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)


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.


Procedure Balance Hessenberg Form Balance Matrix Lower Triangular Form Hessenberg Matrice 
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  1. 1.
    Osborne, E. E.: On pre-conditioning of matrices. J. Assoc. Comput. Mach. 7, 338–345 (i960).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Wilkinson, J. H.: The algebraic eigenvalue problem. London: Oxford University Press 1965-zbMATHGoogle 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.MathSciNetzbMATHCrossRefGoogle 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

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© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • B. N. Parlett
  • C. Reinsch

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