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Verified Numerical Computations for Large-Scale Linear Systems

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Abstract

This paper concerns accuracy-guaranteed numerical computations for linear systems. Due to the rapid progress of supercomputers, the treatable problem size is getting larger. The larger the problem size, the more rounding errors in floating-point arithmetic can accumulate in general, and the more inaccurate numerical solutions are obtained. Therefore, it is important to verify the accuracy of numerical solutions. Verified numerical computations are used to produce error bounds on numerical solutions. We report the implementation of a verification method for large-scale linear systems and some numerical results using the RIKEN K computer and the Fujitsu PRIMEHPC FX100, which show the high performance of the verified numerical computations.

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References

  1. L. S. Blackford, J. Choi, A. Cleary, E. DAzevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, R. C. Whaley: ScaLAPACK — Scalable Linear Algebra PACKage. Available at http://www.netlib.org/scalapack/ (2019).

  2. A. M. Castaldo, R. C. Whaley, A. T. Chronopoulos: Reducing floating point error in dot product using the superblock family of algorithms. SIAM J. Sci. Comput. 31 (2008), 1156–1174.

    Article  MathSciNet  Google Scholar 

  3. FUJITSU: FUJITSU Supercomputer PRIMEHPC FX100. Available at https://www.fujitsu.com/global/products/computing/servers/supercomputer/primehpc-fx100/ (2020).

  4. N. J. Higham: Accuracy and Stability of Numerical Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, 2002.

    Book  Google Scholar 

  5. N. J. Higham, T. Mary: A new approach to probabilistic rounding error analysis. SIAM J. Sci. Comput. 41 (2019), A2815–A2835.

    Article  MathSciNet  Google Scholar 

  6. IEEE Computer Society: IEEE Standard for Floating-Point Arithmetic: IEEE Std 754–2008. IEEE, NewYork, 2008.

    Google Scholar 

  7. C.-P. Jeannerod, S. M. Rump: Improved error bounds for inner products in floating-point arithmetic. SIAM J. Matrix Anal. Appl. 34 (2013), 338–344.

    Article  MathSciNet  Google Scholar 

  8. M. Kolberg, G. Bohlender, L. G. Fernandes: An efficient approach to solve very large dense linear systems with verified computing on clusters. Numer. Linear Algebra Appl. 22 (2015), 299–316.

    Article  MathSciNet  Google Scholar 

  9. X. Li, J. Demmel, D. Bailey, Y. Hida, J. Iskandar, A. Kapur, M. Martin, B. Thompson, T. Tung, D. Yoo: XBLAS — Extra Precise Basic Linear Algebra Subroutines. Available at https://www.netlib.org/xblas/ (2008).

  10. A. Minamihata, K. Sekine, T. Ogita, S. M. Rump, S. Oishi: Improved error bounds for linear systems with H-matrices. Nonlinear Theory Appl., IEICE 6 (2015), 377–382.

    Article  Google Scholar 

  11. Y. Morikura, K. Ozaki, S. Oishi: Verification methods for linear systems using ufp estimation with rounding-to-nearest. Nonlinear Theory Appl., IEICE 4 (2013), 12–22.

    Article  Google Scholar 

  12. M. Nakata: The MPACK: Multiple Precision Arithmetic BLAS (MBLAS) and LAPACK (MLAPACK). Available at http://mplapack.sourceforge.net/ (2011).

  13. A. Neumaier: A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equations. Reliab. Comput. 5 (1999), 131–136.

    Article  MathSciNet  Google Scholar 

  14. T. Ogita, S. Oishi: Fast verification for large-scale systems of linear equations. IPSJ Trans. 46 (2005), 10–18. (In Japanese.)

    Google Scholar 

  15. T. Ogita, S. Oishi, Y. Ushiro: Computation of sharp rigorous componentwise error bounds for the approximate solutions of systems of linear equations. Reliab. Comput. 9 (2003), 229–239.

