Minimax polynomial preconditioning for Hermitian linear systems

This paper explores the use of polynomial preconditioning for Hermitian positive definite and indefinite linear systems $Ax = b$. Unlike preconditioners based on incomplete factorizations, polynomial preconditioners are easy to employ and well suited to vector and/or parallel architectures. It is shown that any polynomial iterative method may be used to define a preconditioning polynomial, and that the optimum polynomial preconditioner is obtained from a minimax approximation problem. A variety of preconditioning polynomials are then surveyed, including the Chebyshev, de Boor and Rice, Grcar, and bilevel polynomials. Adaptive procedures for each of these polynomials are also discussed, and it is shown that the new bilevel polynomial is particularly well suited for use in adaptive CG algorithms.