Block Diagonal and Schur Complement Preconditioners for Block-Toeplitz Systems with Small Size Blocks

Abstract

In this paper we consider the solution of Hermitian positive definite block-Toeplitz systems with small size blocks. We propose and study block diagonal and Schur complement preconditioners for such block-Toeplitz matrices. We show that for some block-Toeplitz matrices, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers where this fixed number depends only on the size of the block. Hence, conjugate gradient type methods, when applied to solving these preconditioned block-Toeplitz systems with small size blocks, converge very fast. Recursive computation of such block diagonal and Schur complement preconditioners is considered by using the nice matrix representation of the inverse of a block-Toeplitz matrix. Applications to block-Toeplitz systems arising from least squares filtering problems and queueing networks are presented. Numerical examples are given to demonstrate the effectiveness of the proposed method.