## Abstract

Let \(\left( x_{i},\rho_{i}\right) ,\ i=1,2,\) be two complete metric space and \(F_{1}:X_{1}\times X_{2}\rightarrow X_{1},\ F_{2}:X_{1}\times X_{2}\rightarrow X_{2}\) two nonlinear mappings. We study the solving of the system \begin{align}

x_{1} & =F_{1}\left( x_{1},x_{2}\right) \label{f.1}\\

x_{2} & =F_{2}\left( x_{1},x_{2}\right) ,\qquad \left( x_{1},x_{2}\right)

\in X.\nonumber

\end{align} by the Gauss-Seidel type method \begin{align}

x_{1}^{\left( n+1\right) } & =F_{1}\left( x_{1}^{\left( n\right)

},x_{2}^{\left( n\right) }\right) \label{f.2}\\

x_{2}^{\left( n+1\right) } & =F_{2}\left( x_{1}^{\left( n+1\right)

},x_{2}^{\left( n\right) }\right) ,\qquad n=0,1,\ldots;\left( x_{1}^{\left(

0\right) },x_{2}^{\left( 0\right) }\right) \in X\nonumber

\end{align} We give sufficient conditions for convergence and some error estimations. We also study the case when the mappings \(F_{1}\) and \(F_{2}\) are replaced by some approximations.

## Authors

Ion Păvăloiu

## Title

### Original title (in French)

*Délimitation des erreur dans la résolution numérique des systèmes d’equations*

### English translation of the title

*Error estimations in the numerical solving of the systems of equations*

## Keywords

nonlinear system in metric space; Gauss-Seidel type method; convergence; approximate value

## References

[1] Pavaloiu, I.,* Introducere in teoria aproximarii solutiilor ecuatiilor*, Editura Dacia, Cluj-Napoca, 1976.

[2] Pavaloiu, I., *La resolution des systemes operationnelles a l’aide des methodes iteratives*, Mathematica, 11(34), (1969), 137–141.

[3] Pavaloiu, I., *Estimation des erreurs dans la resolution numerique des systemes d’equations dans des espaces metriques*, Seminar on Functional Analysis and Numerical Methods, Preprint Nr. 1, (1987), 121–129.

[4] Pavaloiu, I., *La convergence de certaines methodes iteratives pour resoudre certaines equations operatorielles*, Seminar on Functional analysis and Numerical Methods, Preprint Nr. 1 (1986), 127–132.

[5] Traub, J. F., *Iterative Methods for the Solution of Equations*, Prentice Hall Series in Automatic Computation, Englewood Cliffs, N. J. (1964).

[6] Urabe, M., *Convergence of numerical iteration in solution of equations,* J. Sci. Hiroshima Univ. Ser. A, 19 (1956), 479–489.

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

Scanned paper.

## About this paper

##### Cite this paper as:

I. Păvăloiu, *Délimitation des erreur dans la résolution numérique des systèmes d’equations*, Seminar on mathematical analysis, Preprint no. 7 (1988), pp. 167-178 (in French).

##### Journal

Seminar on mathematical analysis,

Preprint

##### Publisher Name

“Babes-Bolyai” University,

Faculty of Mathematics,

Research Seminars

##### DOI

Not available yet.