C.G. Preconditioning for Toeplitz Matrices

We consider the problem of solving a Toeplitz system of equations by conjugate gradient method. When a sequence of nested Toeplitz matrices is associated to a function, the spectral behaviour of the matrices involved is closely related to the analytical properties of the generating function. Thus, it is possible to devise efficient preconditioning techniques by using various functional approximation strategies. This approach leads to attractive results in the case of ill-conditioned matrices, for which a wide class of preconditioners are proposed.