On the iterative methods with high convergence orders

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.

Ion Păvăloiu

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

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

[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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PDF-Latex version of the paper.

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

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