Morgan 9522612 In this project, improvements of the restarted GMRES method for solving large nonsymmetric systems of linear equations will be investigated. The goal is to develop a method that is competitive with short recurrence Lanczos methods such as QMR, but maintains the desirable properties of GMRES. These properties include extraction of the minimum residual solution from the subspace, generally steady convergence, and numerical stability. It is also desired to not significantly raise cost per iteration and storage requirements. The convergence of GMRES can be improved by including approximate eigenvectors in the subspace. Once these eigenvalues are essentially deflated from the spectrum of the matrix, removing even just a few small eigenvalues from the spectrum can significantly improve the convergence. In this project a new and efficient way of including eigenvectors is proposed. Sorensen's implicit-QR Arnoldi method for eigenvalues is combined with an interior eigenvalue version of Arnoldi. The surprising result is that the approximate eigenvectors become part of a new Krylov subspace. This Krylov subspace can be used with GMRES. The expense per iteration is even less than that of regular GMRES for the case where matrix-vector products are expensive, and the number of iterations needed is sometimes greatly reduced. The resulting implicitly restarted GMRES method will be studied and compared with other methods. ***
