Circulant preconditioners for Toeplitz-block matrices

We propose two block preconditioners for Toeplitz-block matrices (i.e. each block is Toeplitz), intended to be used in conjunction with conjugate gradient methods. These preconditioners employ and extend existing circulant preconditioners for point Toeplitz matrices. The two preconditioners differ in whether the point circulant approximation is used once or twice, and also in the cost per step. We discuss efficient implementation of these two preconditioners, as well as some basic theoretical properties (such as preservation of symmetry and positive definiteness). We report results of numerical experiments, including an example from active noise control, to compare their performance.