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Optimization of the Multishift QR Algorithm with Coprocessors for Non-Hermitian Eigenvalue Problems

Published online by Cambridge University Press:  28 May 2015

Takafumi Miyata,
Yusaku Yamamoto,
Takashi Uneyama,
Yoshimasa Nakamura and
Shao-Liang Zhang
Takafumi Miyata*
Affiliation:
Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
Yusaku Yamamoto*
Affiliation:
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
Takashi Uneyama*
Affiliation:
Institute for Chemical Research, Kyoto University, Gokasho, Uji 611-0011, Japan
Yoshimasa Nakamura*
Affiliation:
Graduate School of Informatics, Kyoto University, 36-1 Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan
Shao-Liang Zhang*
Affiliation:
Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
*
Corresponding author. Email: miyata@na.cse.nagoya-u.ac.jp
Corresponding author. Email: yamamoto@cs.kobe-u.ac.jp
Corresponding author. Email: uneyama@scl.kyoto-u.ac.jp
Corresponding author. Email: ynaka@amp.i.kyoto-u.ac.jp
Corresponding author. Email: zhang@na.cse.nagoya-u.ac.jp
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Abstract

The multishift QR algorithm is efficient for computing all the eigenvalues of a dense, large-scale, non-Hermitian matrix. The major part of this algorithm can be performed by matrix-matrix multiplications and is therefore suitable for modern processors with hierarchical memory. A variant of this algorithm was recently proposed which can execute more computational parts by matrix-matrix multiplications. The algorithm is especially appropriate for recent coprocessors which contain many processor-elements such as the CSX600. However, the performance of the algorithm highly depends on the setting of parameters such as the numbers of shifts and divisions in the algorithm. Optimal settings are different depending on the matrix size and computational environments. In this paper, we construct a performance model to predict a setting of parameters which minimizes the execution time of the algorithm. Experimental results with the CSX600 coprocessor show that our model can be used to find the optimal setting.

Keywords

65F1565Y1065Y20Eigenvaluesmultishift QR algorithmbulge-chasingperformance modeling

Information

Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 2 , May 2011, pp. 187 - 196
Copyright
Copyright © Global-Science Press 2011

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