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Accuracy Improvement of the Shifted Block BiCGGR Method for Linear Systems with Multiple Shifts and Multiple Right-Hand Sides

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Book cover 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 consider solving linear systems with multiple shifts and multiple right-hand sides. In order to solve these linear systems efficiently, we develop the Shifted Block BiCGGR method. This method is based on the shift-invariance property of Block Krylov subspaces. Using this property, the Shifted systems can be solved in the process of solving the Seed system without matrix-vector multiplications. The Shifted Block BiCGGR method can generate high accuracy approximate solutions of the Seed system. However, the accuracy of the approximate solutions of the Shifted systems may deteriorate due to the error of the matrix multiplications appearing in the algorithm. In this paper, we improve the accuracy of the approximate solutions of the Shifted systems generated by the Shifted Block BiCGGR method.

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Acknowledgements

This work was partly supported by JSPS KAKENHI Grant Numbers 15K15996, 25286097, 25870099, and University of Tsukuba Basic Research Support Program Type A.

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Authors and Affiliations

  1. University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8573, Japan

    Hiroto Tadano, Shusaku Saito & Akira Imakura

Authors
  1. Hiroto Tadano
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  2. Shusaku Saito
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  3. Akira Imakura
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Corresponding author

Correspondence to Hiroto Tadano .

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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Tadano, H., Saito, S., Imakura, A. (2017). Accuracy Improvement of the Shifted Block BiCGGR Method for Linear Systems with Multiple Shifts and Multiple Right-Hand Sides. 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_12

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