Application of Vector Extrapolation Methods to Consistent Singular Linear Systems

Consider the linear system of equations Bx = f, where B is an N x N singular matrix, but the system is consistent. In this work we show that iterative techniques coupled with vector extrapolation methods can be used to obtain (approximations to) a solution of Bx = f. We do this by extending the results of some previous work on vector extrapolation methods as they apply to nonsingular matrices B. In particular, we show that the minimal polynomial, reduced rank, and modified minimal polynomial extrapolation methods, and the scalar, topological, and vector epsilon algorithms all produce a solution of Bx = f in at most rank(B) < N 1 steps, and that this solution depends on the initial approximation in a simple way. Asymptotic error estimates and error bounds are given for two different limiting procedures that have been considered in previous work. Although we demonstrate all our results for Richardson’s iterative method, they are equally valid for any other iterative method.