Fast Band-Toeplitz Preconditioners for Hermitian Toeplitz Systems

We consider the solutions of Hermitian Toeplitz systems where the Toeplitz matrices are generated by nonnegative functions f. The preconditioned conjugate gradient method with well-known circulant preconditioners fails in the case when f has zeros. In this paper, we employ Toeplitz matrices of xed band-width as preconditioners. Their generating functions g are trigonometric poly-nomials of xed degree and are determined by minimizing the maximum relative error jj(f ?g)=fjj1. We show that the condition number of systems preconditioned by the band-Toeplitz matrices are O(1) for f with or without zeros. When f is positive, our preconditioned systems converge at the same rate as other well-known circulant preconditioned systems. We also give an a priori bound of the number of iterations required for convergence.