder conditions on divided differences; nonlinear equations in Banach spaces; iterative methods
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.
About this paper
Journal
Seminar on mathematical analysis,
Preprint
Publisher Name
“Babes-Bolyai” University,
Faculty of Mathematics,
Research Seminars
DOI
Not available yet.
References
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[5] Ul’m, S., Ob obobscenie metod Steffensen dlea resenia nelineinih operatornih urnavnenii, Journal Vicisl., mat. i mat.-fiz. 4, 6 (1964).