Analysis of an Implicitly Restarted Simpler GMRES Variant of Augmented GMRES

Abstract

We analyze a Simpler GMRES variant of augmented GMRES with implicit restarting for solving nonsymmetric linear systems with small eigenvalues. The use of a shifted Arnoldi process in the Simpler GMRES variant for computing Arnoldi basis vectors has the advantage of not requiring an upper Hessenberg factorization and this often leads to cheaper implementations. However the use of a non-orthogonal basis has been identified as a potential weakness of the Simpler GMRES algorithm. Augmented variants of GMRES also employ non-orthogonal basis vectors since approximate eigenvectors are added to the Arnoldi basis vectors at the end of a cycle and in case the approximate eigenvectors are ill-conditioned, this may have an adverse effect on the accuracy of the computed solution. This problem is the focus of our paper where we analyze the shifted Arnoldi implementation of augmented GMRES with implicit restarting and compare its performance and accuracy with that based on the Arnoldi process. We show that augmented Simpler GMRES with implicit restarting involves a transformation matrix which leads to an efficient implementation and we theoretically show that our implementation generates the same subspace as the corresponding GMRES variant. We describe various numerical tests that indicate that in cases where both variants are successful, our method based on Simpler GMRES keeps comparable accuracy as the augmented GMRES variant. Also, the Simpler GMRES variants perform better in terms of computational time required.