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The Implicit QL Algorithm

  • A. Dubrulle
  • R. S. Martin
  • J. H. Wilkinson
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)

Abstract

In [1] an algorithm was described for carrying out the QL algorithm for a real symmetric matrix using shifts of origin. This algorithm is described by the relations
$$\matrix{ {{Q_s}({A_s} - {k_s}I) = {L_s},} & {{A_{s + 1}} = {L_s}Q_s^T + {k_s}I,} & {{\rm{giving}}} & {{A_{s + 1}} = {Q_s}{A_s}Q_s^T,} \cr } $$
(1)
where Q s is orthogonal, L s is lower triangular and k s is the shift of origin determined from the leading 2×2 matrix of A s .

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References

  1. 1.
    Bowdler, Hilary, R. S. Martin, C. Reinsch, and J. H. Wilkinson. The QR and QL algorithms for symmetric matrices. Numer. Math. 11, 293 -306 (1968). Cf. II/3.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Francis, J. G. F.: The QR transformation, Part I and IL Comput. J. 4, 265–271, 332–345 (1961, 1962).MathSciNetzbMATHGoogle Scholar
  3. 3.
    Givens, J. W.: A method for computing eigenvalues and eigenvectors suggested by classical results on symmetric matrices. Nat. Bur. Standards Appl. Math. Ser. 29, 117–122 (1953).MathSciNetGoogle Scholar
  4. 4.
    Martin, R. S., C. Reinsch, and J. H. Wilkinson. Householder’s tridiagonalization of a symmetric matrix. Numer. Math. 11, 181 -195 (1968). Cf. II/2.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • A. Dubrulle
  • R. S. Martin
  • J. H. Wilkinson

There are no affiliations available

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