Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Linear convergence in the shifted $ QR$ algorithm

Authors: Steve Batterson and David Day
Journal: Math. Comp. 59 (1992), 141-151
MSC: Primary 65F15
MathSciNet review: 1134713
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Global and asymptotic convergence properties for the QR algorithm with Francis double shift are established for certain orthogonal similarity classes of $ 4 \times 4$ real matrices. It is shown that in each of the classes every unreduced Hessenberg matrix will decouple and that the rate of decoupling is almost always linear. The effect of the EISPACK exceptional shift strategy is shown to be negligible.

References [Enhancements On Off] (What's this?)

  • [1] S. Batterson, Convergence of the shifted QR algorithm on $ 3 \times 3$ normal matrices, Numer. Math. 58 (1990), 341-352. MR 1077582 (92b:65027)
  • [2] -, The dynamics of eigenvalue computation, preprint.
  • [3] G. H. Golub and C. F. Van Loan, Matrix computations, 2nd ed., Johns Hopkins Univ. Press, Baltimore, 1989. MR 1002570 (90d:65055)
  • [4] H. Hironaka, Triangulations of algebraic sets, Proc. Sympos. Pure Math., vol. 29, Amer. Math. Soc., Providence, RI, 1975, pp. 165-185. MR 0374131 (51:10331)
  • [5] R. S. Martin, G. Peters, and J. H. Wilkinson, The QR algorithm for real Hessenberg matrices, Numer. Math. 14 (1970), 219-231. MR 1553971
  • [6] B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler, Matrix eigensystem routines--Eispack guide, Lecture Notes in Comput. Sci., vol. 6, Springer-Verlag, 1974. MR 0494879 (58:13662a)
  • [7] G. W. Stewart, Introduction to matrix computations, Academic Press, New York, 1973. MR 0458818 (56:17018)
  • [8] R. van de Geijn, Deferred shifting schemes for parallel QR methods, preprint. MR 1199555 (93k:65031)
  • [9] D. S. Watkins and L. Elsner, Convergence of algorithms of decomposition type for the eigenvalue problem, Linear Algebra Appl. 143 (1991), 19-47. MR 1077722 (91m:65114)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65F15

Retrieve articles in all journals with MSC: 65F15

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society