Numerische Mathematik

, Volume 66, Issue 1, pp 97–122 | Cite as

On the global and cubic convergence of a quasi-cyclic Jacobi method

  • Noah H. Rhee
  • Vjeran Hari


In this paper we consider the global and the cubic convergence of a quasi-cyclic Jacobi method for the symmetric eigenvalue, problem. The method belongs to a class of quasi-cyclic methods recently proposed by W. Mascarenhas. Mascarenhas showed that the methods from his class asymptotically converge cubically per quasi-sweep (one quasi-sweep is equivalent to 1.25 cyclic sweeps) provided the eigenvalues are simple. Here we prove the global convergence of our method and derive very sharp asymptotic convergence bounds in the general case of multiple eigenvalues. We discuss the ultimate cubic convergence of the method and present several numerical examples which all well comply with the theory.

Mathematics Subject Classifications (1991)



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  1. 1.
    Demmel, J. and Veselić, K. (1992): Jacobi's Method is More Accurate than QR, SIAM J. Matrix Anal. Appl.13(4), 1204–1245Google Scholar
  2. 2.
    Forsythe, G.E. and Henrici, P. (1960): The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Am. Math. Soc.94, 1–23Google Scholar
  3. 3.
    Golub, G.H. and Van Loan, C.F. (1989): Matrix Computations, The Johns Hopkins University Press, Baltimore, MarylandGoogle Scholar
  4. 4.
    Hansen, E.R. (1960): On Jacobi Methods and Block Jacobi Methods for Computing Matrix Eigenvalues, Ph.D. thesis, StanfordGoogle Scholar
  5. 5.
    Hansen, E.R. (1991): On cyclic Jacobi methods, SIAM J. Appl. Math.11, 449–459Google Scholar
  6. 6.
    Hari, V. (1991): On pairs of almost diagonal matrices. Linear Algebra Appl.148, 193–223Google Scholar
  7. 7.
    Hari, V. (1991): On sharp quadratic convergence bounds for the serial Jacobi methods. Numer. Math.60, 375–406Google Scholar
  8. 8.
    Henrici, P. and Zimmermann, K. (1968): An estimate for the norms of certain cyclic Jacobi operators. Linear Algebra Appl.1, 489–501Google Scholar
  9. 9.
    Van Kempen, H.P.M. (1966): On the quadratic convergence of the special cyclic Jacobi Method. Numer. Math.9, 19–22Google Scholar
  10. 10.
    Mascarenhas, W.F. (1989): On the Convergtence of the Jacobi Method. Poster Presentation, Fourth SIAM Conference on Parallel Processing for Scientific Computing, Chicago, IllinoisGoogle Scholar
  11. 11.
    Mascarenhas, W.F. (1990): On the convergence of the Jacobi method for arbitrary orderings I. SIAM J. Sci. Stat. Comput (submitted)Google Scholar
  12. 12.
    Nazareth, L. (1975): On the convergence of the cyclic Jacobi method. Linear Algebra Appl.12, 151–164Google Scholar
  13. 13.
    Shroff, G. and Schreiber, R. (1989): On the convergence of the cyclic Jacobi method for parallel block orderings. SIAM J. Matrix Anal. Appl.10, 326–346Google Scholar
  14. 14.
    Veselić, K. and Hari, V. (1989): A note on the one-sided Jacobi algorithm. Numer. Math.56, 627–633Google Scholar
  15. 15.
    Wilkinson, J.H. (1968): Note on the quadratic convergence of the cyclic Jacobi process. Numer. Math.4, 296–300Google Scholar
  16. 16.
    Wilkinson, J.H. (1968): Almost diagonal matrices with multiple or close eigenvalues. Linear Algebra Appl.1, 1–12Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Noah H. Rhee
    • 1
  • Vjeran Hari
    • 2
  1. 1.Department of MathematicsUniversity of Missouri-Kansas CityKansas CityUSA
  2. 2.Department of MathematicsUniversity of ZagrebZagrebCroatia

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