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International Conference on Parallel Processing and Applied Mathematics

PPAM 2017: Parallel Processing and Applied Mathematics pp 600–611Cite as

Structure-Preserving Technique in the Block SS–Hankel Method for Solving Hermitian Generalized Eigenvalue Problems

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10777))

Abstract

The block SS–Hankel method is one of the most efficient methods for solving interior generalized eigenvalue problems (GEPs) when only the eigenvalues are required. However, even if the target GEP is Hermitian, the block SS–Hankel method does not always preserve the Hermitian structure. To overcome this issue, in this paper, we propose a structure-preserving technique of the block SS–Hankel method for solving Hermitian GEPs. We also analyse the error bound of the proposed method and show that the proposed method improves the accuracy of the eigenvalues. The numerical results support the results of the analysis.

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References

  1. ELSES matrix library. http://www.elses.jp/matrix/

  2. FEAST Eigenvalue Solver. http://www.feast-solver.org/

  3. Güttel, S., Polizzi, E., Tang, T., Viaud, G.: Zolotarev quadrature rules and load balancing for the FEAST eigensolver. SIAM J. Sci. Comput. 37, A2100–A2122 (2015). https://doi.org/10.1137/140980090

    Article  MathSciNet  MATH  Google Scholar 

  4. Ide, T., Inoue, Y., Futamura, Y., Sakurai, T.: Highly parallel computation of generalized eigenvalue problem in vibration for automatic transmission of vehicles using the Sakurai–Sugiura method and supercomputers. In: Itou, H., Kimura, M., Chalupecký, V., Ohtsuka, K., Tagami, D., Takada, A. (eds.) Mathematical Analysis of Continuum Mechanics and Industrial Applications. MI, vol. 26, pp. 207–218. Springer, Singapore (2017). https://doi.org/10.1007/978-981-10-2633-1_16

    Chapter  Google Scholar 

  5. Ikegami, T., Sakurai, T.: Contour integral eigensolver for non-Hermitian systems: a Rayleigh-Ritz-type approach. Taiwan. J. Math. 14, 825–837 (2010). https://doi.org/10.11650/twjm/1500405869

    Article  MathSciNet  MATH  Google Scholar 

  6. Ikegami, T., Sakurai, T., Nagashima, U.: A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method. Technical report CS-TR-08-13, Department of Computer Science, University of Tsukuba (2008)

    Google Scholar 

  7. Ikegami, T., Sakurai, T., Nagashima, U.: A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method. J. Comput. Appl. Math. 233, 1927–1936 (2010). https://doi.org/10.1016/j.cam.2009.09.029

    Article  MathSciNet  MATH  Google Scholar 

  8. Imakura, A., Du, L., Sakurai, T.: A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems. Appl. Math. Lett. 32, 22–27 (2014). https://doi.org/10.1016/j.aml.2014.02.007

    Article  MathSciNet  MATH  Google Scholar 

  9. Imakura, A., Du, L., Sakurai, T.: Error bounds of Rayleigh–Ritz type contour integral-based eigensolver for solving generalized eigenvalue problems. Numer. Algorithms 71, 103–120 (2016). https://doi.org/10.1007/s11075-015-9987-4

    Article  MathSciNet  MATH  Google Scholar 

  10. Imakura, A., Du, L., Sakurai, T.: Relationships among contour integral-based methods for solving generalized eigenvalue problems. Jpn. J. Ind. Appl. Math. 33, 721–750 (2016). https://doi.org/10.1007/s13160-016-0224-x

    Article  MathSciNet  MATH  Google Scholar 

  11. Imakura, A., Sakurai, T.: Block Krylov-type complex moment-based eigensolvers for solving generalized eigenvalue problems. Numer. Algorithms 75, 413–433 (2017). https://doi.org/10.1007/s11075-016-0241-5

    Article  MathSciNet  MATH  Google Scholar 

  12. Iwase, S., Futamura, Y., Imakura, A., Sakurai, T., Ono, T.: Efficient and scalable calculation of complex band structure using Sakurai-Sugiura method. In: SCf17 Proceeding of the International Conference for High Performance Computing, Networking, Storage and Analysis, no. 17 (2017, accepted)

    Google Scholar 

  13. Kestyn, J., Kalantzis, V., Polizzi, E., Saad, Y.: PFEAST: a high performance sparse eigenvalue solver using distributed-memory linear solvers. In: SCf16 Proceeding of the International Conference for High Performance Computing, Networking, Storage and Analysis, no. 16 (2016). https://doi.org/10.1109/SC.2016.15

  14. Polizzi, E.: A density matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B 79, 115112 (2009). https://doi.org/10.1103/PhysRevB.79.115112

    Article  Google Scholar 

  15. Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159, 119–128 (2003). https://doi.org/10.1016/S0377-0427(03)00565-X

    Article  MathSciNet  MATH  Google Scholar 

  16. Sakurai, T., Tadano, H.: CIRR: a Rayleigh-Ritz type method with counter integral for generalized eigenvalue problems. Hokkaido Math. J. 36, 745–757 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tang, P.T.P., Polizzi, E.: FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection. SIAM J. Matrix Anal. Appl. 35, 354–390 (2014). https://doi.org/10.1137/13090866X

    Article  MathSciNet  MATH  Google Scholar 

  18. Yamazaki, I., Tadano, H., Sakurai, T., Ikegami, T.: Performance comparison of parallel eigensolvers based on a contour integral method and a Lanczos method. Parallel Comput. 39, 280–290 (2013). https://doi.org/10.1016/j.parco.2012.04.001

    Article  MathSciNet  Google Scholar 

  19. Yin, G., Chan, R.H., Yeung, M.C.: A FEAST algorithm with oblique projection for generalized non-Hermitian eigenvalue problems. arXiv:1404.1768 [math.NA] (2014)

  20. Yin, G.: A randomized FEAST algorithm for generalized eigenvalue problems. arXiv:1612.03300 [math.NA] (2016)

  21. z-Pares: Parallel Eigenvalue Solver. http://zpares.cs.tsukuba.ac.jp/

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Acknowledgements

This work was supported in part by JST/CREST, JST/ACT-I (Grant No. JPMJPR16U6), JSPS KAKENHI (Grant Nos. 17K12690). This research in part used computational resources of COMA provided by Interdisciplinary Computational Science Program in Center for Computational Sciences, University of Tsukuba.

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Authors and Affiliations

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

    Akira Imakura, Yasunori Futamura & Tetsuya Sakurai

  2. JST/CREST, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan

    Tetsuya Sakurai

Authors
  1. Akira Imakura
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  2. Yasunori Futamura
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  3. Tetsuya Sakurai
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Corresponding author

Correspondence to Akira Imakura .

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Editors and Affiliations

  1. Czestochowa University of Technology, Czestochowa, Poland

    Roman Wyrzykowski

  2. University of Tennessee, Knoxville, Tennessee, USA

    Jack Dongarra

  3. University of Southern California, Marina Del Rey, California, USA

    Ewa Deelman

  4. Czestochowa University of Technology, Czestochowa, Poland

    Konrad Karczewski

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Imakura, A., Futamura, Y., Sakurai, T. (2018). Structure-Preserving Technique in the Block SS–Hankel Method for Solving Hermitian Generalized Eigenvalue Problems. In: Wyrzykowski, R., Dongarra, J., Deelman, E., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2017. Lecture Notes in Computer Science(), vol 10777. Springer, Cham. https://doi.org/10.1007/978-3-319-78024-5_52

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  • DOI: https://doi.org/10.1007/978-3-319-78024-5_52

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-78023-8

  • Online ISBN: 978-3-319-78024-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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