The Spectrum of a Family of Circulant Preconditioned Toeplitz Systems

Abstract

The solutions of symmetric positive definite Toeplitz systems $Ax = b$ are studied by the preconditioned conjugate gradient method. The preconditioner is the circulant matrix C that minimizes the Frobenius norm $\| {C - A} \|_F $ [T. Chan, “An Optimal Circulant Preconditioner for Toeplitz Systems,” UCLA Department of Mathematics, CAM Report 87-06, June 1987]. The convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of $C^{ - 1} A$. For Toeplitz matrix A with entries which are Fourier coefficients of a positive function in the Wiener class, this paper establishes the invertibility of C, finds the asymptotic behaviour of the eigenvalues of the preconditioned matrix $C^{ - 1} A$ as the dimension increases and proves that they are clustered around 1.