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Fast Verification for Respective Eigenvalues of Symmetric Matrix

  • Shinya Miyajima
  • Takeshi Ogita
  • Shin’ichi Oishi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)

Abstract

A fast verification algorithm of calculating guaranteed error bounds for all approximate eigenvalues of a real symmetric matrix is proposed. In the proposed algorithm, Rump’s and Wilkinson’s bounds are combined. By introducing Wilkinson’s bound, it is possible to improve the error bound obtained by the verification algorithm based on Rump’s bound with a small additional cost. Finally, this paper includes some numerical examples to show the efficiency of the proposed algorithm.

Keywords

Symmetric Matrix Error Bound Real Symmetric Matrix Approximate Eigenvalue Respective Eigenvalue 
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.

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References

  1. 1.
    ANSI/IEEE: IEEE Standard for Binary Floating-Point Arithmetic: ANSI/IEEE Std 754-1985. IEEE, New York (1985)Google Scholar
  2. 2.
    Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  3. 3.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM Publications, Philadelphia (2002)zbMATHCrossRefGoogle Scholar
  4. 4.
    Ipsen, I.C.F.: Relative Perturbation Bounds for Matrix Eigenvalues and Singular Values. Acta Numerica 7, 151–201 (1998)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Miyajima, S., Ogita, T., Ozaki, K., Oishi, S.: Fast error estimation for eigenvalues of symmetric matrix without directed rounding. In: Proc. 2004 International Symposium on Nonlinear Theory and its Applications, Fukuoka, Japan, pp. 167–170 (2004)Google Scholar
  6. 6.
    Oishi, S.: Fast enclosure of matrix eigenvalues and singular values via rounding mode controlled computation. Linear Alg. Appl. 324, 133–146 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Parlett, B.N.: The Symmetric Eigenvalue Problem. Classics in Applied Mathematics 20. SIAM Publications, Philadelphia (1997)Google Scholar
  8. 8.
    Rump, S.M.: Computational error bounds for multiple or nearly multiple eigenvalues. Linear Alg. Appl. 324, 209–226 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Rump, S.M.: Private communication (2004)Google Scholar
  10. 10.
    Rump, S.M., Zemke, J.: On eigenvector bounds. BIT 43, 823–837 (2004)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Stewart, G.W., Sun, J.-G.: Matrix Perturbation Theory. Academic Press, New York (1990)zbMATHGoogle Scholar
  12. 12.
    The MathWorks Inc.: MATLAB User’s Guide Version 7 (2004)Google Scholar
  13. 13.
    Wilkinson, J.H.: Rigorous error bounds for computed eigensystem. Computer J. 4, 230–241 (1961)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Yamamoto, T.: Error bounds for computed eigenvelues and eigenvectors. Numer. Math. 34, 189–199 (1980)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Shinya Miyajima
    • 1
  • Takeshi Ogita
    • 1
    • 2
  • Shin’ichi Oishi
    • 1
  1. 1.Faculty of Science and EngineeringWaseda UniversityShinjuku-ku TokyoJapan
  2. 2.CREST, Japan Science and Technology Agency 

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