A polynomial preconditioner for the GMRES algorithm

Summary and conclusions In this paper a least-squares polynomial preconditioner of low degree for use with the GMRES algorithm is discussed. The parameters of the polynomial preconditioner are computed from eigenvalue estimates that can be obtained from the GMRES iterative process. Since we have considered polynomials of low degree only stability is not a major problem. The algorithm is simple and therefore will applicable in real life. Two different methods for obtaining eigenvalue estimates are discussed. A simple and stable approach is to calculate the Ritz values with Arnoldi’s method. In literature it was reported that the zeroes of the GMRES residual polynomial may yield more robust eigenvalue estimates. We describe in this paper a stable strategy for computing these zeroes. An experiment where polynomial preconditioners based on these two eigenvalue estimates are compared renders insufficient information to draw soundly based conclusions, and more research on this topic will.be necessary. Experiments indicate that a considerable reduction of the CPU-time can be reached if the polynomial preconditioner is applied in combination with GMRES with restarts after cycles of iterations. The experiments do not indicate that a significantly greater reduction of the CPU-time can be achieved if a polynomial preconditioner of high degree is applied. We observed only a marginal reduction of the CPU-time if a polynomial preconditioner of degree higher than two was applied. For full GMRES no reduction of the number of matrix-vector products can be achieved. However, since the orthogonalization of the basis for the Krylov subspace becomes increasingly expensive an important reduction of the CPU-time can be achieved by applying the polynomial preconditioner. Moreover, since the number of iterations is decreased if the polynomial preconditioner is applied, far less basis vectors for the Krylov subspace need to be stored. We have