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

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
(Tiberiu Popoviciu Institute of Numerical Analysis)

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

nonlinear equation in Banach space; iterative method; error estimation

Scanned paper.

PDF-Latex version of the paper. (in English)

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).

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