Multigrid Method for Ill-Conditioned Symmetric Toeplitz Systems

In this paper, we consider solutions of Toeplitz systems A n u = b where the Toeplitz matrices A n are generated by nonnegative functions with zeros. Since the matrices A n are ill-conditioned, the convergence factor of classical iterative methods, such as the damped Jacobi method, will approach 1 as the size n of the matrices becomes large. Here we propose to solve the systems by the multigrid method. The cost per iteration for the method is of O(n log n) operations. For a class of Toeplitz matrices which includes weakly diagonally dominant Toeplitz matrices, we show that the convergence factor of the two-grid method is uniformly bounded below 1 independent of n and the full multigrid method has convergence factor depends only on the number of levels. Numerical results are given to illustrate the rate of convergence.