• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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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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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: bestellung.verlag@oeaw.ac.at
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Convergence rates of individual Ritz values in block preconditioned gradient-type eigensolvers

    Ming Zhou, Klaus Neymeyr

ETNA - Electronic Transactions on Numerical Analysis, pp. 597-620, 2023/12/05

doi: 10.1553/etna_vol58s597

doi: 10.1553/etna_vol58s597


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doi:10.1553/etna_vol58s597



doi:10.1553/etna_vol58s597

Abstract

Many popular eigensolvers for large and sparse Hermitian matrices or matrix pairs can be interpreted as accelerated block preconditioned gradient (BPG) iterations for the purpose of analyzing their convergence behavior by composing known estimates. An important feature of the BPG method is the cluster robustness, i.e., that reasonable performance for computing clustered eigenvalues is ensured by a sufficiently large block size. Concise estimates reflecting this feature can easily be derived for exact-inverse (exact shift-inverse) preconditioning. Therein, the BPG method is compatible with an abstract block iteration analyzed by Knyazev [Soviet J. Numer. Anal. Math. Modelling, 2 (1987), pp. 371–396]. An adaptation to more general preconditioning is difficult as some orthogonality properties cannot be preserved. Another analysis by Ovtchinnikov [Linear Algebra Appl., 415 (2006), pp. 140–166] provides sumwise estimates for Ritz values containing elegant convergence factors. However, additional technical terms lead to cumbersome bounds and could cause overestimations in the first steps. We expect to improve the existing results by deriving concise estimates for individual Ritz values. A mid-term goal has been achieved for the BPG iteration with fixed step sizes by the authors in [Math. Comp., 88 (2019), pp. 2737–2765]. The present paper deals with the more practical case that the step sizes are implicitly optimized by the Rayleigh–Ritz method.

Keywords: preconditioned subspace eigensolvers, Ritz values, cluster robustness