Rational Approximation Preconditioners for General Sparse Linear Systems

This paper presents a class of preconditioning techniques which exploit rational function approximations to the original matrix. The matrix is rst shifted and then an incomplete LU factorization of the resulting matrix is computed. The resulting factors are then used to compute a better preconditioner to the original matrix. Since the incomplete factorization is made on a shifted matrix, a good LU factorization is obtained without allowing much ll-in. The result needs to be extrapolated to the non-shifted 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. An error analysis for the conjugate gradient algorithm gives some guidance for choosing the shift of the matrix, in the special case where the shifted system is solved exactly.