Incomplete factorization methods for banded toeplitz matrices

We describe two incomplete factorization methods which can be applied to solve banded Toeplitz (or dose to Toeplitz) linear systems. This approach leads to efficient sequential and parallel algorithms for the solution and/or the preconditioning of such systems. We discuss the question of the existence of the proposed factorizations, which are related to the existence of a solvent of suitable matrix equations of size dependent on the bandwidth of T. The application of these techniques to the solution of linear systems arising from the discrete approximation of Poisson and Biharmonic equations is also considered.