article

An Implicit Multishift $QR$-Algorithm for Hermitian Plus Low Rank Matrices

Authors:

Gianna M. Del CorsoAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 32, Issue 4

Pages 2190 - 2212

https://doi.org/10.1137/090754522

Published: 01 July 2010 Publication History

Abstract

Hermitian plus possibly non-Hermitian low rank matrices can be efficiently reduced into Hessenberg form. The resulting Hessenberg matrix can still be written as the sum of a Hermitian plus low rank matrix. In this paper we develop a new implicit multishift $QR$-algorithm for Hessenberg matrices, which are the sum of a Hermitian plus a possibly non-Hermitian low rank correction. The proposed algorithm exploits both the symmetry and low rank structure to obtain a $QR$-step involving only $\mathcal{O}(n)$ floating point operations instead of the standard $\mathcal{O}(n^2)$ operations needed for performing a $QR$-step on a Hessenberg matrix. The algorithm is based on a suitable $\mathcal{O}(n)$ representation of the Hessenberg matrix. The low rank parts present in both the Hermitian and low rank part of the sum are compactly stored by a sequence of Givens transformations and a few vectors. Due to the new representation, we cannot apply classical deflation techniques for Hessenberg matrices. A new, efficient technique is developed to overcome this problem. Some numerical experiments based on matrices arising in applications are performed. The experiments illustrate effectiveness and accuracy of both the $QR$-algorithm and the newly developed deflation technique.

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Cited By

Bevilacqua RDel Corso GGemignani L(2020)Fast QR iterations for unitary plus low rank matricesNumerische Mathematik10.1007/s00211-019-01080-4144:1(23-53)Online publication date: 1-Jan-2020
https://dl.acm.org/doi/10.1007/s00211-019-01080-4
Aurentz JVandebril RWatkins D(2014)Fast computation of eigenvalues of companion, comrade, and related matricesBIT10.1007/s10543-013-0449-x54:1(7-30)Online publication date: 1-Mar-2014
https://dl.acm.org/doi/10.1007/s10543-013-0449-x
Aishima KMatsuo TMurota KSugihara M(2012)A Wilkinson-like multishift QR algorithm for symmetric eigenvalue problems and its global convergenceJournal of Computational and Applied Mathematics10.1016/j.cam.2011.04.012236:15(3556-3560)Online publication date: 1-Sep-2012
https://dl.acm.org/doi/10.1016/j.cam.2011.04.012

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cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 32, Issue 4

June 2010

752 pages

ISSN:1064-8275

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 July 2010

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Cited By

Bevilacqua RDel Corso GGemignani L(2020)Fast QR iterations for unitary plus low rank matricesNumerische Mathematik10.1007/s00211-019-01080-4144:1(23-53)Online publication date: 1-Jan-2020
https://dl.acm.org/doi/10.1007/s00211-019-01080-4
Aurentz JVandebril RWatkins D(2014)Fast computation of eigenvalues of companion, comrade, and related matricesBIT10.1007/s10543-013-0449-x54:1(7-30)Online publication date: 1-Mar-2014
https://dl.acm.org/doi/10.1007/s10543-013-0449-x
Aishima KMatsuo TMurota KSugihara M(2012)A Wilkinson-like multishift QR algorithm for symmetric eigenvalue problems and its global convergenceJournal of Computational and Applied Mathematics10.1016/j.cam.2011.04.012236:15(3556-3560)Online publication date: 1-Sep-2012
https://dl.acm.org/doi/10.1016/j.cam.2011.04.012

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