Skip to main content
SpringerLink
Log in

Asymptotic quadratic convergence of the serial block-Jacobi EVD algorithm for Hermitian matrices

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We provide the proof of the asymptotic quadratic convergence of the classical serial block-Jacobi EVD algorithm for Hermitian matrices with well-separated eigenvalues (including the multiple ones) as well as clusters of eigenvalues. At each iteration step, two off-diagonal blocks with the largest Frobenius norm are eliminated which is an extension of the original Jacobi approach to the block case. Numerical experiments illustrate and confirm the developed theory.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Bečka, M., Okša, G., Vajteršic, M.: Dynamic ordering for a parallel block-Jacobi SVD algorithm. Parallel Comput. 28, 243–262 (2002)

    Article  MATH  Google Scholar 

  2. Davis, C., Kahan, W.: The rotation of eigenvectors by a perturbation: III. SIAM J. Numer. Anal. 7, 1–46 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  3. Demmel, J.W., Veselić, K.: Jacobi’s method is more accurate than QR. SIAM J. Math. Anal. Appl. 13, 1204–1245 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dopico, F.M., Molera, J.M., Moro, J.: An orthogonal high relative accuracy algorithm for symmetric eigenproblem. SIAM J. Math. Anal. Appl. 25, 301–351 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Drmač, Z., Veselić, K.: New fast and accurate Jacobi SVD algorithm: I. SIAM J. Matrix Anal. Appl. 29, 1322–1342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Drmač, Z., Veselić, K.: New fast and accurate Jacobi SVD algorithm: II. SIAM J. Matrix Anal. Appl. 29, 1343–1362 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Drmač, Z.: A global convergence proof for cyclic Jacobi methods with block rotations. SIAM J. Matrix Anal. Appl. 31, 1329–1350 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins UP, Baltimore (2013)

    MATH  Google Scholar 

  9. Hari, V.: On sharp quadratic convergence bounds for the serial Jacobi methods. Numer. Math. 60, 375–406 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hari, V.: Convergence to diagonal form of block Jacobi-type methods. Numer. Math. (2014). doi:10.1007/s00211-014-0647-8

    MathSciNet  MATH  Google Scholar 

  11. Hoffman, A.J., Wielandt, H.W.: The variation of the spectrum of a normal matrix. Duke Math. J. 20, 37–39 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kudo, S., Yamamoto, Y., Bečka, M., Vajteršic, M.: Performance of the parallel one-sided block-Jacobi SVD algorithm on a modern distributed-memory parallel computer. LNCS 9573, 594–604 (2016)

    MathSciNet  Google Scholar 

  13. Luk, F., Park, H.: On parallel Jacobi orderings. SIAM J. Sci. Stat. Comput. 10, 18–26 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Matejaš, J., Hari, V.: Accuracy of the Kogbetliantz method for the scaled diagonally dominant triangular matrices. Appl. Math. Comput. 217, 3726–3746 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Mathias, R.: Accurate eigensystem computations by Jacobi methods. SIAM J. Math. Anal. Appl. 16, 977–1003 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Parlett, B.N.: The Symmetric Eigenvalue Problem. SIAM, Philadelphia (1987)

    MATH  Google Scholar 

  17. Sameh, A.H.: On Jacobi and Jacobi-like algorithms for parallel computer. Math. Comput. 25, 579–590 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schönhage, A.: Zur Konvergenz des Jacobi-Verfahrens. Numer. Math. 3, 374–380 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schönhage, A.: Zur quadratischen Konvergenz des Jacobi-Verfahrens. Numer. Math. 6, 410–412 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  20. Singer, S., Singer, S., Novaković, V., Davidović, D., Bokulić, K., Ušćumlić, A.: Three-level parallel J-Jacobi algorithms for Hermitian matrices. Appl. Math. Comput. 218, 5704–5725 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Sugihara, M., Murota, K.: Mathematics of Numerical Linear Algebra. Iwanami-Shoten, Tokyo (2009). (in Japanese)

    Google Scholar 

  22. Yamamoto, Y., Lan, Z., Kudo, S.: Convergence analysis of the parallel classical block Jacobi method for the symmetric eigenvalue problem. JSIAM Lett. 6, 57–60 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are grateful for valuable recommendations to prof. Hans-Joachim Bungartz and prof. Vjeran Hari who read the first draft of this paper. We also thank both anonymous referees for their comments and suggestions that improved the paper’s quality.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Slovak Academy of Sciences, Bratislava, Slovak Republic

    Gabriel Okša & Marián Vajteršic

  2. Department of Communication Engineering and Informatics, University of Electro-Communications, Tokyo, Japan

    Yusaku Yamamoto

  3. Department of Computer Sciences, University of Salzburg, Salzburg, Austria

    Marián Vajteršic

Authors
  1. Gabriel Okša
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yusaku Yamamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marián Vajteršic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriel Okša.

Additional information

Gabriel Okša and Marián Vajteršic were supported by the VEGA Grant no. 2/0026/14. Yusaku Yamamoto was supported in part by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (Nos. 26286087, 15H02708, 15H02709), Core Research for Evolutional Science and Technology (CREST) Program ”Highly Productive, High Performance Application Frameworks for Post Petascale Computing” of Japan Science and Technology Agency (JST).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Okša, G., Yamamoto, Y. & Vajteršic, M. Asymptotic quadratic convergence of the serial block-Jacobi EVD algorithm for Hermitian matrices. Numer. Math. 136, 1071–1095 (2017). https://doi.org/10.1007/s00211-016-0863-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-016-0863-5

Mathematics Subject Classification

Search

Navigation