Skip to main content
SpringerLink
Log in

Block GPBi-CG method for solving nonsymmetric linear systems with multiple right-hand sides and its convergence analysis

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, the block generalized product-type bi-conjugate gradient (GPBi-CG) method for solving large, sparse nonsymmetric linear systems of equations with multiple right-hand sides is proposed. The new algorithm is based on the block BiCG process. We analyze the convergence behavior of this method and present a bound for the residual norm of block GPBi-CG according to the residual norm of Bl-GMRES method. In addition, we prove that convergence is guaranteed when A is positive real. The numerical experiments show the efficiency of the new method and confirm the theoretical results.

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

Similar content being viewed by others

References

  1. Addam, M., Heyouni, M., Sadok, H.: The block Hessenberg process for matrix equations. ElectronTrans. Numer. Anal. 46, 460–473 (2017)

    MathSciNet  MATH  Google Scholar 

  2. Amini, S. , Toutounian, F., Gachpazan, M.: The block CMRH method for solving nonsymmetric linear systems with multiple right-hand sides. J. Comput. Appl. Math. 337, 166–174 (2018)

    Article  MathSciNet  Google Scholar 

  3. Calvetti, D., Golub, G.H., Reichel, L.: Adaptive Chebyshev iterative methods for nonsymetric linear systems based on modified moments. Numer. Math. 67, 21–40 (1994)

    Article  MathSciNet  Google Scholar 

  4. Davis, T., Hu, Y.: The university of Florida sparse matrix collection. ACM Trans. Math. Softw. 38, 1–25 (2011). Avaiable online at http://www.cise.ufl.edu/research/sparse/matrices/list$($_$)by$($_$)id.html

    MathSciNet  MATH  Google Scholar 

  5. Freund, R. W., Malhotra, M.: A Block-QMR algorithm for non-hermitian linear systems with multiple right-hand sides. Linear Algeb. Appl. 254, 119–157 (1997)

    Article  MathSciNet  Google Scholar 

  6. Frommer, A., Lund, K., Szyld, D. B.: Block Krylov subspace methods for functions of matrices. Electron. Transact. Numer. Anal. 47, 100–126 (2017). https://doi.org/10.1553/etna-vol47s100

    MathSciNet  MATH  Google Scholar 

  7. Frommer, A., Lund, K., Szyld, D.B.: Block Krylov subspace methods for functions of matrices II: Modified block FOM, tech. report, MATHICSE cublens. https://doi.org/10.5075/ep-MATHICSE-265508 (2019)

  8. El Guennouni, A., Jbilou, K., Sadok, H.: The block Lanczos method for linear systems with multiple right-hand sides. J. App. Numer. Math. 51, 243–256 (2004)

    Article  MathSciNet  Google Scholar 

  9. 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 

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

    Article  MathSciNet  Google Scholar 

  11. Karimi, S., Toutounian, F.: The block least squares method for solving nonsymmetric linear systems with multiple right-hand sides. Appl. Math. Comput. 177, 852–862 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Koulaei, M. H., Toutounian, F.: On computing of block ILU preconditioner for block tridiagonal systems. J. Comput. Appl. Math. 202, 248–257 (2007)

    Article  MathSciNet  Google Scholar 

  13. Kub\(\acute {\mathrm {i}}\textit {nov}\acute {\mathrm {a}}\), M., Soodhalter, K.M.: Admissible and attainable convergence behavior of block Arnoldi and GMRES. Siam J. Matrix Anal. Appl. 41, 464–486 (2019)

  14. Nikishin, A., Yeremin, A.: Variable block cg algorithms for solving large sparse symmetric positive definite linear systems on parallel computers, I: General iterative scheme. SIAM J. Matrix Anal. Appl. 16, 1135–1153 (1995)

    Article  MathSciNet  Google Scholar 

  15. OLeary, D.: The block conjugate gradient algorithm and related methods. Linear Algeb. Appl. 29, 293–332 (1980)

    Article  MathSciNet  Google Scholar 

  16. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    Book  Google Scholar 

  17. Simoncini, V.: A stabilized QMR version of block BiCG. SIAM J. Matrix Anal. Appl. 18, 419–434 (1997)

    Article  MathSciNet  Google Scholar 

  18. Simoncini, V., Gallopoulos, E.: Convergence properties of block GMRES and matrix polynomials. Linear Algeb. Appl. 247, 97–119 (1996)

    Article  MathSciNet  Google Scholar 

  19. Sonneveld, P.: CGS, A fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 10, 36–52 (1989)

    Article  MathSciNet  Google Scholar 

  20. Stiefel, E.L.: Kernel polynomial in linear algebra and their numerical applications. In: Further contributions to the determination of eigenvalues, vol. 49, pp 1–22 (1958)

  21. Tong, C. H., Ye, Q.: Analysis Of the finite precision bi-Conjugate gradient algorithm for nonsymmetric linear systems. Math. Comput. 69, 1559–1575 (1999)

    Article  MathSciNet  Google Scholar 

  22. Toutounian, F., Mojarrab, M.: The block LSMR method: A novel efficient algorithm for solving non-symmetric linear systems with multiple right-hand sides. Iran J. Sci. Technol. 39, 69–78 (2015)

    MathSciNet  MATH  Google Scholar 

  23. 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. Statist. Comput. 13, 631–644 (1992)

    Article  MathSciNet  Google Scholar 

  24. 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  Google Scholar 

Download references

Acknowledgements

We would like to thank the referees for their valuable remarks and helpful suggestions.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

    A. Taherian & F. Toutounian

  2. The Center of Excellence on Modeling and Control Systems, Ferdowsi University of Mashhad, Mashhad, Iran

    F. Toutounian

Authors
  1. A. Taherian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Toutounian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Toutounian.

Additional information

Publisher’s note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Taherian, A., Toutounian, F. Block GPBi-CG method for solving nonsymmetric linear systems with multiple right-hand sides and its convergence analysis. Numer Algor 88, 1831–1850 (2021). https://doi.org/10.1007/s11075-021-01097-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01097-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation