A Comparison of Adaptive Chebyshev and Least Squares Polynomial Preconditioning for Hermitian Positive Definite Linear Systems

Abstract

This paper explores the use of adaptive polynomial preconditioning for Hermitian positive definite linear systems, $Ax = b$. Such preconditioners are easy to employ and well suited to vector and/or parallel machines. After examining the role of polynomial preconditioning in conjugate gradient methods, the least squares and Chebyshev preconditioning polynomials are discussed. Eigenvalue distributions for which each is well suited are then determined. An adaptive procedure for dynamically computing the best Chebyshev polynomial preconditioner is also described. Finally, the effectiveness of adaptive polynomial preconditioning is demonstrated in a variety of numerical experiments on a Cray X-MP/48 and Alliant FX/8. The results suggest that relatively low degree (2–16) polynomials are usually best.