Skip to main content
SpringerLink
Log in

Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

This paper describes a simultaneous iteration technique for computing a nested sequence of orthonormal bases for the dominant invariant subspaces of a non-Hermitian matrix. The method is particularly suited to large sparse eigenvalue problems, since it requires only that one be able to form the product of the matrix in question with a vector. A convergence theory for the method is developed and practical details are discussed.

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

Similar content being viewed by others

References

  1. Bauer, F. L.: Das Verfahren der Treppeniteration und verwandte Verfahren zur Lösung algebraischer Eigenwertprobleme. Z. Angew. Math. Phys.8, 214–235 (1957)

    Google Scholar 

  2. Clint, M., Jennings, A.: The evaluation of eigenvalues and eigenvectors of real symmetric matrices by simultaneous iteration. Comp. J.13, 76–80 (1970)

    Google Scholar 

  3. Clint, M., Jennings, A.: A simultaneous iteration method for the unsymmetric eigenvalue problem, J. Inst. Math. Appl.8, 111–121 (1971)

    Google Scholar 

  4. Jennings, A.: A direct iteration method of obtaining latent roots and vectors of a symmetric matrix. Proc. Cambridge Philos. Soc.63, 755–765 (1967)

    Google Scholar 

  5. Parlett, B. N., Poole, W. G.: A geometric theory for the QR, LU, and power iterations, SIAM J. Numer. Anal.10, 389–412 (1973)

    Google Scholar 

  6. Rutishauser, H.: Computational aspects of F. L. Bauer's simultaneous iteration method, Numer. Math.13, 4–13 (1969)

    Google Scholar 

  7. Rutishauser, H.: Simultaneous iteration method for symmetric matrices. Numer. Math.16, 205–223 (1970)

    Google Scholar 

  8. Smith, B. T., Boyle, J. M., Garbow, B. S., Ikebe, Y., Klema, V. C., Moler, C. B.: Lecture Notes in Computer Science V6: Matrix Eigensystem Routines-EISPACK Guide. New York: Springer 1974

    Google Scholar 

  9. Stewart, G. W.: Accelerating the orthogonal iteration for the eigenvectors of a Hermitian matrix. Numer. Math.13, 362–376 (1969)

    Google Scholar 

  10. Stewart, G. W.: Error bounds for approximate invariant subspaces of closed linear operators. SIAM J. Numer. Anal.8, 796–808 (1971)

    Google Scholar 

  11. Stewart, G. W.: Introduction to Matrix Computations. New York: Academic Press 1973

    Google Scholar 

  12. Stewart, G. W.: Methods of simultaneous iteration for calculating eigenvectors of matrices. In: Topics in Numerical Analysis II (John J. H. Miller, ed.). New York: Academic Press 1975, pp. 185–189

    Google Scholar 

  13. Stewart, G. W.: Perturbation bounds for the QR factorization of a matrix. University of Maryland Computer Science Department Technical Report TR-323 (1974). To appear in SIAM F. Numer. Anali

  14. Vandergraft, J. S.: Generalized Rayleigh methods with applications to finding eigenvalues of large matrices. Lin. Alg. and Appl. (1971) p. 353–368

  15. Wilkinson, J. H., Reinsch, C. (eds.): Handbook for Automatic Computation VII Linear Algebra. New York: Springer 1971

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Computer Science, University of Maryland, 20742, College Park, Ma., USA

    G. W. Stewart

Authors
  1. G. W. Stewart
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported in part by the Office of Naval Research under Contract No. N00014-67-A-0128-0018.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stewart, G.W. Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices. Numer. Math. 25, 123–136 (1976). https://doi.org/10.1007/BF01462265

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01462265

Keywords

Search

Navigation