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 $A_n$. In this paper we are interested in solving $g(A_n)\mathbf{x}=\mathbf{b}$ by the preconditioned conjugate method or the preconditioned minimal residual method, namely in the cases when $g(z)$ are the functions $e^z$, $\sin z$ and $\cos z$. Numerical results are given to show the effectiveness of the proposed preconditioners.