BIT Numerical Mathematics

, Volume 36, Issue 3, pp 542–562

Invariant subspaces for tightly clustered eigenvalues of tridiagonals

  • B. N. Parlett
Article

Abstract

The invariant subspace of a real symmetric tridiagonal matrixT associated with a tight cluster ofm eigenvalues has a special structure. This structure is revealed by the envelope of the subspace (defined in Section 3) havingm high hills separated bym − 1 low valleys.

This paper describes a long technical report that shows the existence ofm submatrices ofT each one having a single eigenvalue in the cluster interval whose normalized eigenvector has small entries in the first and last positions. These little eigenvectors determine a distinguished basis for an approximating subspace whose overlap matrix is tridiagonal and close to the identity. The only communication needed among the submatrices to make an orthogonal basis is between nearest neighbors.

A variety of examples illustrate the theory.

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References

  1. 1.
    W. B. Gragg and W. J Harrod,The numerically stable reconstruction of Jacobi spectral data, Numer. Math., 44 (1984), pp. 317–336.Google Scholar
  2. 2.
    B. N. Parlett,The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.Google Scholar
  3. 3.
    B. N. Parlett,The construction of orthogonal eigenvectors for tight clusters by use of submatrices, Technical Report PAM-664, Center for Pure and Applied Mathematics, University of California at Berkeley, 1996.Google Scholar
  4. 4.
    J. H. Wilkinson,The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.Google Scholar
  5. 5.
    Q. Ye,On close eigenvalues of tridiagonal matrices, Numer. Math., 70 (1995), pp. 507–514.Google Scholar

Copyright information

© BIT Foundation 1996

Authors and Affiliations

  • B. N. Parlett
    • 1
  1. 1.Mathematics Department and Computer Science Division, EECS DepartmentUniversity of CaliforniaBerkeleyUSA

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