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)


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
$$\matrix{ {{A_s} - {k_s}I = {Q_s}{R_s},} & {{R_s}{Q_s} = {A_{s + 1}}} & {(s = 1,2, \ldots )} \cr } $$
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.


Input Matrix National Physical Laboratory Minor Step Band Matrix Band Matrice 
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  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

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© 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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