Can symmetric Toeplitz solvers be strongly stable?

Abstract

We describe symmetric positive definite Toeplitz systems for which the floating point vector that is closest to the actual solution is not the solution of a nearby symmetric Toeplitz system. With these systems we are able to show that a large class of Toeplitz solvers are not strongly stable for solving symmetric (or symmetric positive definite) Toeplitz systems; i.e., the computed solution is not necessarily the solution of a nearby symmetric Toeplitz system. This class of algorithms includes Gaussian elimination and seems to include all known fast and superfast Toeplitz solvers; certainly, it includes the better known ones. These results strongly suggest that all symmetric Toeplitz solvers are not strongly stable.