ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis ISBN 978-3-7001-8258-0 Online Edition Research Article
Yuli Eidelman,
Iulian Haimovici
S. 316 - 347 doi:10.1553/etna_vol58s316 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol58s316
Abstract: We apply a general bisection eigenvalue algorithm, developed forHermitian matrices with quasiseparable representations, to the particular caseof real band symmetric Toeplitz matrices. We show that every band symmetricToeplitz matrix $T_q$ with bandwidth $q$ admits the representation$T_q=A_q+H_q$, where the eigendata of $A_q$ are obtained explicitly andthe matrix $H_q$ has nonzero entries only in two diagonal blocksof size $(q-1)\times (q-1)$. Based on this representation, one obtainsan interlacing property of the eigenvalues of the matrix $T_q$ and the knowneigenvalues of the matrix $A_q$. This allows us to essentially improve the performance of thebisection eigenvalue algorithm. We also present an algorithm tocompute the corresponding eigenvectors. Keywords: Toeplitz, quasiseparable, banded matrices, eigenstructure, inequalities, Sturm with bisection Published Online: 2023/03/30 09:23:06 Object Identifier: 0xc1aa5572 0x003e286a 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 1068-9613. …
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |