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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
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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.

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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