On Adaptive Weighted Polynomial Preconditioning for Hermitian Positive Definite Matrices

The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bounds for the extreme eigenvalues. Here a different approach is proposed, which aims at adapting the preconditioner to the eigenvalue distribution of the coefficient matrix. The technique is based on the observation that good estimates for the eigenvalue distribution can be derived after only a few steps of the Lanczos process. This information is then used to construct a weight function for a suitable Chebyshev approximation problem. The solution of this problem yields the polynomial preconditioner. In particular, we investigate the use of Bernstein-Szeg_ weights. Key words, linear systems, Hermitian positive definite matrices, conjugate gradient algorithm, polynomial preconditioning, Chebyshev approximation problem, Bernstein-Szeg/_ weights AMS(MOS) subject classifications. 65F10