## Abstract

We study the nonlinear equations of the form \[x=\lambda D\left( x\right) +y,\] where \(\lambda \in \mathbb{R}\) and \(y\in E\) are fixed, and \(D:E\rightarrow E,\) with \(D\left( 0\right) =0\) a nonlinear mapping on the Banach space \(E\). We consider the iterative method \[\xi_{n+1}=\lambda D_{\varepsilon}\left( \xi_{n}\right) +y_{\varepsilon},\] where \(D_{\varepsilon}\) is an operator which approximates \(D\) and \(y_{\varepsilon}\) is an approximation for \(y\). We obtain an evaluation for \(\left \Vert \bar{x}-\xi_{n+1}\right \Vert \) in terms of \(\left \Vert D_{\varepsilon}\left( x\right) -D\left( x\right) \right \Vert \) and \(\left \Vert y-y_{\varepsilon}\right \Vert \).

## Authors

Ion Păvăloiu

## Title

### Original title (in French)

*Sur l’estimation des erreurs en convergence numérique de certaines méthodes d’iteration*

### English translation of the title

*On the error estimation in the numerical convergence of certain iterative methods*

## Keywords

nonlinear equation in Banach space; iterative method; error estimation

## References

[1] Babici, D.M., Ivanov, V.N., *Otenca polnoi progresnosti prireshenia nelineinyh operatornyh uravnenii metodov prostei iteratii*. Jurnal vycislitelnoi matematiki i matematiceskoi fisiki 7, 5 (1967), 988–1000.

[2] Pavaloiu, I., *Introduction in the Theory of Approximating the Solutions of Equations*, Ed. Dacia 1976 (in Romanian).

[3] Urabe, M., *Error estimation in numerical solution of equations by iteration process*, J. Sci. Hiroshima Univ. Ser. A-I, 26 (1962), 77–91

## About this paper

##### Cite this paper as:

I. Păvăloiu, *Sur l’estimation des erreurs en convergence numérique de certaines méthodes d’iteration*, Seminar on functional analysis and numerical methods, Preprint no. 1 (1986), pp. 133-136 (in French, English translation provided).

##### Journal

Seminar on functional analysis and numerical methods,

Preprint

##### Publisher Name

“Babes-Bolyai” University,

Faculty of Mathematics and Physics,

Research Seminars

##### DOI

Not available yet.