Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Convergence of the shifted $QR$ algorithm for unitary Hessenberg matrices

Authors: Tai-Lin Wang and William B. Gragg
Journal: Math. Comp. 71 (2002), 1473-1496
MSC (2000): Primary 65F15, 15A18
Published electronically: November 30, 2001
MathSciNet review: 1933041
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper shows that for unitary Hessenberg matrices the $QR$algorithm, with (an exceptional initial-value modification of) the Wilkinson shift, gives global convergence; moreover, the asymptotic rate of convergence is at least cubic, higher than that which can be shown to be quadratic only for Hermitian tridiagonal matrices, under no further assumption. A general mixed shift strategy with global convergence and cubic rates is also presented.

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

  • 1. H. Bowdler, R. S. Martin, C. Reinsch, and J. H. Wilkinson, The $QR$ and $QL$ algorithms for symmetric matrices, Numer. Math. 11 (1968), 293-306.
  • 2. H. J. Buurema, A geometric proof of convergence for the $QR$ method, doctoral dissertation, Rijksuniversiteit Te Groningen, Groningen, The Netherlands, 1970. MR 52:4597
  • 3. T. J. Dekker and J. F. Traub, The shifted $QR$ algorithm for Hermitian matrices, Linear Algebra Appl. 4 (1971), 137-154. MR 44:1207
  • 4. P. J. Eberlein and C. P. Huang, Global convergence of the $QR$ algorithm for unitary matrices with some results for normal matrices, SIAM J. Numer. Anal. 12 (1975), 97-104. MR 50:8948
  • 5. J. G. F. Francis, The $QR$ transformation. I, II, Comput. J. 4 (1961-1962), 265-271, 332-345. MR 25:744
  • 6. W. B. Gragg, The $QR$ algorithm for unitary Hessenberg matrices, J. Comput. Appl. Math. 16 (1986), 1-8.
  • 7. W. B. Gragg, Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle, J. Comput. Appl. Math. 46 (1993), 183-198. MR 94e:65046
  • 8. W. Hoffmann and B. N. Parlett, A new proof of global convergence for the tridiagonal $QL$ algorithm, SIAM J. Numer. Anal. 15 (1978), 929-937. MR 80a:65075
  • 9. C. P. Huang, On the convergence of the QR algorithm with origin shifts for normal matrices, IMA J. Numer. Anal. 1 (1981), 127-133. MR 87d:65130
  • 10. E. Jiang and Z. Zhang, A new shift of the $QL$ algorithm for irreducible symmetric tridiagonal matrices, Linear Algebra Appl. 65 (1985), 261-272. MR 86g:65082
  • 11. B. N. Parlett, Singular and invariant matrices under the $QR$ transformation, Math. Comp. 20 (1966), 611-615. MR 35:3870
  • 12. B. N. Parlett, The Rayleigh quotient iteration and some generalizations for nonnormal matrices, Math. Comp. 28 (1974), 679-693. MR 53:9615
  • 13. B. N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, N.J., 1980. MR 81j:65063
  • 14. T.-L. Wang, Convergence of the $QR$ algorithm with origin shifts for real symmetric tridiagonal and unitary Hessenberg matrices, Ph.D. thesis, University of Kentucky, Lexington, Kentucky, 1988.
  • 15. T.-L. Wang and W. B. Gragg, Convergence of the shifted $QR$ algorithm for unitary Hessenberg matrices, Technical Report NPS-53-90-007, Naval Postgraduate School, Monterey, California, 1990.
  • 16. D.S. Watkins and L. Elsner, Convergence of algorithms of decomposition type for the eigenvalue problem, Linear Algebra Appl. 143 (1991), 19-47. MR 91m:65114
  • 17. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.
  • 18. J. H. Wilkinson, Global convergence of tridiagonal $QR$ algorithm with origin shifts, Linear Algebra Appl. 1 (1968), 409-420. MR 38:2938

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65F15, 15A18

Retrieve articles in all journals with MSC (2000): 65F15, 15A18

Additional Information

Tai-Lin Wang
Affiliation: Department of Mathematical Sciences, National Chengchi University, Taipei, Taiwan, Republic of China

William B. Gragg
Affiliation: Department of Mathematics, Naval Postgraduate School, Monterey, California 93943

Keywords: $QR$ algorithm, shift strategy, unitary Hessenberg matrices
Received by editor(s): March 9, 1999
Received by editor(s) in revised form: January 6, 2000, and November 7, 2000
Published electronically: November 30, 2001
Additional Notes: The research of the first author was supported by the Center for Computational Sciences at the University of Kentucky
The research of the second author was supported in part by the National Science Foundation under grant DMS-8704196
Dedicated: Dedicated to the memory of James H. Wilkinson
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society