ETNA  Electronic Transactions on Numerical Analysis

Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A1011 Wien, Dr. Ignaz SeipelPlatz 2
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ETNA  Electronic Transactions on Numerical Analysis ISBN 9783700182580 Online Edition Research Article
Yuli Eidelman,
Iulian Haimovici
The bisection eigenvalue method for unitary Hessenberg matrices via their quasiseparable structure ()
S. 60  88doi:10.1553/etna_vol59s60 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol59s60
Abstract: If $N_0$ is a normal matrix, then the Hermitian matrices $\\frac{1}{2}(N_0+N_0^*)$ and $\\frac{i}{2}(N_0^*N_0)$have the same eigenvectors as $N_0$. Their eigenvalues are the real part and the imaginary part of the eigenvalues of $N_0$, respectively.If $N_0$ is unitary, then only the real part of each of its eigenvalues and the sign ofthe imaginary part is needed to completely determine the eigenvalue, sincethe sum of the squares of these two parts is known to be equal to $1$.Since a unitary upper Hessenberg matrix $U$ has a quasiseparable structure of order oneand we express the matrix $A=\frac{1}{2}(U+U^*)$ as quasiseparable matrix of order two , we can findthe real part of the eigenvalues and, when needed, a corresponding eigenvector $x$, by using techniquesthat have been established in the paper by Eidelman and Haimovici [Oper. Theory Adv. Appl., 271 (2018), pp. 181–200]. We describe here a fast procedure, which takes only $1.7\%$ of the bisection method time, to find the signof the imaginary part.For instance, in the worst case only, we build one rowof the quasiseparable matrix $U$ and multiply it by a known eigenvector of$A$, as the main part of the procedure.This case occurs for our algorithm when amongthe $4$ numbers $\pm\cos t\pm i \sin t$ there are exactly $2$ eigenvalues andthey are opposite, so that we have to distinguish between the case $\lambda,\lambda$and the case $\overline\lambda,\overline\lambda$. The performance of the developedalgorithm is illustrated by a series of numerical tests. The algorithm is more accurate and many times faster(when executed in Matlab) than forgeneral Hermitian matrices of quasiseparable order two, because the action of the quasiseparable generators,which are small matrices in the previous cited paper, can be replaced by scalars, most of them real numbers. Keywords: quasiseparable, eigenstructure, Sturm property, bisection, unitary Hessenberg Published Online: 2023/06/05 12:38:14 Object Identifier: 0xc1aa5572 0x003e486c Rights: . Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 10689613. …

Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A1011 Wien, Dr. Ignaz SeipelPlatz 2
Tel. +431515 81/DW 3420, Fax +431515 81/DW 3400 https://verlag.oeaw.ac.at, email: verlag@oeaw.ac.at 