Multigrid Methods for Symmetric Positive Definite Block Toeplitz Matrices with Nonnegative Generating Functions

Abstract

In this paper we introduce a generalized multigrid method for solving linear systems ${\bf T}_{N, M} {\bf x} = {\bf b}$ where ${\bf T}_{N, M} \in \Re^{NM \times NM}$ is a symmetric block Toeplitz matrix with symmetric Toeplitz blocks. We use a special choice of the projection operator whose coefficients simply depend on the generating functions associated with the proposed class of matrices. This choice leads to iterative methods with convergence rates independent of the Euclidean condition number $\kappa_2 ({\bf T}_{N, M})$ and of the dimension of the involved matrices. The total arithmetic cost is $O(NM \log(NM))$ for dense matrices and $O(NM)$ for band matrices with band blocks; in the PRAM model only $O(\log(NM))$ parallel steps are required. This algorithm is therefore competitive with the preconditioned conjugate gradient methods proposed by Chan and Jin [SIAM J. Sci. Statist. Comput., 13 (1992), pp. 1218--1235], Ku and Kuo [SIAM J. Sci. Statist. Comput., 13 (1992), pp. 948--966], Di Benedetto [SIAM J. Sci. Comput., 17 (1996)], and Serra [BIT, 34 (1994), pp. 579--594] for dense matrices and improves those results for band block Toeplitz matrices with band Toeplitz blocks.