Annals of Operations Research

, Volume 103, Issue 1–4, pp 351–358 | Cite as

The Successive Over Relaxation Method (SOR) and Markov Chains

  • Wilhelm Niethammer


In the sixties SOR has been the working horse for the numerical solution of elliptic boundary problems; classical results for chosing the relaxation parameter have been derived by D. Young and R.S. Varga.

In the last fifteen years SOR has been examined for the computation of the stationary distribution of Markov chains. In the paper there are pointed out similarities and differences compared with the application of SOR for elliptic boundary problems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P.J. Courtois and P. Semal, Block iterative algorithms for stochastic matrices, Linear Algebra Appl. 20 (1986) 59–76.Google Scholar
  2. [2]
    M. Eiermann, W. Niethammer and R.S. Varga, A study of semiiterative methods for nonsymmetric systems of linear equations, Numer. Math. 47 (1985) 503–533.Google Scholar
  3. [3]
    F.-J. Fritz, B. Huppert and W. Willems, Stochastische Matrizen (Springer, Berlin, 1979).Google Scholar
  4. [4]
    A. Greenbaum, Iterative Methods for Solving Linear Systems (SIAM, Philadelphia, 1997).Google Scholar
  5. [5]
    A. Hadjidimos, On the optimization of the classical iterative schemes for the solution of complex singular linear systems, SIAM J. Algebra Disc. Methods 6 (1985) 555–565.Google Scholar
  6. [6]
    W. Kahan, Gauss-Seidel methods of solving large systems of linear equations, Thesis, University of Toronto (1958).Google Scholar
  7. [7]
    F.I. Karpelevich, On the characteristic roots of matrices with non-negative elements, Izvestija Akad. Nauk SSSR Ser. Mat. 15 (1951) 361–383 (in Russian).Google Scholar
  8. [8]
    K. Kontovasalis, R.J. Plemmons and W.J. Stewart, Block cyclic SOR forMarkov chains with p-cyclic infinitesimal generator, Linear Algebra Appl. 154-156 (1991) 145–223.Google Scholar
  9. [9]
    D.P. O'Leary, Iterative methods for finding the stationary vector for Markov chains, in: Linear Algebra, Markov Chains and Queuing Models, eds. C.D. Mayer and R.J. Plemmons, IMA Volumes in Mathematics and its Applications, Vol. 48 (Springer, Berlin, 1993) pp. 125–136.Google Scholar
  10. [10]
    T.L. Markham, M. Neumann and R.J. Plemmons, Convergence of a direct-iterative method applied to sparse least squares problems, Linear Algebra Appl. 69 (1985) 155–167.Google Scholar
  11. [11]
    W. Niethammer, A note on the extended convergence of SOR for two-periodic Markov chains, Linear Algebra Appl. 287 (1999) 315–322.Google Scholar
  12. [12]
    R.V. Southwell, Relaxation Methods in Engineering Science (Oxford University Press, Oxford, 1940).Google Scholar
  13. [13]
    R.V. Southwell, Relaxation Methods in Theoretical Physics (Oxford University Press, Oxford, 1946).Google Scholar
  14. [14]
    W.J. Stewart, Introduction to the Numerical Solution of Markov Chains (Princeton University Press, Princeton, 1994).Google Scholar
  15. [15]
    R.S. Varga, p-cyclic matrices: a generalization of the Young-Frankel successive overrelaxation scheme, Pacific J. Math. 9 (1959) 925–939.Google Scholar
  16. [16]
    R.S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, 1962).Google Scholar
  17. [17]
    P. Wild and W. Niethammer, Over-and underrelaxation for linear systems with weakly cyclic Jacobi matrices of index p, Linear Algebra Appl. 91 (1987) 29–52.Google Scholar
  18. [18]
    D.M. Young, Iterative methods for solving partial differenced equations of elliptic type, Trans. Amer. Math. Soc. 76 (1954) 92–101.Google Scholar
  19. [19]
    D.M. Young, Iterative Solution of Large Linear Systems (Academic Press, New York, 1971).Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Wilhelm Niethammer
    • 1
  1. 1.Institut für Praktische MathematikUniversität KarlsruheGermany

Personalised recommendations