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The QR and QL Algorithms for Symmetric Matrices

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

Abstract

The QR algorithm as developed by Francis [2] and Kublanovskaya [4] is conceptually related to the LR algorithm of Rutishauser [7]. It is based on the observation that if
$$A = QR{\text{ and }}B{\text{ = }}RQ{\text{,}}$$
(1)
where Q is unitary and R is upper-triangular then
$$B = RQ = {Q^H}AQ,$$
(2)
that is, B is unitarily similar to A. By repeated application of the above result a sequence of matrices which are unitarily similar to a given matrix A 1 may be derived from the relations
$${A_s} = {Q_s}{R_s},{\rm{ }}{A_{s + 1}} = {R_s}{Q_s} = Q_s^H{A_s}{Q_s}$$
(3)
and, in general, A s tends to upper-triangular form.

Keywords

Diagonal Element Tridiagonal Matrix Multiple Eigenvalue Tridiagonal Matrice Compute 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

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • H. Bowdler
  • R. S. Martin
  • C. Reinsch
  • J. H. Wilkinson

There are no affiliations available

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