    Article  MathSciNet  Google Scholar 

  16. T. Ogita, S. M. Rump, S. Oishi: Accurate sum and dot product. SIAM J. Sci. Comput. 26 (2005), 1955–1988.

    Article  MathSciNet  Google Scholar 

  17. T. Ogita, S. M. Rump, S. Oishi: Verified Solution of Linear Systems Without Directed Rounding: Technical Report No. 2005–04. Advanced Research Institute for Science and Engineering, Waseda University, Tokyo, 2005.

    Google Scholar 

  18. S. Oishi, S. M. Rump: Fast verification of solutions of matrix equations. Numer. Math. 90 (2002), 755–773.

    Article  MathSciNet  Google Scholar 

  19. K. Ozaki, T. Ogita: Generation of linear systems with specified solutions for numerical experiments. Reliab. Comput. 25 (2017), 148–167.

    MathSciNet  Google Scholar 

  20. K. Ozaki, T. Ogita, S. Miyajima, S. Oishi, S. M. Rump: A method of obtaining verified solutions for linear systems suited for Java. J. Comput. Appl. Math. 199 (2007), 337–344.

    Article  MathSciNet  Google Scholar 

  21. K. Ozaki, T. Ogita, S. Oishi: An algorithm for automatically selecting a suitable verification method for linear systems. Numer. Algorithms 56 (2011), 363–382.

    Article  MathSciNet  Google Scholar 

  22. A. Petitet: PBLAS — Parallel Basic Linear Algebra Subprograms. Available at http://www.netlib.org/scalapack/pblas_qref.html.

  23. RIKEN Center for Computational Science: What is K? Available at https://www.r-ccs.riken.jp/en/k-computer/about/ (2019).

  24. S. M. Rump: Kleine Fehlerschranken bei Matrixproblemen: PhD Thesis. Universität Karlsruhe, Karlsruhe, 1980. (In German.)

  25. S. M. Rump: Accurate solution of dense linear systems I: Algorithms in rounding to nearest. J. Comput. Appl. Math. 242 (2013), 157–184.

    Article  MathSciNet  Google Scholar 

  26. S. M. Rump: Accurate solution of dense linear systems II: Algorithms using directed rounding. J. Comput. Appl. Math. 242 (2013), 185–212.

    Article  MathSciNet  Google Scholar 

  27. R. D. Skeel: Iterative refinement implies numerical stability for Gaussian elimination. Math. Comput. 35 (1980), 817–832.

    Article  MathSciNet  Google Scholar 

  28. V. Strassen: Gaussian elimination is not optimal. Numer. Math. 13 (1969), 354–356.

    Article  MathSciNet  Google Scholar 

  29. N. Yamanaka, T. Ogita, S. M. Rump, S. Oishi: A parallel algorithm for accurate dot product. Parallel Comput. 34 (2008), 392–410.

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors wish to thank the anonymous reviewers for comments on earlier version of this paper and suggestions for future work. We sincerely express our thank to Fujitsu Limited for developing the function of switches of rounding modes and giving us the fruitful information of BLAS functions. Thanks to Mr. Ryota Ochiai, Mr. Atsushi Sakamoto, and Mr. Ryota Kobayashi, former students in Shibaura Institute of Technology for the assistance of coding and numerical tests.

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Authors and Affiliations

  1. Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama, 337-8570, Japan

    Katsuhisa Ozaki & Takeshi Terao

  2. Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo, 167-8585, Japan

    Takeshi Ogita

  3. Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi, 464-8601, Japan

    Takahiro Katagiri

Authors
  1. Katsuhisa Ozaki
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  2. Takeshi Terao
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  3. Takeshi Ogita
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  4. Takahiro Katagiri
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Corresponding author

Correspondence to Katsuhisa Ozaki.

Additional information

This research was supported by MEXT as “Exploratory Challenge on Post-K computer” (Development of verified numerical computations and super high-performance computing environment for extreme researches) using computational resources of the K computer at RIKEN R-CCS and Fujitsu FX100 at Nagoya University through the HPCI System Research Project (Project ID: hp190192).

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Ozaki, K., Terao, T., Ogita, T. et al. Verified Numerical Computations for Large-Scale Linear Systems. Appl Math 66, 269–285 (2021). https://doi.org/10.21136/AM.2021.0318-19

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  • DOI: https://doi.org/10.21136/AM.2021.0318-19

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