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Convergence of the unitary $QR$ algorithm with a unimodular Wilkinson shift

Authors: Tai-Lin Wang and William B. Gragg
Journal: Math. Comp. 72 (2003), 375-385
MSC (2000): Primary 65F15, 15A18
Published electronically: June 25, 2002
MathSciNet review: 1933826
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Abstract: In applying the $QR$ algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a projected Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the $QR$ iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.

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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): May 5, 1999
Received by editor(s) in revised form: April 3, 2001
Published electronically: June 25, 2002
Additional Notes: The first author’s research was supported by the Center for Computational Sciences at the University of Kentucky
The second author’s research was supported in part by the National Science Foundation under grant DMS-8704196
Article copyright: © Copyright 2002 American Mathematical Society

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