Numerische Mathematik

, Volume 53, Issue 5, pp 571–593 | Cite as

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

  • Gene H. Golub
  • Michael L. Overton


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): 65F10 CR: G1.3 


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  1. Concus, P., Golub, G.H.: A generalized conjugate gradient method for nonsymmetric systems of equations. Proc. Second Internat. Symp. on Computing Methods in Applied Sciences and Engineering, IRIA (Paris, Dec. 1975), Lecture Notes in Economics and Mathematical Systems, vol. 134, R. Glowinski and J.L. Lions, eds.), Springer, Berlin Heidelberg New York 1976Google Scholar
  2. Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Number. Anal.19, 400–408 (1982)Google Scholar
  3. Eiermann, M., Niethammer, W., Varga, R.S.: A study of semiiterative methods for nonsymmetric systems of linear equations. Numer. Math.47, 505–533 (1985)Google Scholar
  4. Freund, R., Ruscheweyh, S.: On a class of Chebyshev approximation problems which arise in connection with a conjugate gradient type method. Numer. Math.48, 525–542 (1986)Google Scholar
  5. Glowinski, R., Lions, J., Trémolières, R.: Analyse numerique des inéquations variationelles, Vol. I. Paris: Dunod 1976Google Scholar
  6. Golub, G.H.: The use of Chebyshev matrtix polynomials in the iterative solution of linear equations compared with the method of successive overrelaxation. Ph.D. thesis, University of Illinois 1959Google Scholar
  7. Golub, G.H.: Bounds for the round-off errors in the Richardson second-order method. BIT2, 212–223 (1962)Google Scholar
  8. Golub, G.H., Overton, M.L.: Convergence of a two-stage Richardson iterative procedure for solving systems of linear equations. In: Numerical analysis (Proceedings of the Ninth Biennial Conference, Dundee, Scotland, 1981) (G.A. Watson, ed.). Lect. Notes Math. 912, Springer, New York Heidelberg Berlin, pp. 128–139Google Scholar
  9. Golub, G.H., Varga, R.S.: Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods, Parts I and II. Numer. Math.3, 147–168 (1961)Google Scholar
  10. Greenbaum, A.: Behavior of slightly perturbed Lanczos and conjugate gradient recurrences. Linear Algebra Appl. 1988 (to appear)Google Scholar
  11. Gunn, J.E.: The numerical solution ofV·aV u=f by a semi-explicit alternating-direction iterative technique. Numer. Math.6, 181–184 (1964)Google Scholar
  12. Manteuffel, T.A.: The Tchebyshev iteration for nonsymmetric linear systems. Numer. Math.28, 307–327 (1977)Google Scholar
  13. Nichols, N.K.: On the convergence of two-stage iterative processes for solving linear equations. SIAM J. Numer. Anal.10, 460–469 (1973)Google Scholar
  14. NAG Library Manual (1984). Numerical Algorithms Group, 256 Banbury Rd., OxfordGoogle Scholar
  15. Nicolaides, R.A.: On the local convergence of certain two step iterative procedures. Numer. Math.24, 95–101 (1975)Google Scholar
  16. Niethammer, W., Varga, R.S.: The analysis ofk-step iterative methods for linear systems from summability theory. Numer. Math.41, 177–206 (1983)Google Scholar
  17. Pereyra, V.: Accelerating the convergence of discretization algorithms. SIAM J. Numer. Anal.4, 508–533 (1967)Google Scholar
  18. Varga, R.S.: Matrix iterative analysis. Englewood Cliffs, NJ: Prentice-Hall 1962Google Scholar
  19. Widlund, O.: A Lanczos method for a class of non symmetric systems of linear equations. SIAM J. Numer. Anal.15, 801–812 (1978)Google Scholar
  20. Woźniakowski, H.: Numerical stability of the Chebyshev method for the solution of large linear systems. Numer. Math.28, 191–209 (1977)Google Scholar
  21. Young, D.M.: Iterative solution of large linear systems. New York, London: Academic Press 1971Google Scholar

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

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