Posts by Ion Păvăloiu

Abstract

Let \(X\) be a Banach space and \(Y\) a normed space, and \(P:X\rightarrow Y\) a nonlinear operator. In order to solve the equation \(P\left( x\right)=0\), we consider the iterative method \(x_{n+1}=x_{n}+\varphi \left(x_{n}\right) \), where \(\varphi:X\rightarrow X\). We give some sufficient semilocal conditions relating \(\varphi\) and \(P\) for these iterations to converge to a solution with a given convergence order. As particular instances, we obtain convergence results for the Newton, Chebyshev and Steffensen mehods.

Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Title

Original title (in French)

Sur les procedées itératifs à un ordre élevé de convergence

English translation of the title

On the iterative methods with high convergence orders

Keywords

iterative methods in normed spaces; convergence order; Newton type method; Chebyshev type method; Steffensen type method; semilocal convergence

PDF

Cite this paper as:

I. Păvăloiu, Sur les procedées itérative à un order élevé de convergence, Mathématica, 12(35) (1970) no. 2, pp. 309-324 (in French).

About this paper

Journal

Mathematica

Publisher Name

Academia Republicii S.R.

DOI

Not available yet.

Print ISBN

Not available yet.

Online ISBN

Not available yet.

References

[1] Kantorovici, L. V., Functionalıi analiz i pricladnaia matematica. U.M.N., 28, 3 (1948).

[2] Kantorovici, L. V., O metodı Niutona. Trudı mat. i-ta im. Steklova. 28, pp. 104–144 (1949).

[3] Janko, B, si Goldner, G., Despre rezolvarea ecuatiilor operationale cu metoda lui Cebısev. (II), Studia Univ. Babes-Bolyai Cluj 2, pp. 55–58 (1968).

[4] Ghinea, Monique, Sur la resolution des equations operationnelles dans les espaces de Banach, Revue Francaise de traitement de l’information 8, pp. 3–22 (1965).

[5] Ostrowski, A. M., Resenie uravnenii i sistem uravnenii. Izd. inost. lit., Moskva, (1963).

[6] Pavaloiu, I., Asupra rezolvarii ecuatiilor operationale prin metode de iteratie de ordin superior. Lucrarile colocviului de teoria aproximarii functiilor (rezumat), Cluj 15-20 septembrie 1967.

[7] Pavaloiu, I., Sur la methode de Steffensen pour la resolution des equations operationnelles non lineaires. Revue Roumaine des Mathematiques Pures et Appliquees, XIII, (1968) 6, pp. 857–861.

[8] Pavaloiu, I., Asupra operatorilor iterativi, Studii si Cercetari matematice (in print); appeared as: 23 (1971) no. 10, pp. 1567–1574.

[9] Pavaloiu, I., Interpolation dans des espaces lineaires normes et applications. Mathematica, Cluj, vol. 12 (35), 1, 1970, pp. 149–158.

[10] Stein, M. L., Sufficient conditions for the Banach spaces. Proc. Amer. Math. Soc. 3, pp. 858–863 (1952).

[11] Traub, J, F., Iterative methods for the solution of equations. Prentice-Hall. Inc. Englewood Cliffs N. J. 1964.

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