Convergence and Comparison Theorems for Various Splittings of Matrices Based on Generalized Inverses

Abstract

The convergence of iterative methods for numerically solving the linear systems of equations associated with different types of splittings has been well studied in the literature. In this dissertation, we define new types of splittings based on generalized inverses (Moore-Penrose inverse, Drazin inverse, Group inverse) and present convergence results based on those splittings, namely, Weak Nonnegative Proper Splitting, Proper Regular Splitting, Index Proper Regular Splitting, Weak Nonnegative Index Proper Splitting, Group Proper Regular Splitting, and Weak Nonnegative Group Proper Splitting. We also present some new comparison theorems between the spectral radii of matrices, which are useful in the analysis of the rate of convergence of iterative methods for the different types of splittings of matrices.