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The Q R Algorithm for Band Symmetric Matrices

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

Abstract

The Q R algorithm with shifts of origin may be used to determine the eigenvalues of a band symmetric matrix A. The algorithm is described by the relations
$$\matrix{ {{A_s} - {k_s}I = {Q_s}{R_s},} & {{R_s}{Q_s} = {A_{s + 1}}} & {(s = 1,2, \ldots )} \cr } $$
(1)
where the Q s are orthogonal and the R s are upper triangular. It has been described in some detail by Wilkinson [5, pp. 557–561]. An essential feature is that all the A s remain of band symmetric form and R s is also a band matrix, so that when the width of the band is small compared with the order of the A s , the volume of computation in each step is quite modest. If the shifts ks are appropriately chosen the off-diagonal elements in the last row and column tend rapidly to zero, thereby giving an eigenvalue.

Keywords

Input Matrix National Physical Laboratory Minor Step Band Matrix Band Matrice 
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

  1. 1.
    Bowdler, Hilary, Martin, R. S., Reinsch, C, Wilkinson, J. H.: The QR and QL algorithms for symmetric matrices. Numer. Math. 11, 293–306 (1968). Cf. II/3.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Martin, R. S., Wilkinson, J. H.: Solution of symmetric and unsymmetric band equations and the calculation of eigenvectors of band matrices. Numer. Math. 9, 279–301 (1967). Cf. I/6.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Martin, R. S., Wilkinson, J. H.The implicit QL algorithm. Numer. Math. 12, 377–383 (1968). Cf. II/4.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Schwarz, H. R.: Tridiagonalization of a symmetric band matrix. Numer. Math. 12, 231–241 (1968). Cf. II/8.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Wilkinson, J. H.: The algebraic eigenvalue problem, 662 p. Oxford: Clarendon Press 1965.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

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

There are no affiliations available

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