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)


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
$$\matrix{ {{Q_s}({A_s} - {k_s}I) = {L_s},} & {{A_{s + 1}} = {L_s}Q_s^T + {k_s}I,} & {{\rm{giving}}} & {{A_{s + 1}} = {Q_s}{A_s}Q_s^T,} \cr } $$
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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  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
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    Francis, J. G. F.: The QR transformation, Part I and IL Comput. J. 4, 265–271, 332–345 (1961, 1962).MathSciNetzbMATHGoogle Scholar
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    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

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© 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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