BIT Numerical Mathematics

, Volume 26, Issue 3, pp 369–376 | Cite as

Generalized consistent orderings and the Accelerated Overrelaxation method

  • Yiannis G. Saridakis
Part II Numerical Mathematics


In this paper, the behavior of the block Accelerated Overrelaxation (AOR) iterative method, when applied to linear systems with a generalized consistently ordered coefficient matrix, is investigated. An equation, relating the eigenvalues of the block Jacobi iteration matrix to the eigenvalues of its associated block AOR iteration matrix, as well as sufficient conditions for the convergence of the block AOR method, are obtained.

Subject Classifications AMS (MOS)

65F10 65F15 65F40 CR: G.1.3 


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  1. 1.
    C. G. Broyden,Some generalizations of the theory of successive over-relaxation, Num. Math. 6, pp. 269–284, 1964.CrossRefGoogle Scholar
  2. 2.
    C. G. Broyden,Some aspects of consistent ordering, Num. Math. 12, pp. 47–56, 1968.Google Scholar
  3. 3.
    S. P. Frankel,Convergence rates of iterative treatments of partial differential equations, MTAC 4, pp. 65–75, 1950.Google Scholar
  4. 4.
    A. Hadjidimos,Accelerated overrelaxation method, Math. Comp. 32, pp. 149–157, 1978.Google Scholar
  5. 5.
    G. Kjellberg,On the successive over-relaxation method for cyclic operators, Num. Math. 3, pp. 87–91, 1961.CrossRefGoogle Scholar
  6. 6.
    N. M. Missirlis,Convergence theory of extrapolated iterative methods for a certain class of non-symmetric linear systems, Num. Math. 45, pp. 447–458, 1984.CrossRefGoogle Scholar
  7. 7.
    N. K. Nichols and L. Fox,Generalized consistent ordering and optimum successive overrelaxation factor, Num. Math. 13, pp. 425–433, 1969.Google Scholar
  8. 8.
    W. Niethammer,On different splittings and associated iteration methods, SIAM J. Numer. Anal. 16, pp. 186–200, 1979.CrossRefGoogle Scholar
  9. 9.
    T. S. Papatheodorou,Block AOR iteration for nonsymmetric matrices, Math. Comp. 41, pp. 511–525, 1983.Google Scholar
  10. 10.
    Y. G. Saridakis, Ph.D. Dissertation, Clarkson University, 1985.Google Scholar
  11. 11.
    M. Sisler,Über ein zweiparametriges Iterations-Verfahrens, Appl. Mat. 18, pp. 325–332, 1973.Google Scholar
  12. 12.
    R. S. Varga,p-cyclic matrices: A generalization of the Young-Frankel successive overrelaxation, Pac. J. Math. 9, pp. 617–628, 1959.Google Scholar
  13. 13.
    R. S. Varga,Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, N.Y. 1962.Google Scholar
  14. 14.
    R. S. Varga, W. Niethammer and D. Y. Cai,p-cyclic matrices and the symmetric successive overrelaxation method, Lin. Alg. Appl. 58, pp. 425–439, 1984.CrossRefGoogle Scholar
  15. 15.
    J. H. Verner and M. J. M. Bernal,On generalizations of the theory of consistent orderings for successive overrelaxation methods, Num. Math. 12, pp. 215–222, 1968.Google Scholar
  16. 16.
    D. M. Young,Iterative methods for solving partial difference equations of elliptic type, Trans. Amer. Math. Soc. 76, pp. 92–111, 1954.Google Scholar
  17. 17.
    D. M. Young,Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.Google Scholar

Copyright information

© BIT Foundations 1986

Authors and Affiliations

  • Yiannis G. Saridakis
    • 1
  1. 1.Department of Mathematics and Computer ScienceClarkson UniversityPotsdamUSA

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