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
(Tiberiu Popoviciu Institute of Numerical Analysis)
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
PDF-Latex version of the paper. (in English)
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).
About this paper
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.
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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Sur l’estimation des erreurs en convergence numérique de certaines méthodes d’itération (English translation) On the estimation of errors in numerical convergence of certain iteration methods
Sur l’estimation des erreurs en convergence numérique de certaines méthodes d’itération
"Babeş-Bolyai" University
Faculty of Mathematics and Physics
Research Seminars
Seminar on Functional Analysis and Numerical Methods
Preprint nr.1, 1986, pp.133-136
Sur l’estimation des erreurs en convergence numérique de certaines méthodes d’itération
(English translation)
On the estimation of errors in numerical convergence of certain iteration methods
Let be a Banach space and
| (1) |
an operatorial equation, where and is the null element of the space
In order to solve equation (1) we consider the following iterative process:
| (2) |
We denote by the ball of radius and center .
Regarding the convergence of iterations (2) we shall consider the following theorem
Theorem 1.
If the application and the element from equation (1) verify the following conditions:
-
i.
for all where is a functional;
-
ii.
-
iii.
then equation (1) admits a unique solution
This solution is obtained as the limit of the sequence generated by method (2), and the following estimation holds:
| (3) |
Let be an application which verifies the conditions:
-
i
where and
-
ii
for every where
-
iii
whereand
-
iv
We consider an element for which whereand
In order to solve equation (1) we consider, instead of iterative method (2) the following iterative procedure:
| (4) |
Regarding the convergence of method (4) we obtain the following theorem:
Theorem 2.
If the conditions of Theorem 1 are fulfilled, the operator and the element verify conditions – and if
| (5) |
then there exists an so that for all we have
| (6) |
and
| (7) |
Proof.
Indeed, from the fact that and it follows that there exists a number so that for we have
and
for because for and as .
If we take now
then the theorem is proved. ∎
We show now that in the conditions of Theorem 2 we have the following estimation:
| (8) |
References
- [1]
- [2] Babici, D.M., Ivanov, V.N., Oţenca polnoi progresnosti prireshenia nelineinyh operatornyh uravnenii metodov prostei iteraţii. Jurnal vycislitelnoi matematiki i matematiceskoi fisiki 7, 5 (1967), 988–1000.
- [3] Păvăloiu, I., ††margin: clickable Introduction in the Theory of Approximating the Solutions of Equations, Ed. Dacia 1976 (in Romanian).
- [4] Urabe, M., Error estimation in numerical solution of equations by iteration process, J. Sci. Hiroshima Univ. Ser. A-I, 26 (1962), 77–91.