The $QR$ Steps with Perfect Shifts

Mastronardi, Nicola;Van Dooren, Paul
(2018) SIAM Journal on Matrix Analysis and Applications — Vol. 39, n° 4, p. 1591-1615 (2018)

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  • Mastronardi, Nicolaorcid-logoIstituto per le Applicazioni del Calcolo ``M. Picone," Bari, Italy
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  • Van Dooren, PaulUCLouvain
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Abstract
In this paper we revisit the problem of performing a QR step on an unreduced Hessenberg matrix H when we know an ``exact"" eigenvalue \lambda 0 of H. In exact arithmetic, this eigenvalue will appear on the diagonal of the transformed Hessenberg matrix H\~ and will be decoupled from the remaining part of the Hessenberg matrix, thus resulting in a deflation. But it is well known that in finite-precision arithmetic the so-called perfect shift can get blurred and that the eigenvalue \lambda 0 can then not be deflated and/or is perturbed significantly. In this paper, we develop a new strategy for computing such a QR step so that the deflation is almost always successful. We also show how to extend this technique to double QR steps with complex conjugate shifts
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Mastronardi, N., & Van Dooren, P. (2018). The $QR$ Steps with Perfect Shifts. SIAM Journal on Matrix Analysis and Applications, 39(4), 1591-1615. https://doi.org/10.1137/17m1139321 (Original work published 2018)