Skip to main content
Log in

On the convergence rate of SOR: A worst case estimate

Über die Konvergenzrate von SOR—Eine worst-case-Analyse

  • Published:
Computing Aims and scope Submit manuscript

Abstract

LetA be any real symmetric positive definiten×n matrix, and κ(A) its spectral condition number. It is shown that the optimal convergence rate

$$\rho _{SOR}^* = \mathop {\min }\limits_{0< \omega< 2} \rho (M_{SOR,\omega } )$$

of the successive overrelaxation (SOR) method satisfies

$$\rho _{SOR}^* \leqslant 1 - \frac{1}{{\alpha _n \kappa (A)}}, \alpha _n \approx \log n.$$

This worst case estimate is asymptotically sharp asn→∞. The corresponding examples are given by certain Toeplitz matrices.

Zusammenfassung

Sei A eine reelle symmetrische positiv definiten×n Matrix mit Spektralkonditionszahl κ. Wir zeigen, daß für die optimale Konvergenzrate

$$\rho _{SOR}^* = \mathop {\min }\limits_{0< \omega< 2} \rho (M_{SOR,\omega } )$$

der Methode der sukzessiven Überrelaxation (SOR) gilt:

$$\rho _{SOR}^* \leqslant 1 - \frac{1}{{\alpha _n \kappa (A)}}, \alpha _n \approx \log n.$$

Diese worst-case-Abschätzung ist asymptotisch scharf fürn→∞. Die entsprechenden Beispiele werden von gewissen Toeplitz-Matrizen geliefert.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Böttcher, A., Silbermann, B.: Invertibility and asymptotics of Toeplitz matrices Berlin: Akademie-Verlag 1983.

    Google Scholar 

  2. Böttcher, A., Silbermann, B.: Analysis of Toeplitz operators. Berlin: Akademie-Verlag 1989.

    Google Scholar 

  3. Dryja, M., Smith, B. F., Widlund, O.: Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. Preprint Courant Institute, New York Univ., May 1993.

  4. Griebel, M., Oswald, P.: Remarks on the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. (submitted, also: Report TUM-I 9314, TU Munich July 1993).

  5. Hackbusch, W.: Iterative Lösung großer schwachbesetzter Gleichungssysteme, Stuttgart: Teubner 1991. English translation solution of large sparse systems of equations. New York: Springer 1994.

    Google Scholar 

  6. Hardy, G. M., Littlewood, J. E., Polya, G.: Inequalities Cambridge: Cambridge University Press 1959.

    Google Scholar 

  7. Kwapien, S., Pelzcynski, A.: The main triangle projection in matrix spaces and its applications. Studià Math.34, 43–68 (1970).

    Google Scholar 

  8. Xu, J.: Iterative methods by space decomposition and subspace corrections: a unifying approach. SIAM Rev.34, 581–613 (1992).

    Google Scholar 

  9. Young, D. M.: Iterative solution of large linear systems. New York: Academic Press 1971.

    Google Scholar 

  10. Yserentant, H.: Old and new convergence proofs for multigrid method. Acta Numerica 1993, 285–326.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Oswald, P. On the convergence rate of SOR: A worst case estimate. Computing 52, 245–255 (1994). https://doi.org/10.1007/BF02246506

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02246506

AMS Subject Classifications

Key words

Navigation