Analysis of Some Vector Extrapolation Methods for Solving Systems of Linear Equations

In this paper, we consider some vector extrapolation methods for solving nonsymmetric systems of linear equations. When applied to sequences generated linearly, these methods namely the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE), are Krylov subspaces methods and are respectively equivalent to the method of Arnoldi and to the GCR and GMRES. By considering the geometrical aspect of these methods, we derive new expressions for their residual norms and give a relationship between them; this allows us to compare the two methods. Using this new approach, we will show that for nonsingular skew symmetric matrices the GMRES stagnates every two iterations and the restarted version GMRES(m) (m 2) is always convergent. Finally, the incomplete forms are considered and some convergence results are given.