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

## 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
(Tiberiu Popoviciu Institute of Numerical Analysis)

## 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

## PDF

##### 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.

## References

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