## Abstract

Let \(X\) be a Banach space, \(Y\) a normed space and \(P:X\rightarrow Y\) a nonlinear operator. We study the convergence of the following method for solving the equation \(P\left( x\right) =0\) \[ x_{n+1}=Q\left( x_{n}\right) -\left[ P^{\prime}\left( x_{n}\right) \right] ^{-1}P\left( Q(x_n)\right),\ n=0,1,…, \ x_{0}\in X \] where \(Q\) is a nonlinear operator associated to the nonlinear equation \(P\left( x\right) =0\). We show that if the successive approximations of \(Q\) converge with order \(k\geq2\), there the above sequence converge to the solution with order \(k+1\).

## Authors

Ion Păvăloiu

## Title

### Original title (in French)

* Sur la convergence d’une classe de méthodes itératives de J.F. Traub*

### English translation of the title

*On the convergence of a class of a iteration methods of J.F. Traub*

## Keywords

Traub method; iterative method; nonlinear operator equation; convergence order; semilocal convergence

## References

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

[2] Pavaloiu, I., *Sur les procedes iteratifs a un ordre eleve de convergence*. Mathematica Cluj, 12 (35), 12 , 309-324 (1970).

[3] Pavaloiu, I., *Asupra operatorilor iterativi*, Studii si cercetari matematice, 10, 23, 1537–1544 (1971).

[4] Traub, J. F., *Iterative Methods for the Solution of Equations*, Prentice-Hall. Inc. Englewood Cliffs N. J., 1964.

Scanned paper.

## About this paper

##### Cite this paper as:

I. Păvăloiu, *Sur le convergence d’une classe de méthodes itératives de J. Traub*, Rev. Anal. Numér. Théor. Approx., **2** (1973), pp. 99-104 (in French).

##### Journal

Revue d’analyse numerique et de la Theorie de l’Approximation

##### Publisher Name

Academia R.S. Romane

##### Journal website

##### Print ISSN

0301-9241

##### Online ISSN

2457-810X