Development of iterative techniques and extrapolation methods for Drazin inverse solution of consistent or inconsistent singular linear systems

Abstract

Consider the linear system of equations Bx=ƒ, where B is an Nx N singular matrix. In an earlier work by the author it was shown that iterative techniques coupled with standard vector extrapolation methods can be used to obtain or approximate a solution of this system when it is consistent. In the present work we expand on that approach to treat the case in which this system is in general inconsistent. Starting with Richardson's iterative method, we develop a family of new iterative techniques and vector extrapolation methods that enable us to obtain or approximate the Drazin inverse solution of this system whether the index of B is 1 or greater than 1. We show that the Drazin inverse solution can be constructed from a finite number of iterations, this number being at most N+2. We also provide detailed convergence analyses of the new iterative techniques and vector extrapolation methods and give their precise rates of convergence.