## 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).

## About this paper

##### 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

##### Publisher Name

##### DOI

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