# Posts by Ion Păvăloiu

## 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

[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.

## 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…

## An algorithm in the solving of equations by interpolation

Abstract Consider the nonlinear equation in $$R$$, $$f\left( x\right) =0$$, where $$f:A\rightarrow B$$, $$(A,B\subseteq \mathbb{R})$$ which is assumed bijective. The Lagrange…

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

Abstract Let $$X_{1},X_{2}$$ be two complete metric spaces, $$X=X_{1}\times X_{2}$$ and the nonlinear mappings $$F_{1}:X\rightarrow X_{1},\ F_{2}:X\rightarrow X_{2}$$. In order…

## 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 $$y\in E$$…

## Solving equations with the aid of inverse rational interpolation functions

Abstract We study the convergence of an iterative method for solving the equation $$f\left( x\right) =0,\ f:A\rightarrow B$$, $$A,B\subseteq \mathbb{R}$$, $$f$$…

## Solving equations by Hermite type inverse interpolation

Abstract We study the convergence of an iterative method for solving the equation (fleft( xright) =0, f:Isubseteq mathbb{Rrightarrow R}). The…

## Solving equations with the aid of inverse interpolation spline functions

Abstract We consider the solving of a nonlinear equation in $$\mathbb{R}$$. We construct a spline function which approximates the nonlinear…

## Optimal Steffensen type iterative methods obtained by inverse interpolation

Abstract Let $$f:I\subset \mathbb{R\rightarrow R}$$ be a nonlinear mapping and the equation $$f\left( x\right) =0$$ with solution $$x^{\ast}$$; consider the…