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Accelerating the Reorthogonalization of Singular Vectors with a Multi-core Processor

  • Hiroki Toyokawa
  • Hiroyuki Ishigami
  • Kinji Kimura
  • Masami Takata
  • Yoshimasa Nakamura
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7851)

Abstract

The dLV twisted factorization is an algorithm to compute singular vectors for given singular values fast and in parallel. However the orthogonality of the computed singular vectors may be worse if a matrix has clustered singular values. In order to improve the orthogonality, reorthogonalization by, for example, the modified Gram-Schmidt algorithm should be done. The problem is that this process takes a longer time. In this paper an algorithm to accelerate the reorthogonalization of singular vectors with a multi-core processor is devised.

Keywords

Singular Value Decomposition Greedy Algorithm Hybrid Algorithm Singular Vector Task Schedule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Demmel, J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)zbMATHCrossRefGoogle Scholar
  2. 2.
    Golub, G., Van Loan, C.: Matrix Computation, 3rd edn. John Hopkins Univ. Press, Baltimore (1996)Google Scholar
  3. 3.
    Ishigami, H., Kimura, K., Nakamura, Y.: Implementation and performance evaluation of inverse iteration with new reorthogonalization algorithm. In: Proceedings of International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA 2011), vol. II, pp. 775–780 (2011)Google Scholar
  4. 4.
    Iwasaki, M., Nakamura, Y.: Accurate computation of singular values in terms of shifted integrable schemes. Japan Journal of Industrial and Applied Mathematics 23, 239–259 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Iwasaki, M., Nakamura, Y.: Positivity of dLV and mdLVs algorithms for computing singular values. Electronic Transactions on Numerical Analysis 38, 184–201 (2011)MathSciNetGoogle Scholar
  6. 6.
    Iwasaki, M., Sakano, S., Nakamura, Y.: Accurate twisted factorization of real symmetric tridiagonal matrices and its application to singular value decomposition. Transactions of the Japan Society for Industrial and Applied Mathematics 15, 461–481 (2005) (in Japanese)Google Scholar
  7. 7.
    Konda, T., Takata, M., Iwasaki, M., Nakamura, Y.: A new singular value decomposition algorithm suited to parallelization and preliminary results. In: Proceedings of IASTED International Conference on Advances in Computer Science and Technology (ACST 2006), pp. 79–85 (2006)Google Scholar
  8. 8.
    Nakamura, Y.: Functionality of Integrable System. Kyoritsu Publishing, Tokyo (2006) (in Japanese)Google Scholar
  9. 9.
    Parlett, B.N., Dhillon, I.S.: Relatively robust representations of symmetric tridiagonals. Linear Algebra Appl. 309, 121–151 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Takata, M., Kimura, K., Iwasaki, M., Nakamura, Y.: Algorithms for generating bidiagonal test matrices. In: Proceedings of International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA 2007), pp. 732–738 (2007)Google Scholar
  11. 11.
    Takata, M., Kimura, K., Nakamura, Y.: Generating algorithms for matrices with large condition number to evaluate singular value decomposition. In: Proceedings of International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA 2010), pp. 619–625 (2010)Google Scholar
  12. 12.
    Takata, M., Kimura, K., Iwasaki, M., Nakamura, Y.: Performance of a new scheme for bidiagonal singular value decomposition of large scale. In: Proceedings of IASTED International Conference on Parallel and Distributed Computing and Networks (PDCN 2006), pp. 304–309 (2006)Google Scholar
  13. 13.
    Takata, M., Kimura, K., Iwasaki, M., Nakamura, Y.: Implementation of library for high speed singular value decomposition. Journal of Information Processing Society of Japan 47 SIG7(ACS 14), 91–104 (2006)Google Scholar
  14. 14.
    Toyokawa, H., Kimura, K., Takata, M., Nakamura, Y.: On parallelism of the I-SVD algorithm with a multi-core processor. JSIAM Letters 1, 48–51 (2009)MathSciNetGoogle Scholar
  15. 15.
    Toyokawa, H., Kimura, K., Takata, M., Nakamura, Y.: On parallelization of the I-SVD algorithm and its evaluation for clustered singular values. In: Proceedings of International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA 2009), pp. 711–717 (2009)Google Scholar
  16. 16.
    Yamamoto, Y., Hirota, Y.: A parallel algorithm for incremental orthogonalization based on the compact WY representation. JSIAM Letters 3, 89–92 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Hiroki Toyokawa
    • 1
    • 2
  • Hiroyuki Ishigami
    • 2
  • Kinji Kimura
    • 2
  • Masami Takata
    • 3
  • Yoshimasa Nakamura
    • 2
  1. 1.CyberAgent, Inc.Shibuya-kuJapan
  2. 2.Graduate School of InformaticsKyoto UniversitySakyo-kuJapan
  3. 3.Academic Group of Information and Computer SciencesNara Women’s UniversityNara-cityJapan

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