Skip to main content
Log in

An improved generalized conjugate residual squared (IGCRS2) algorithm suitable for distributed parallel computing

  • Original Paper
  • Area 2
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, based on generalized conjugate residual squared (GCRS2) algorithm in Zhang et al. (2010 Third International Conference on Information and Computing, pp 326–329, 2010) and the ideas in Gu et al. (Appl Math Comput 186:1243–1253, 2007), we present an improved generalized conjugate residual squared (IGCRS2) algorithm, which is designed for distributed parallel environments. The improved algorithm reduces two global synchronization points to one by changing the computation sequence in the GCRS2 algorithm and all inner products per iteration are independent and communication time required for inner product can be overlapped with useful computation. Theoretical analysis and numerical comparison about isoefficiency analysis show that the IGCRS2 method has better parallelism and scalability than the GCRS2 method and the parallel performance can be improved by a factor of about 2.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bücker, H.M., Sauren, M.: A parallel version of the quasi-minimal residual method based on coupled two-term recurrences. In: Proceedings of Workshop on Applied Parallel computing in Industrial Problems and Optimization (Para96), Technical University of Denmark, Springer, Lyngby (1996)

  2. Chi, L.H., Liu, J., Liu, X.P., Hu, Q.F., Li, X.M.: An improved conjugate residual algorithm for large symmetric linear systems. In: Computational Physics, Proceedings of the Joint Conference of ICCP6 and CCP2003, pp. 325–328. Rinton Press, New Jersey (2005)

  3. Freund, R.W., Nachtigal, N.M.: QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60, 315–339 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Grama, A., Gupta, A., Kumar, V.: Isoefficiency function: a scalability metric for parallel algorithms and architectures. IEEE Parallel Distrib. Technol. 1(3), 12–21 (1993)

    Article  Google Scholar 

  5. Gu, T.X., Liu, X.P., Mo, Z.Y.: Multiple search direction conjugate gradient method I: methods and their propositions. Int. J. Comput. Math. 81(9), 1133–1143 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gu, T.X., Zuo, X.Y., Zhang, L.T., Zhang, W.Q., Sheng, Z.Q.: An improved Bi-Conjugate residual algorithm suitable for distributed parallel computing. Appl. Math. Comput. 186, 1243–1253 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Jiang, D.D., Xu, Z.Z., Chen, Z.H., Han, Y., Xu, H.W.: Joint time-frequency sparse estimation of large-scale network traffic. Comput. Netw. 55(10), 3533–3547 (2011)

    Article  Google Scholar 

  8. Jiang, D.D., Xu, Z.Z., Xu, H.W., Han, Y., Chen, Z.H., Yuan, Z.: An approximation method of origin-destination flow traffic from link load counts. Comput. Electr. Eng. 37(6), 1106–1121 (2011)

    Article  Google Scholar 

  9. Liu, X.P., Gu, T.X., Hang, X.D., Sheng, Z.Q.: A parallel version of QMRCGSTAB method for large linear systems in distributed parallel environments. Appl. Math. Comput. 172(2), 744–752 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston (1996)

    MATH  Google Scholar 

  11. Sogabe, T., Zhang, S.L.: Extended conjugate residual methods for solving nonsymmetric linear systems. In: Yuan, Y.-X. (ed.) Numerical Linear Algebra and Optimization, pp. 88–99. Science Press, Beijing (2003)

    Google Scholar 

  12. Sonneveld, P.: CGS: a fast lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 10(1), 36–52 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  13. de Sturler, E., van der Vorst, H.A.: Reducing the effect of the global communication in GMRES(m) and CG on parallel distributed memory computers. Appl. Numer. Math. 18, 441–459 (1995)

    Article  MATH  Google Scholar 

  14. de Sturler, E.: A performance model for Krylov subspace methods on mesh-based parallel computers. Parallel Comput. 22, 57–74 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)

    Article  MATH  Google Scholar 

  16. Yang, T.R.: The improved CGS method for large and sparse linear systems on bulk synchronous parallel architectures. In: 2002 5th Intern, Conf. Algorithms and Architectures for Parallel Processing, pp. 232–237. IEEE Computer Society (2002)

  17. Yang, T.R., Brent, R.P.: The improved BiCGSTAB method for large and sparse nonsymmetric linear systems on parallel distributed memory architectures, In: 2001 5th Intern, Conf. Algorithms and Architectures for Parallel Processing, pp. 324–328. IEEE Computer Society (2002)

  18. Yang, T.R., Brent, R.P.: The improved BiCG method for large and sparse linear systems on parallel distributed memory architectures. Inf. J. 6, 349–360 (2003)

    MATH  MathSciNet  Google Scholar 

  19. Yang, T.R., Lin, H.X.: The improved quasi-minimal residual method on massively distributed memory computers. In: Proceedings of The International Conference on High Performance Computing and Networking (HPCN-97), April 1997

  20. Zhang, L.T., Huang, T.Z., Gu, T.X., Zuo, X.Y.: An improved conjugate residual squared algorithm suitable for distributed parallel computing. Microelectron. Comput. 25(10), 12–14 (2008)

    Google Scholar 

  21. Zhang, J.H., Zhao, J.: A generalized conjugate residual squared algorithm for solving nonsymmetric linear systems. In: 2010 Third International Conference on Information and Computing, pp. 326–329 (2010)

  22. Zhang, L.T., Zuo, X.Y., Gu, T.X., Huang, T.Z.: Conjugate residual squared method and its improvement for non-symmetric linear systems. Int. J. Comput. Math. 87(7), 1578–1590 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the referees and Editor for their helpful and detailed suggestions for revising this manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Li-Tao Zhang.

Additional information

This research of this author is supported by NSFC Tianyuan Mathematics Youth Fund (11226337), NSFC(11471098,61203179,61202098,61170309,91130024,61272544,61472462 and 11171039), Aeronautical Science Foundation of China (2013ZD55006), Project of Youth Backbone Teachers of Colleges and Universities in Henan Province(2013GGJS-142), ZZIA Innovation team fund (2014TD02), Major project of development foundation of science and technology of CAEP (2012A0202008), Defense Industrial Technology Development Program, Basic and Advanced Technological Research Project of of Henan Province (122300410181,142300410333), China Postdoctoral Science Foundation (2014M552001), Henan Province Postdoctoral Science Foundation (2013031), Natural Science Foundation of Henan Province (13A110399,14A630019,14B110023), Natural Science Foundation of Zhengzhou City (141PQYJS560).

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, LT., Dong, XN., Gu, TX. et al. An improved generalized conjugate residual squared (IGCRS2) algorithm suitable for distributed parallel computing. Japan J. Indust. Appl. Math. 32, 143–155 (2015). https://doi.org/10.1007/s13160-014-0163-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-014-0163-3

Keywords

Mathematics Subject Classification

Navigation