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

## 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 $$\left \Vert y-y_{\varepsilon}\right \Vert$$.

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

## PDF

##### 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.