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)


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
$$A = QR{\text{ and }}B{\text{ = }}RQ{\text{,}}$$
where Q is unitary and R is upper-triangular then
$$B = RQ = {Q^H}AQ,$$
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
$${A_s} = {Q_s}{R_s},{\rm{ }}{A_{s + 1}} = {R_s}{Q_s} = Q_s^H{A_s}{Q_s}$$
and, in general, A s tends to upper-triangular form.


Diagonal Element Tridiagonal Matrix Multiple Eigenvalue Tridiagonal Matrice Compute Eigenvalue 
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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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