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)
Sur la méthode de Steffensen pour la résolution des équations operationnelles nonlinéaires
Authors
Keywords
Steffensen method; divided differences; Banach space; semilocal convergence.
Scanned paper (translation in English)
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).
About this paper
Journal
Revue Roumaine des Mathématiques pures et appliquées
Publisher Name
Editura Academiei Republicii Socialiste Romane
DOI
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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