Preconditioning for nonsymmetry and time-dependence

For symmetric systems, rigorous convergence bounds can be obtained which depend only on the eigenvalues of the system. However for nonsymmetric systems, no generally descriptive convergence bounds are known and therefore the development of preconditioners for these problems is typically heuristic. In this paper, we describe one simple but frequently occurring example of nonsymmetric Toeplitz matrices, where we are able to guarantee rapid convergence of an appropriate iterative method. The method employs a simple trick of reordering the variables to rewrite the system as a symmetric one. A symmetric positive definite absolute value preconditioner is also proposed which is used within a standard symmetric solver such as minres. We also show how this can be applied to time-dependent linear ODEs which are inherently nonsymmetric.