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A Parallel Bisection and Inverse Iteration Solver for a Subset of Eigenpairs of Symmetric Band Matrices

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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))

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Abstract

The tridiagonalization and its back-transformation for computing eigenpairs of real symmetric dense matrices are known to be the bottleneck of the execution time in parallel processing owing to the communication cost and the number of floating-point operations. To overcome this problem, we focus on real symmetric band eigensolvers proposed by Gupta and Murata since their eigensolvers are composed of the bisection and inverse iteration algorithms and do not include neither the tridiagonalization of real symmetric band matrices nor its back-transformation. In this paper, the following parallel solver for computing a subset of eigenpairs of real symmetric band matrices is proposed on the basis of Murata’s eigensolver: the desired eigenvalues of the target band matrices are computed directly by using parallel Murata’s bisection algorithm. The corresponding eigenvectors are computed by using block inverse iteration algorithm with reorthogonalization, which can be parallelized with lower communication cost than the inverse iteration algorithm. Numerical experiments on shared-memory multi-core processors show that the proposed eigensolver is faster than the conventional solvers.

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Acknowledgements

The authors would like to express their gratitude to reviewers for their helpful comments. In this work, we used the supercomputer of ACCMS, Kyoto University.

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Authors and Affiliations

  1. Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto-shi, Kyoto, Japan

    Hiroyuki Ishigami, Kinji Kimura & Yoshimasa Nakamura

  2. Yahoo Japan Corporation, Kioi Tower, 1-3 Kioicho, Chiyoda-ku, Tokyo, Japan

    Hiroyuki Ishigami

  3. Faculty of Library, Information and Media Science, University of Tsukuba, Kasuga 1-2, Tsukuba, Japan

    Hidehiko Hasegawa

Authors
  1. Hiroyuki Ishigami
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  2. Hidehiko Hasegawa
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  3. Kinji Kimura
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  4. Yoshimasa Nakamura
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Corresponding author

Correspondence to Hiroyuki Ishigami .

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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Ishigami, H., Hasegawa, H., Kimura, K., Nakamura, Y. (2017). A Parallel Bisection and Inverse Iteration Solver for a Subset of Eigenpairs of Symmetric Band Matrices. 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_3

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