The spectrum of circulant-like preconditioners for some general linear multistep formulas for linear boundary value problems

The spectrum of the eigenvalues, the conditioning, and other related properties of circulant-like matrices used to build up block preconditioners for the nonsymmetric algebraic linear equations of time-step integrators for linear boundary value problems are analyzed. Moreover, results concerning the entries of a class of Toeplitz matrices related to the latter are proposed. Generalizations of implicit linear multistep formulas in boundary value form are considered in more detail. It is proven that there exists a new class of approximations which are well conditioned and whose eigenvalues have positive and bounded real and bounded imaginary part. Moreover, it is observed that preconditioners based on other circulant-like approximations, which are well suited for Hermitian linear systems, can be severely ill conditioned even if the matrices of the nonpreconditioned system are well conditioned.