# Error estimation in numerical solution of equations and systems of equations

## Abstract

In \cite{7 Urabe}, \cite{8 Urabe} M. Urabe studies the numerical convergence and error estimation in the case of operatorial equation solution by means of iteration methods. Urabe’s results refer to operatorial equations in complete metric spaces, while as application the numerical convergence of Newton’s method in Banach spaces is studied. Using Urabe’s results, M. Fujii \cite{1 Fujii} studies the same problems for Steffensen’s method and the chord method applied to equations with real functions. In \cite{6 Pavaloiu} Urabe’s method is applied to a large class of iteration methods with arbitrary convergence order. We propose further down to extend Urabe’s results to the case of the Gauss-Seidel method for systems of equations in metric spaces.

Ion Păvăloiu

## Keywords

systems of equations in metric spaces; error estimations; Gauss-Seidel method

## References

 Fujii, M., Remarks to accelerated iterative processes for numerical solution of equations, J. Sci. Hiroshima Univ. Ser. A-I, 27 (1963), 97–118.

 Lanncaster, P., Error for the Newton-Raphson method, Numerische Mathematik, 9, 1, (1966), 55–68.

 Ostrowski, A., The round of stability of iterations, Z.A.M.M. 47, (1967), 77–81

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

 Pavaloiu, I., Delimitations des erreurs dans la resolution numerique des systemes d’equations, Research Seminars. Seminar on Mathematical analysis. Preprint Nr. 7, (1988), 167–178.

 Pavaloiu, I., Introducere in teoria aproximarii solutiilor ecuatiilor. Ed. Dacia, Cluj- Napoca, (1976).

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

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

 Varga, R.S., Matrix Iterative Analysis, Englewood Cliffs H.J. Prentice Hall (1962).

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##### Cite this paper as:

I. Păvăloiu, Error estimation in numerical solution of equations and systems of equations, Rev. Anal. Numér. Théor. Approx., 21 (1992) no. 2, pp. 153-165.

1222-9024

2457-8126
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