Skip to main content
SpringerLink
Log in

Development of the Block BiCGGR2 method for linear systems with multiple right-hand sides

  • Special Feature: Original Paper
  • International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2018)
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

Block Krylov subspace methods are efficient iterative methods for solving linearsystems with multiple right-hand sides. The Block BiCGGR method is one of the Block Krylov subspace methods, and can generate highly accurate approximatesolutions. However, when the number of right-hand sides is large, this method becomes numerically unstable. In this paper, the Block BiCGGR2 method is proposed. This method is a numerically stable and highly accurate method than the Block BiCGGR method and the Block BiCGSTAB method. Numerical results illustrate the robustness and the accuracy of the approximate solutions of the Block BiCGGR2 method.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Aoki, S., Ishikawa, K.-I., Ishizuka, N., Kadoh, D., Kanaya, K., Kuramashi, Y., Namekawa, Y., Okawa, M., Taniguchi, Y., Ukawa, A., Ukita, N., Yoshie, T.: \(2+1\) flavor lattice QCD toward the physical point. Phys. Rev. D 79, 034503 (2009)

    Article  MATH  Google Scholar 

  2. CCS Matrix Forum. http://ccsmf.ccs.tsukuba.ac.jp

  3. Dubrulle, A.A.: Retooling the method of block conjugate gradients. Electron. Trans. Numer. Anal. 12, 216–233 (2001)

    MathSciNet  MATH  Google Scholar 

  4. Freund, R.W., Malhotra, M.: A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sides. Lin. Alg. Appl. 254, 119–157 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. El Guennouni, A., Jbilou, K., Sadok, H.: A block version of BiCGSTAB for linear systems with multiple right-hand sides. Electron. Trans. Numer. Anal 16, 129–142 (2003)

    MathSciNet  MATH  Google Scholar 

  6. Gutknecht, M.H., Schmelzer, T.: The block grade of a block Krylov space. Lin. Alg. Appl. 430, 174–185 (2009)

    Article  MathSciNet  MATH  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jbilou, K., Messaoudi, A., Sadok, H.: Global FOM and GMRES algorithms for matrix equations. Appl. Numer. Math. 31, 49–63 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jbilou, K., Sadok, H., Tinzefte, A.: Oblique projection methods for linear systems with multiple right-hand sides. Electron. Trans. Numer. Anal. 20, 119–138 (2005)

    MathSciNet  MATH  Google Scholar 

  10. O’Leary, D.P.: The block conjugate gradient algorithm and related methods. Linear Algebra Appl. 29, 293–322 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  11. Stathopoulos, A., Wu, K.: A block orthogonalization procedure with constant synchronization requirements. SIAM J. Sci. Comput. 23, 2165–2182 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. SuiteSparse Matrix Collection. https://sparse.tamu.edu

  13. Tadano, H., Saito, S., Imakura, A.: Accuracy improvement of the shifted Block BiCGGR method for linear systems with multiple shifts and multiple right-hand sides. In: Sakurai, T. (eds.) Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing (EPASA2015). Lecture Notes in Computational Science and Engineering, Vol. 117, pp. 171–185 (2017)

  14. Tadano, H., Sakurai, T., Kuramashi, Y.: Block BiCGGR: a new block krylov subspace method for computing high accuracy solutions. JSIAM Lett. 1, 44–47 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Vital, B.: Etude de quelques méthodes de résolution de problèmes linéaires de grande taille sur multiprocesseur. Ph.D. thesis, Université de Rennes I, Rennes, France (1990)

  16. Zhang, J., Dai, H., Zhao, J.: A new family of global methods for linear systems with multiple right-hand sides. J. Comput. Appl. Math. 236, 1562–1575 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, S.-L.: GPBi-CG: generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems. SIAM J. Sci. Comput. 18, 537–551 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Computational Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8577, Japan

    Hiroto Tadano

Authors
  1. Hiroto Tadano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroto Tadano.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tadano, H. Development of the Block BiCGGR2 method for linear systems with multiple right-hand sides. Japan J. Indust. Appl. Math. 36, 563–577 (2019). https://doi.org/10.1007/s13160-019-00354-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-019-00354-6

Keywords

Mathematics Subject Classification

Search

Navigation