A New Preconditioner for Toeplitz Matrices

In this paper we introduce and analyze a new preconditioner for Toeplitz matrices that exhibits excellent spectral properties: the eigenvalues of the preconditioned matrix are highly clustered around the unity. As a result, it yields very rapid convergence when used to solve Toeplitz equations via the preconditioned conjugate gradient method. The new preconditioner can be regarded as a refinement of preconditioners built by embedding the Toeplitz matrix in a positive definite circulant. Necessary and sufficient conditions that ensure that the positive definite embedding is possible are given.