Optimal preconditioners for systems defined by functions of Toeplitz matrices

We propose several circulant preconditioners for systems defined by some functions g of Toeplitz matrices An. In this paper we are interested in solving g(An)x=b by the preconditioned conjugate method or the preconditioned minimal residual method, namely in the cases when g(z) are the functions ez, sin⁡z and cos⁡z. Numerical results are given to show the effectiveness of the proposed preconditioners.