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
Karsten Kahl,
Bruno Lang
S. 51 - 67 doi:10.1553/etna_vol54s51 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol54s51
Abstract: It is customary to identify sparse matrices with the corresponding adjacency or incidence graphs. For the solution of a linear system of equations using Gaussian elimination, the representation by its adjacency graph allows a symbolic factorization that can be used to predict memory footprints and enables the determination of near-optimal elimination orderings based on heuristics. The Hermitian eigenvalue problem on the other hand seems to evade such treatment at first glance due to its inherent iterative nature. In this paper we prove this assertion wrong by revealing a tight connection of Hermitian eigensolvers based on rank-1 modifications with a symbolic edge elimination procedure. A symbolic calculation based on the incidence graph of the matrix can be used in analogy to the symbolic phase of Gaussian elimination to develop heuristics which reduce memory footprint and computations. Yet, we also show that the question of an optimal elimination strategy remains NP-complete, in analogy to the linear systems case. Keywords: symmetric eigenvalue problem, hypergraphs, Gaussian elimination Published Online: 2020/11/13 08:48:17 Object Identifier: 0xc1aa5576 0x003bfef6 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 |