Circulant Preconditioners for Ill-Conditioned Hermitian Toeplitz Matrices

In this paper, we propose a new family of circulant preconditioners for solving ill-conditioned Hermitian Toeplitz systems Ax = b. The eigenval-ues of the preconditioners are given by the convolution products of the generating function f of A with some summation kernels. When f is a nonnegative 2-periodic continuous function deened on ?; ] with a zero of order 2p, we show that the circulant preconditioners are positive deenite and the spectrum of the preconditioned matrix is uniformly bounded in a; b] with at most 2p+2 outliers where 0 < a < b < 1. Hence the linear system can be solved by the preconditioned conjugate gradient method eeciently. We emphasize that the construction of the circulant preconditioner does not require the explicit knowledge of the generating function. Numerical results are included.