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Preconditioned Iterative Methods for Eigenvalue Counts

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Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing (EPASA 2015)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 117))

Abstract

We describe preconditioned iterative methods for estimating the number of eigenvalues of a Hermitian matrix within a given interval. Such estimation is useful in a number of applications. It can also be used to develop an efficient spectrum-slicing strategy to compute many eigenpairs of a Hermitian matrix. Our method is based on the Lanczos- and Arnoldi-type of iterations. We show that with a properly defined preconditioner, only a few iterations may be needed to obtain a good estimate of the number of eigenvalues within a prescribed interval. We also demonstrate that the number of iterations required by the proposed preconditioned schemes is independent of the size and condition number of the matrix. The efficiency of the methods is illustrated on several problems arising from density functional theory based electronic structure calculations.

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Notes

  1. 1.

    The variance of a Gaussian trace estimator applied to a Hermitian matrix also depends on the magnitude of the diagonal elements and can be expressed only in terms of the eigenvalues of A.

  2. 2.

    Available in the PARSEC group of the University of Florida Sparse Matrix Collection at https://www.cise.ufl.edu/research/sparse/matrices/.

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Acknowledgements

Support for this work was provided through Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research.

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Authors and Affiliations

  1. Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA

    Eugene Vecharynski & Chao Yang

Authors
  1. Eugene Vecharynski
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  2. Chao Yang
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Corresponding author

Correspondence to Eugene Vecharynski .

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Tetsuya Sakurai

  2. Department of Electrical Engineering and Computer Science, Nagoya University, Nagoya, Aichi, Japan

    Shao-Liang Zhang

  3. RIKEN Advanced Institute for Computational Science, Kobe, Hyogo, Japan

    Toshiyuki Imamura

  4. Department of Communication Engineering and Informatics, University of Electro-Communications, Graduate School of Informatics and Engineering, Tokyo, Japan

    Yusaku Yamamoto

  5. Department of Particle Physics, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Yoshinobu Kuramashi

  6. Department of Applied Mathematics and Physics, Tottori University, Tottori, Japan

    Takeo Hoshi

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Vecharynski, E., Yang, C. (2017). Preconditioned Iterative Methods for Eigenvalue Counts. In: Sakurai, T., Zhang, SL., Imamura, T., Yamamoto, Y., Kuramashi, Y., Hoshi, T. (eds) Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing. EPASA 2015. Lecture Notes in Computational Science and Engineering, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-62426-6_8

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