Numerische Mathematik

, Volume 53, Issue 5, pp 571–593

The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems

  • Gene H. Golub
  • Michael L. Overton

DOI: 10.1007/BF01397553

Cite this article as:
Golub, G.H. & Overton, M.L. Numer. Math. (1988) 53: 571. doi:10.1007/BF01397553


The Chebyshev and second-order Richardson methods are classical iterative schemes for solving linear systems. We consider the convergence analysis of these methods when each step of the iteration is carried out inexactly. This has many applications, since a preconditioned iteration requires, at each step, the solution of a linear system which may be solved inexactly using an “inner” iteration. We derive an error bound which applies to the general nonsymmetric inexact Chebyshev iteration. We show how this simplifies slightly in the case of a symmetric or skew-symmetric iteration, and we consider both the cases of underestimating and overestimating the spectrum. We show that in the symmetric case, it is actually advantageous to underestimate the spectrum when the spectral radius and the degree of inexactness are both large. This is not true in the case of the skew-symmetric iteration. We show how similar results apply to the Richardson iteration. Finally, we describe numerical experiments which illustrate the results and suggest that the Chebyshev and Richardson methods, with reasonable parameter choices, may be more effective than the conjugate gradient method in the presence of inexactness.

Subject Classifications

AMS(MOS): 65F10CR: G1.3

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Gene H. Golub
    • 1
  • Michael L. Overton
    • 2
  1. 1.Computer Science DepartmentStanford UniversityStanfordUSA
  2. 2.Computer Science Department, Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA