Skip to main content
SpringerLink
Log in

Parallel Algorithms for Dense Eigenvalue Problems

  • Conference paper
Workshop on High Performance Computing and Gigabit Local Area Networks

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 226))

  • 118 Accesses

Summary

Parallel solutions of dense eigenvalue problems have been active research topics since the implementation of the first parallel eigenvalue algorithm in 1971. In this paper we review Jacobi methods and spectral division methods in respective frameworks, introduce the issues arising for high performance implementation on contemporary computers, especially, parallel computers, and address problems that remain open.

This work was in part supported by the Advanced Research Projects Agency, under contract P-95006

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. E. Anderson and et al. LAPACK User’s Guide. SIAM, Philadelphia, 2nd edition, 1995.

    Google Scholar 

  2. Z. Bai, J. W. Demmel, and M. Gu. Inverse free parallel spectral divide and conquer algorithms for nonsymmetric eigenproblems. Research Report 94–01, Department of Mathematics, University of Kentucky, 1994.

    Google Scholar 

  3. C. H. Bischof. The two-sided block Jacobi method on hypercube architectures. In M. T. Heath, editor, Hypercube Multiprocessors. SIAM Publications, Philadelphia, 1987.

    Google Scholar 

  4. C. H. Bischof and G. Quintana-Orti. Computing rank-revealing QR factorizations of dense matrices, to appear on ACM: Trans, of Math. Soft., 1996.

    Google Scholar 

  5. C. H. Bischof and X. Sun. A framework for band reduction and tridiagonaliza-tion of symmetric matrices. T.R. MCS-P298–0392, Argonne National Laboratory, Mathematics and Computer Science Division, 1992.

    Google Scholar 

  6. R. P. Brent and F. T. Luk. The solution of singular-value and symmetric eigenvalue problems on multiprocessor arrays. SIAM Journal on Scientific and Statistical Computing, 6: 69–84, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Chan. Rank-revealing QR factorizations. Lin. Alg. Appl., 88 /89: 67–82, 1987.

    Google Scholar 

  8. J. Demmel and K. Veselic. Jacobi’s method is more accurate than QR. SIAM J. Matrix Anal. Appl., 13: 1204–1245, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. J. Dongara, J. DuCros, I. Duff, and S. Hammarling. A set of level 3 Basic Linear Algebra Subprograms. ACM Trans. Math. Softw., 16: 1–17, 1990.

    Article  Google Scholar 

  10. J. J. Dongarra and D. C. Sorensen. A fully parallel algorithm for the symmetric eigenvalue problem. SIAM Journal on Scientific and Statistical Computing, 8:sl39-sl54, 1987.

    Google Scholar 

  11. J. J. Dongarra and R. A. van de Geijn. Reduction to condensed form for the eigenvalue problem on distributed memory architectures. Report ORNL/TM- 12006, Mathematical Sciences Section, Oak Ridge National Laboratory, 1991.

    Google Scholar 

  12. G. Fan, R. J. Littlefield, and D. Elwood. Performance of a fully parallel dense real symmetric eigensolver in quantum chemistry applications. In Proceedings of High Performance Computing ‘85, Phoenix, AZ, Apr. 1995. Society for Computer Simulation.

    Google Scholar 

  13. G. H. Golub and C. F. Van Loan. Matrix Computations. The Johns Hopkins University Press, 2nd edition, 1989.

    Google Scholar 

  14. M. Gu and S. Eisenstat. A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. SIAM J. Matrix Anal. Appl., 16: 171–191, 1995.

    Google Scholar 

  15. B. Henderckson, E. Jessup, and C. Smith. A parallel eigensolver for dense symmetric matrices. Technical report, Sandia National Labs, Univ. of Colorado, and Advanced Product and Process Tech., March 1996.

    Google Scholar 

  16. Vicente Hernandez, Robert A. van de Geijn, and Antonio M. Vidal. A Ja-cobi method by blocks on a mesh of processors. Concurrency: Practice and Experience, 1996. to appear.

    Google Scholar 

  17. P. Hong and C. T. Pan. The rank revealing QR and SVD. Math. Comp., 58: 213–232, 1992.

    MATH  MathSciNet  Google Scholar 

  18. C. G. J. Jacobi. Uber ein leichtes Verfahren die in der Theorie der Säculärstörungen vorkommenden Gleichungen numerisch aufzulösen. Journal für die reine und angewandte Mathematik, 30: 51 — s94, 1846.

    Article  MATH  Google Scholar 

  19. F. T. Luk. Computing the singular-value decomposition on the ILLIAC IV. ACM Transactions on Mathematical Software, 6: 524–539, 1983.

    Article  MathSciNet  Google Scholar 

  20. F. T. Luk and H. Park. On parallel jacobi orderings. SIAM Journal on Scientific and Statistical Computing, 10: 18–26, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  21. Niloufer Mackey. Hamilton and Jacobi meet again: quaternions and the eigenvalue problem. SIAM J. Matrix Annal Appl, 16: 421–435, Apr. 1995.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. N. Malyshev. Parallel algorithm for solving some spectral problems of linear algebra. Lin. Alg. Appl, 188, 189: 489–520, 1993.

    Article  MathSciNet  Google Scholar 

  23. G. Quintana, X. Sun, and C. H. Bischof. A BLAS-3 version of the QR factorization with column pivoting. Technical Report MCS-P551–1295, Mathematics and Computer Science Division, Argonne National Laboratory, 1996. To appear in SIAM J. Sei. Comp.

    Google Scholar 

  24. G. Quintana-Orti and E. S. Quintana-Orti. Parallel bidimensional algorithms for computing rank-revealing QR factorizations. In in the same proceeding, 1996.

    Google Scholar 

  25. A. Sameh. On Jacobi and Jacobi-like algorithms for a parallel computer. Mathematics of Computation, 25: 579–590, 1971.

    Article  MATH  MathSciNet  Google Scholar 

  26. D. S. Scott, M. T. Heath, and R. C. Ward. Parallel block Jacobi eigenvalue algorithms using systolic arrays. Linear Algebra and Its Applications, 77: 345–355, 1986.

    Article  MATH  Google Scholar 

  27. G. Shroff and R. Schreiber. On the convergence of the cyclic Jacobi method for parallel block ordering. SIAM J. on Matrix Anal. Appl., 10 (3): 326–346, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  28. G. W. Stewart. An updating algorithm for subspace tracking. IEEE Trans. Signal Processing, 40: 1535–1541, 1992.

    Article  Google Scholar 

  29. X. Sun. Scaling and squaring in invariant subspace decomposition methods. In Large-Scale Matrix Diagonalization Methods. Argonne national Laboratory, Chemistry Theory Institute, May 1996.

    Google Scholar 

  30. X. Sun and E. S. Quintana-Orti. Spectral division methods for block generalized Schur decompositions. Technical Report CS-1996–13, Duke University, Department of Computer Science, August 1996.

    Google Scholar 

  31. J. H. Wilkinson. The Algebraic Eigenvalue Problem. Claredon Press, Oxford, England, 1965.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Duke University, Durham, NC, 27708, USA

    Xiaobai Sun

Authors
  1. Xiaobai Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. College of Computer Science, Northeastern University, M/S 215 CN, 02115, Boston, MA, USA

    G. Cooperman

  2. Institut für Experimentelle Mathematik, Universität GH Essen, Ellenstr. 29, D-45326, Essen, Germany

    G. Michler  & H. Vinck  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag London

About this paper

Cite this paper

Sun, X. (1997). Parallel Algorithms for Dense Eigenvalue Problems. In: Cooperman, G., Michler, G., Vinck, H. (eds) Workshop on High Performance Computing and Gigabit Local Area Networks. Lecture Notes in Control and Information Sciences, vol 226. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3540761691_14

Download citation

  • DOI: https://doi.org/10.1007/3540761691_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-76169-3

  • Online ISBN: 978-3-540-40937-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation