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Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors

  • B. N. Parlett
  • C. Reinsch
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)

Abstract

This algorithm is based on the work of Osborne [1]. He pointed out that existing eigenvalue programs usually produce results with errors at least of order ε‖A E , where is the machine precision and ‖A E is the Euclidean (Frobenius) norm of the given matrix A**. Hence he recommends that one precede the calling of such a routine by certain diagonal similarity transformations of A designed to reduce its norm.

Keywords

Procedure Balance Hessenberg Form Balance Matrix Lower Triangular Form Hessenberg Matrice 
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.
    Osborne, E. E.: On pre-conditioning of matrices. J. Assoc. Comput. Mach. 7, 338–345 (i960).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Wilkinson, J. H.: The algebraic eigenvalue problem. London: Oxford University Press 1965-MATHGoogle Scholar
  3. 3.
    Martin, R. S., and J. H. Wilkinson. Similarity reduction of general matrices to Hessenberg form. Numer. Math. 12, 349–368 (1969). Cf. 11/13-MathSciNetCrossRefGoogle Scholar
  4. 4.
    Martin, R. S., and J. H. Wilkinson The modified LR algorithm for complex Hessenberg matrices. Numer. Math. 12, 369–376 (1969). Cf. II/16.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Martin, R. S., and J. H. WilkinsonG. Peters, and J. H. Wilkinson. The QR algorithm for real Hessenberg matrices. Numer. Math. 14, 219–231 (1970). Cf. II/14.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Peters, G., and J. H. Wilkinson. Eigenvectors of real and complex matrices by LR and QR triangularizations. Numer. Math. 16, 181–204 (1970). Cf. 11/15.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • B. N. Parlett
  • C. Reinsch

There are no affiliations available

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