Convergence of Matrix Iterations Subject to Diagonal Dominance

TLDR: In this paper, the convergence conditions for stationary relaxation processes of Gauss-Seidel and Jacobi types were established for linear systems of linear equations with respect to necessary and sufficient conditions for convergence.

Abstract: Cyclic iterative methods of solving systems of linear equations are investigated with reference to necessary and sufficient conditions for convergence. A new and concise method of proof is given for the convergence of point Gauss–Seidel and Jacobi iterations subject to strict and irreducible weak diagonal dominance. The method is developed to obtain convergence conditions, not previously established, for stationary relaxation processes of Gauss–Seidel and Jacobi type. Bounds are obtained on the spectral radius of the iteration matrix, hence a range of values of the relaxation parameter is derived sufficient to guarantee convergence.