Remarks on solving the systems of equations by iterative methods

In this paper we give a condition for the convergence of an iterative method of Gauss-Seidel type  \[ x_i = \varphi (x_{i-1}, y_{i-1}) \\ y_i = \psi (x_i, y_{i-1})\] for solving the nonlinear system of equations of the type: \[ x = \varphi (x, y) \\ y = \psi (x, y). \] We then provide some error evaluations in case of exact and approximate computations. We also provide a sufficient condition when the system contains \(k\) equations with \(k\) unknowns \[x_i = \varphi_i (x_1,x_2,…,x_k), \quad i=1,…,k.\]

Ion Păvăloiu  – Tiberiu Popoviciu Institute of Numerical Analysis

Original Title (in Romanian)

Observații asupra rezolvării sistemelor de ecuații cu ajutorul procedeelor iterative

English Translation of the Title

Remarks on solving the systems of equations by iterative methods

nonlinear system of equations; fixed point problem; Gauss-Seidel method.

1. B.P. Demidovici, I.A. Maron, Osnovi vacislietel’noi matematiki, Gas. izd. fiz. mat. lit., Moskva, 1960, pp. 148–151.
2. J.F. Traub, Iterative Methods for the Solution of Equations. Prentice Hall, Inc., Englewwod Cliffs, N.J., 1964, 99.38-39.
3. A.M. Ostrowski, Resenie uravnenii i sistem uravnenii, Izd. inostr. lit., Moskva, 1963, pp. 83–94.
4. I. Pavaloiu, On some recurrent inequalities and some of their applications, Communication at the Scientific Session of the Institute for Mining Petrosani, February 7–10, 1966.

LaTeX-PDF version of the paper (En)

Published version (scanned) (Ro)

I. Păvăloiu, Observaţii asupra rezolvării sistemelor de ecuaţii cu ajutorul procedeelor iterative, Studii şi cercetări matematice, 19 (1967) no. 9, pp. 1289-1298.

Google scholar – not available yet