Circulant preconditioners for complex Toeplitz matrices

The solution of n-by-n complex Toeplitz systems $A_n x = b$ by the preconditioned conjugate gradient method is studied. The preconditioner $C_n$ is the circulant matrix that minimizes $||B_n - A_n ||_F $ over all circulant matrices $B_n$ . The authors prove that if the generating function of $A_n$ is a $2\mu $-periodic continuous complex-valued function without any zeros, then the spectrum of the normalized preconditioned matrix $(C_n^{ - 1} A_n )^ * (C_n^{ - 1} A_n )$ will be clustered around one. Hence they show that if the condition number of $A_n$ is of $O(n^\alpha )$, the conjugate gradient method, when applied to solving the normalized preconditioned system, converges in at most $O(\alpha \log n + 1)$ steps. Thus the total complexity of the algorithm is $O(\alpha n\log ^2 n + 1\log n)$.