Convergence of the shifted 𝑄𝑅 algorithm for unitary Hessenberg matrices

Wang, Tai-Lin; Gragg, William

doi:10.1090/S0025-5718-01-01387-4

Convergence of the shifted $QR$ algorithm for unitary Hessenberg matrices
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by Tai-Lin Wang and William B. Gragg PDF

Math. Comp. 71 (2002), 1473-1496 Request permission

Abstract:

This paper shows that for unitary Hessenberg matrices the $Q\!R$ 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

H. Bowdler, R. S. Martin, C. Reinsch, and J. H. Wilkinson, The $Q\!R$ and $Q\!L$ algorithms for symmetric matrices, Numer. Math. 11 (1968), 293-306.
Hendrik Jan Buurema, A geometric proof of convergence for the $QR$ method, Rijksuniversiteit te Groningen, Groningen, 1970. Doctoral dissertation, University of Groningen. MR 0383717
T. J. Dekker and J. F. Traub, The shifted $QR$ algorithm for Hermitian matrices, Linear Algebra Appl. 4 (1971), 137–154. MR 283977, DOI 10.1016/0024-3795(71)90035-8
P. J. Eberlein and C. P. Huang, Global convergence of the $\textrm {QR}$ algorithm for unitary matrices with some results for normal matrices, SIAM J. Numer. Anal. 12 (1975), 97–104. MR 356478, DOI 10.1137/0712009
J. G. F. Francis, The $QR$ transformation. II, Comput. J. 4 (1961/62), 332–345. MR 137289, DOI 10.1093/comjnl/4.4.332
W. B. Gragg, The $Q\!R$ algorithm for unitary Hessenberg matrices, J. Comput. Appl. Math. 16 (1986), 1-8.
William 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), no. 1-2, 183–198. Computational complex analysis. MR 1222480, DOI 10.1016/0377-0427(93)90294-L
W. Hoffmann and B. N. Parlett, A new proof of global convergence for the tridiagonal QL algorithm, SIAM J. Numer. Anal. 15 (1978), no. 5, 929–937. MR 507555, DOI 10.1137/0715060
Hou De Han, Nonconforming elements in the mixed finite element method, J. Comput. Math. 2 (1984), no. 3, 223–233. MR 815417
Er Xiong Jiang and Zhen Yue Zhang, A new shift of the $QL$ algorithm for irreducible symmetric tridiagonal matrices, Linear Algebra Appl. 65 (1985), 261–272. MR 774356, DOI 10.1016/0024-3795(85)90101-6
Beresford Parlett, Singular and invariant matrices under the $QR$ transformation, Math. Comp. 20 (1966), 611–615. MR 213005, DOI 10.1090/S0025-5718-1966-0213005-9
B. N. Parlett, The Rayleigh quotient iteration and some generalizations for nonnormal matrices, Math. Comp. 28 (1974), 679–693. MR 405823, DOI 10.1090/S0025-5718-1974-0405823-3
Beresford N. Parlett, The symmetric eigenvalue problem, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. MR 570116
T.-L. Wang, Convergence of the $Q\!R$ algorithm with origin shifts for real symmetric tridiagonal and unitary Hessenberg matrices, Ph.D. thesis, University of Kentucky, Lexington, Kentucky, 1988.
T.-L. Wang and W. B. Gragg, Convergence of the shifted $Q\!R$ algorithm for unitary Hessenberg matrices, Technical Report NPS-53-90-007, Naval Postgraduate School, Monterey, California, 1990.
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, DOI 10.1016/0024-3795(91)90004-G
J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.
J. H. Wilkinson, Global convergence of tridiagonal $\textrm {QR}$ algorithm with origin shifts, Linear Algebra Appl. 1 (1968), 409–420. MR 234622, DOI 10.1016/0024-3795(68)90017-7