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A Householder-Based Algorithm for Hessenberg-Triangular Reduction
Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia.
Umeå University, Faculty of Science and Technology, Department of Computing Science.ORCID iD: 0000-0002-4675-7434
Institute of Mathematics, EPFL, Lausanne, Switzerland.
2018 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 39, no 3, p. 1270-1294Article in journal (Refereed) Published
Abstract [en]

The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil $A - \lambda B$ requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of choice for HT reduction relies entirely on Givens rotations regrouped and accumulated into small dense matrices which are subsequently applied using matrix multiplication routines. A nonvanishing fraction of the total flop-count must nevertheless still be performed as sequences of overlapping Givens rotations alternately applied from the left and from the right. The many data dependencies associated with this computational pattern leads to inefficient use of the processor and poor scalability. In this paper, we therefore introduce a fundamentally different approach that relies entirely on (large) Householder reflectors partially accumulated into block reflectors, by using (compact) WY representations. Even though the new algorithm requires more floating point operations than the state-of-the-art algorithm, extensive experiments on both real and synthetic data indicate that it is still competitive, even in a sequential setting. The new algorithm is conjectured to have better parallel scalability, an idea which is partially supported by early small-scale experiments using multithreaded BLAS. The design and evaluation of a parallel formulation is future work.

Place, publisher, year, edition, pages
SIAM Publications , 2018. Vol. 39, no 3, p. 1270-1294
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-154648DOI: 10.1137/17M1153637ISI: 000453716400010OAI: oai:DiVA.org:umu-154648DiVA, id: diva2:1273439
Funder
EU, Horizon 2020, 671633Available from: 2018-12-21 Created: 2018-12-21 Last updated: 2019-01-08Bibliographically approved

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Karlsson, Lars

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