Improved Schur complement preconditioners for block-Toeplitz systems with small size blocks

In this paper, we employ the preconditioned conjugate gradient method with the Improved Schur complement preconditioners for Hermitian positive definite block-Toeplitz systems with small size blocks. Schur complement preconditioners have been proved to be an effective method for such block-Toeplitz systems (Ching et al. 2007). The modification is based on Taylor expansion approximation. We prove that the matrices preconditioned by improved Schur preconditioners have more clustered spectra compared to that of the Schur complement preconditioners. Hence, preconditioned conjugate gradient type methods will converge faster. Numerical examples are given to demonstrate the efficiency of the proposed method.