On the Steffensen method for solving nonlinear operator equations

Abstract

We consider the equation \[F\left( x\right) =x-A\left( x\right)=0,\] where \(A\) is an operator from a Banach space \(X\) to itself. The generalized Steffensen method has the form

$$ x_{n+1}=x_{n}-\left[ x_{n},A\left( x_{n}\right) ;F\right] ^{-1}F\left(
x_{n}\right) $$
which is equivalent to
$$
x_{n+1}=A\left( x_{n}\right) -\left[ x_{n},A\left( x_{n}\right)
;F\right] ^{-1}F\left( A\left( x_{n}\right) \right) \label{f.1.4}%
$$

In this paper we give new semilocal convergence conditions which ensure the convergence of the method.

Original title (in French)

Authors

Ion Păvăloiu

Keywords

Steffensen method; divided differences; Banach space; semilocal convergence.

References

[1] J. W. Schmith, Konvergenzgeschwindigkeit der im Banachraum. ZAMM, 1966, 46, 2, 146-148.

[2] S. ULM, Obobscenie metoda Steffensena dlja resenija nelinejnah operatornıh uravneij.”Jur. vacisl. mat. mat. fiziki”, 1964, 4, 6, 1093-1097.

[3] A. M. OSTROVSKI, Resenie uravnenij i sistema uravnenij. ”Mat. izd-vo in. lit.”, 1963.

[4] L. V. KANTOROVICI, Funktional’naj analiz i prikladnaja matematika. ”UMN”, 1948
(28), 3,

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About this paper

Cite this paper as:

I. Păvăloiu, Sur la méthode de Steffensen pour la résolution des équations operationnelles nonlinéaires, Revue Roumaine des Mathématiques pures et appliquées, 13 (1968) no. 1, pp. 857-861 (in French).

Journal

Revue Roumaine des Mathématiques pures et appliquées

Publisher Name

Editura Academiei Republicii Socialiste Romane

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