Inverse generating function approach for the preconditioning of Toeplitz-block systems: Inverse generating function approach for Toeplitz-block systems

As proposed by R. H. Chan and M. K. Ng (1993), linear systems of the form T [f] x = b, where T [f] denotes the n x n Toeplitz matrix generated by the function f, can be solved using iterative solvers with T[1/f] as a preconditioner. This article aims at generalizing this approach to the case of Toeplitz-block matrices and matrix-valued generating functions F. We prove that if F is Hermitian positive definite, most eigenvalues of the preconditioned matrix T [F-1] T[F] are clustered around one. Numerical experiments demonstrate the performance of this preconditioner.