A generalization of the Newton method

Abstract

Let \(X\) be a Banach space, \(Y\) a normed space, \(G:X\rightarrow Y\) a nonlinear operator, and \(G\left( x\right) =0\) a nonlinear equation. We denote by \(F:X^{2}\rightarrow Y\) a nonlinear operator for which the restriction to the diagonal of \(X^{2}\) coincide with \(G\). We first prove a Taylor type formula for operators with two variables. Next we consider the following two-step Newton type method: \[F\left( x_{n},x_{n-1}\right) +F_{x}^{\prime}\left( x_{n},x_{n-1}\right) \left( x_{n+1}-x_{n}\right) +F_{y}^{\prime}\left( x_{n},x_{n-1}\right) \left( x_{n}-x_{n-1}\right)=0.\] We study the convergence to the solution of the above sequence.

Authors

Ion Păvăloiu

Title

Original title (in French)

Une généralisation de methode de Newton

English translation of the title

A generalization of the Newton method

Keywords

Taylor polynomial with two variables; two-step Newton type method

References

[1] Kantorovici, L.V., Functionalnîi analiz i prikladnaia matematika, UMN, 28, 89-185 (1948).

[2] Pavaloiu, I., Sur les procédés iteratif à un ordere élevé de convergence. Mathematica, 12,  (35), 2 309-324 (1970).

[3] Weinisckhe, J. H., Über eine Klasse von Iterationsverfahren. Numeriche Mathematik , 6, 395-404, (1964).

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Cite this paper as:

I. Păvăloiu, Une généralisation de methode de Newton, Mathematica, 20(43) (1978) no. 1, pp. 45-52 (in French).

Journal

Mathematica

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