Error estimations in the numerical solving of the systems of equations

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.

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

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

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

Scanned paper.

PDF-Latex version of the paper.

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).

[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.