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 2pi-periodic continuous complex-valued function without any zeros, then the spectrum of the normalized preconditioned matrix (C(n)-1A(n))*(C(n)-1A(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(alphan log2 n + n log n).