The Chebyshev iteration revisited

Compared to Krylov space methods based on orthogonal or oblique projection, the Chebyshev iteration does not require inner products and is therefore particularly suited for massively parallel computers with high communication cost. Here, six different algorithms that implement this method are presented and compared with respect to roundoff effects, in particular, the ultimately achievable accuracy. Two of these algorithms replace the three-term recurrences by more accurate coupled two-term recurrences and seem to be new. It is also shown that, for real data, the classical three-term Chebyshev iteration is never seriously affected by roundoff, in contrast to the corresponding version of the conjugate gradient method. Even for complex data, strong roundoff effects are seen to be limited to very special situations where convergence is anyway slow. The Chebyshev iteration is applicable to symmetric definite linear systems and to nonsymmetric matrices whose eigenvalues are known to be confined to an elliptic domain that does not include the origin. Also considered is a corresponding stationary 2-step method, which has the same asymptotic convergence behavior and is additionally suitable for mildly nonlinear problems. 2002 Elsevier Science B.V. All rights reserved. MSC: 65F10; 65G05; 65H10