|Title:||Application of Vector Extrapolation Methods to Consistent Singular Linear Systems
|Abstract:||Consider the linear system of equations Bx=f , where B is an NxN 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 previoU$ work on vector extrapolation methods as they apply to nonsingular B. In particular, we sbow 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 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.|
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