Preconditioned iterative methods for nonsymmetric Toeplitz and block Toeplitz matrices

Linear systems with nonsingular Toeplitz or block Toeplitz matrices arise in many applications, notably when discretizing partial differential, fractional differential or integral equations using constant time steps. These linear systems are amenable to solution by iterative methods, e.g., Krylov subspace methods, but to keep the number of iterations low preconditioning is typically required.

The goal of preconditioning is form an equivalent linear system to the original that is somehow easier to solve. When the (block) Toeplitz matrix is symmetric, descriptive convergence theory guides the choice of preconditioner, but in the nonsymmetric case preconditioning is largely heuristic. In this talk we show how to symmetrize (block) Toeplitz matrices, so that the descriptive convergence theory for symmetric problems can be applied in order to design preconditioners that are guaranteed to be effective. Our numerical experiments validate the efficiency and robustness of the proposed approach.