Rational approximation preconditioners for sparse linear systems

This paper presents a class of preconditioning techniques which exploit rational function approximations to the inverse of the original matrix. The matrix is first shifted and then an incomplete LU factorization of the resulting matrix is computed. The resulting factors are then used to compute a better preconditioner for the original matrix. Since the incomplete factorization is made on a shifted matrix, a good LU factorization is obtained without allowing much fill-in. The result needs to be extrapolated to the nonshifted matrix. Thus, the main motivation for this process is to save memory. The method is useful for matrices whose incomplete LU factorizations are poor, e.g., unstable.