Simultaneous Newton’s Iteration for the Eigenproblem

  • Françoise Chatelin
Part of the Computing Supplementum book series (COMPUTING, volume 5)


For an ill-conditioned eigenproblem (close eigenvalues and/or almost parallel eigenvectors) it is advisable to group some eigenvalues and to compute a basis of the corresponding invariant subspace. We show how Newton’s method may be used for the iterative refinement of an approximate invariant subspace.


Eigenvalue Problem Invariant Subspace Bounded Linear Operator Multigrid Method Iterative Refinement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ahués, M.: Raffinement d’éléments propres approchés d’un opérateur compact. Dr. Ing. Thesis, Université de Grenoble, 1983.Google Scholar
  2. [2]
    Ahués, M., Chatelin, F.: The use of defect correction to refine the eigenelements of compact integral operator. SIAM J. Numer. Anal. 20, 1087–1093 (1983).MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Anselone, P. M., Rall, L. B.: The solution of characteristic value-vector problems by Newton’s method. Numer. Math. 11, 38–45 (1968).MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Bartels, R. H., Stewart, G. W.: Algorithm 432, solution of the matrix equation AX+XB=C. Comm. ACM 15, 820–826 (1972).CrossRefGoogle Scholar
  5. [5]
    Brandt, A., Mc Cormick, S., Ruge, J.: Multigrid methods for differential eigenproblems. SIAM J. S. S. C. 4, 244–260 (1983).MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Chatelin, F.: Spectral approximation of linear operators. New York: Academic Press 1983.MATHGoogle Scholar
  7. [7]
    Chatelin, F.: Valeurs propres de matrices. Paris: Masson (to appear).Google Scholar
  8. [8]
    Chatelin, F., Miranker, W. L.: Aggregation/disaggregation for eigenvalue problems. SIAM. J. Num. Anal. 21, 567–582 (1984).MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Davis, C., Kahan, W.: The rotation of eigenvectors by a perturbation. III. SIAM. J. Num. Anal. 7, 1–46 (1968).MathSciNetCrossRefGoogle Scholar
  10. [10]
    Gantmacher, F. R.: Theérie des matrices, Tome I. Paris: Dunod 1966.Google Scholar
  11. [11]
    Dongarra, J. J., Moler, C. B., Wilkinson, J. H.: Improving the accuracy of computed eigenvalues and eigenvectors. SIAM. J. Num. Anal. 20, 23–45 (1983).MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Golub, G. H., Nash, S., Van Loan, C.: A Hessenberg-Schur form for the problem AX+XB=C. IEEE Trans. Autom. Control AC-24, 909–913 (1979).CrossRefGoogle Scholar
  13. [13]
    Hackbusch, W.: On the computation of approximate eigenvalue and eigenfunctions of elliptic operators by means of a multigrid method. SIAM J. Num. Anal. 16, 201–215 (1979).MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Lin Qun: Iterative refinement of finite element approximations for elliptic problems. RAIRO Anal. Numer. 16, 39–47 (1982).MathSciNetMATHGoogle Scholar
  15. [15]
    Parlett, B. N.: The symmetric eigenvalue problem. Englewood Cliffs, N. J.: Prentice-Hall 1980.MATHGoogle Scholar
  16. [16]
    Peters, G., Wilkinson, J. H.: Inverse iteration, ill-conditioned equations and Newton’s method. SIAM Rev. 21, 339–360 (1979).MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Rosenblum, M.: On the operator equation BXXA = Q. Duke Math. J. 23, 263–269 (1956).MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Stewart, G. W.: Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Rev. 15, 727–764 (1973).MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Neumaier, A.: Residual inverse iteration for the nonlinear eigenvalue problem. Preprint, Univ. Freiburg, 1984.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Françoise Chatelin
    • 1
  1. 1.Université de Paris IX - Dauphine visiting IBM Développement ScientifiqueParisFrance

Personalised recommendations