Circulant preconditioners for solving singular perturbation delay differential equations

We consider the solution of singular perturbation delay di erential equations (SPDDEs) by using boundary value methods (BVMs). These methods require the solution of some nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed for solving these linear systems. We prove that if an Ak1 ; k2 -stable BVM is used for solving a system of SPDDEs, then our preconditioner is invertible and the eigenvalues of the preconditioned system are clustered. When the GMRES method is applied to the preconditioned systems, the method would converge fast. Numerical results are given to show the e ectiveness of our methods. Copyright ? 2004 John Wiley & Sons, Ltd.