On some iterative methods for solving operator equations

Abstract

Let \(X,Y\) be two Banach spaces and \(P:X\rightarrow Y\) a nonlinear operator. We study the semilocal convergence of the Newton, chord and Steffensen methods for which the derivative \(P^{\prime}\left( x\right) \) or the divided differences from each iteration step are approximated by a sequence of operators obtained with the Schultz method:
\begin{equation}
\left\{
\begin{array}
[c]{l}%
x_{n+1}=x_{n}-A_{n}P\left( x_{n}\right) \\
A_{n+1}=A_{n}\left( 2E-\left[ x_{n},x_{n+1};P\right] A_{n}\right)
,\qquad n=0,1,\ldots
\end{array}
\right. \label{f.1.6}%
\end{equation}
and considering the Steffensen method:%
\begin{equation}
\left\{
\begin{array}
[c]{l}%
x_{n+1}=x_{n}-A_{n}P\left( x_{n}\right) \\
A_{n+1}=A_{n}\left( 2E-\left[ x_{n+1},Q\left( x_{n+1}\right) ;P\right]
A_{n}\right) ,\qquad n=0,1,\ldots
\end{array}
\right. \label{f.1.7}%
\end{equation}

Authors

A. Diaconu, I. Păvăloiu

Title

Originnal title (in French)

Sur quelques méthods itératives pour la résolution des equations operationnelles

English translation of the title

On some iterative methods for solving operator equations

Keywords

Newton method; chord method; Steffensen method; Schultz method; semilocal convergence

References

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[2] Diaconu, A., Pavaloiu, I., Asupra unor metode iterative pentru rezolvarea ecuatiilor operationale neliniare (I), Revista de analiza numerica si teoria aproximatiei, sous presse 2 (1973), nr. 1, pp. 61–79.

[3] Janko, B., Sur la theorie unitaire des methdes d’iteration pour la resolution des equations operationelles non lineaires. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, II, Ser. A, 302–311 (1961).

[4] Janko, B., Rezolvarea ecuatiilor operationale neliniare in spatii Banach. Bucuresti, Editura Academiei R.S.R. (1969).

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[10] Sergheev, A. S., O metode hord. Sibirski mat. jurnal, XI, 2, 282–289 (1961)

[11] Traub, J. F., Iterative Methods for the Solution of Equations. Prentice-Hall Inc. Englewood
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Izv. Akad. Nauk Estonskoi S.S.R., 16, 4, 403–411 (1967)

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Cite this paper as:

A. Diaconu, I. Păvăloiu, Sur quelques méthods itératives pour la résolution des equations operationnelles, Rev. Anal. Numér. Théor. Approx., 1 (1972), pp. 45-61 (in French).

Journal

Revue d’Analyse Numérique et de Théorie de l’ Approximation

Publisher Name

Academia Republicii S.R.

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