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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 ![]()
Andreas Frommer,
Kathryn Lund,
Daniel B. Szyld
S. 100 - 126 doi:10.1553/etna_vol47s100 ![]() Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol47s100
Abstract: A variety of block Krylov subspace methods have been successfully developed for linear systems and matrix equations. The application of block Krylov methods to compute matrix functions is, however, less established, despite the growing prevalence of matrix functions in scientific computing. Of particular importance is the evaluation of a matrix function on not just one but multiple vectors. The main contribution of this paper is a class of efficient block Krylov subspace methods tailored precisely to this task. With the full orthogonalization method (FOM) for linear systems forming the backbone of our theory, the resulting methods are referred to as B(FOM)2: block FOM for functions of matrices. Many other important results are obtained in the process of developing these new methods. Matrix-valued inner products are used to construct a general framework for block Krylov subspaces that encompasses already established results in the literature. Convergence bounds for B(FOM)2 are proven for Stieltjes functions applied to a class of matrices which are self-adjoint and positive definite with respect to the matrix-valued inner product. A detailed algorithm for B(FOM)2 with restarts is developed, whose efficiency is based on a recursive expression for the error, which is also used to update the solution. Numerical experiments demonstrate the power and versatility of this new class of methods for a variety of matrix-valued inner products, functions, and matrices. Keywords: matrix functions, restarted Krylov subspace methods, block Krylov subspace methods, global methods Published Online: 2017/11/15 13:02:52 Object Identifier: 0xc1aa5576 0x0037106a 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 |