A Multishift, Multipole Rational QZ Method with Aggressive Early Deflation
Abstract
In the article “A Rational QZ Method” by D. Camps, K. Meerbergen, and R. Vandebril [SIAM J. Matrix Anal. Appl., 40 (2019), pp. 943--972], we introduced rational QZ (RQZ) methods. Our theoretical examinations revealed that the convergence of the RQZ method is governed by rational subspace iteration, thereby generalizing the classical QZ method, whose convergence relies on polynomial subspace iteration. Moreover the RQZ method operates on a pencil more general than Hessenberg---upper triangular, namely, a Hessenberg pencil, which is a pencil consisting of two Hessenberg matrices. However, the RQZ method can only be made competitive to advanced QZ implementations by using crucial add-ons such as small bulge multishift sweeps, aggressive early deflation, and optimal packing. In this paper we develop these techniques for the RQZ method. In the numerical experiments we compare the results with state-of-the-art routines for the generalized eigenvalue problem and show that the presented method is competitive in terms of speed and accuracy.
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Information & Authors
Information
Published In
SIAM Journal on Matrix Analysis and Applications
Pages: 753 - 774
DOI: 10.1137/19M1249631
ISSN (online): 1095-7162
Copyright
© 2021, Society for Industrial and Applied Mathematics.
History
Submitted: 12 March 2019
Accepted: 19 February 2021
Published online: 27 May 2021
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Funding Information
Onderzoeksraad, KU Leuven https://doi.org/10.13039/501100004497 : C14/16/056, OT/14/074
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