Upper Bounds of the Spectral Radii of Some Iterative Matrices

Abstract

In this paper, the concept of optimally scaled matrix and the estimate of $\|M^{-1}N\|_{\infty}$ in our previous paper are used to find the upper bounds of the spectral radii of the iterative matrices SOR, SSOR, AOR and SAOR. The sharpness of the upper bounds of the spectral radii of SOR and AOR is established. The proofs are very intuitive and may be considered as the geometrical interpretations of our theorems.