A Note on Preconditioned Block Toeplitz Matrices

Abstract

In [SIAM J. Sci. Statist. Comput., 13 (1992), pp. 948–966], Ku and Kuo proposed and analysed a block circulant preconditioner $R_{mn} $ for solving a family of block Toeplitz systems $T_{mn} v = b$. For a special class of block matrices called the quadrantally symmetric Toeplitz matrices, they proved that the eigenvalues of $R_{mn}^{ - 1} T_{mn} $ are clustered around one except at most $O(m + n)$ outliers with $T_{mn} $ generated by a two-dimensional rational function. The superior convergence rate of the preconditioned conjugate gradient (PCG) method is explained by the clustering property of the spectrum of $R_{mn}^{ - 1} T_{mn} $. However, in their analysis, there is no discussion on the positive definiteness of the matrix $T_{mn} $, and the preconditioner $R_{mn} $ is assumed to be invertible. In this paper, we give some results on these two aspects. Under the assumptions in the above-referenced paper, we prove that if the generating function $f(x,y)$ of $T_{mn} $ is positive, then $T_{mn} $ is positive definite. Moreover, we show that $R_{mn} $ is uniformly invertible when m and n are sufficiently large.