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

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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.