On the Steffensen method for solving nonlinear operator equations


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)


Ion Păvăloiu


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


[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,


Scanned paper (soon)

PDF-Latex version (in French)

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).


Revue Roumaine des Mathématiques pures et appliquées

Publisher Name

Editura Academiei Republicii Socialiste Romane


Not available yet.

Print ISSN

Not available yet.

Online ISSN

Not available yet.

Google Scholar Profile

Related Posts

No results found.