Skip to main content
SpringerLink
Log in

An improved block splitting preconditioner for complex symmetric indefinite linear systems

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

Abstract

In this paper, an improved block splitting preconditioner for a class of complex symmetric indefinite linear systems is proposed. By adopting two iteration parameters and the relaxation technique, the new preconditioner not only remains the same computational cost with the block preconditioners but also is much closer to the original coefficient matrix. The theoretical analysis shows that the corresponding iteration method is convergent under suitable conditions and the preconditioned matrix can have well-clustered eigenvalues around (0,1) with a reasonable choice of the relaxation parameters. An estimate concerning the dimension of the Krylov subspace for the preconditioned matrix is also obtained. Finally, some numerical experiments are presented to illustrate the effectiveness of the presented preconditioner.

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

Similar content being viewed by others

References

  1. Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, Z.-Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.-Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1–32 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.: Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM. J. Sci. Comput. 26, 844–863 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Bai, Z.-Z., Golub, G.H.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA. J. Numer. Anal. 27, 1–23 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bai, Z.-Z.: Several splittings for non-Hermitian linear systems. Sci. China Ser. A: Math. 51, 1339–1348 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algor. 56, 297–317 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bai, Z.-Z.: Block preconditioners for elliptic PDE-constrained optimization problems. Computing. 91, 379–395 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with application to distributed control problems. IMA J. Numer. Anal. 33, 343–369 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bai, Z.-Z.: Rotated block triangular preconditioning based on PMHSS. Sci. China Math. 56, 2523–2538 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93, 41–60 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA. J. Numer. Anal. 28, 598–618 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan, H.-T., Zhu, X.-Y.: A generalized relaxed positive-definite and skew-Hermitian splitting preconditioner for non-Hermitian saddle point problems. Appl. Math. Comput. 58, 36–48 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Fan, H.-T., Zhu, X.-Y.: A modified relaxed splitting preconditioner for generalized saddle point problems from the incompressible Navier-Stokes equations. Appl. Math. Lett. 55, 18–26 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Golub, G.H., Van Loan, C.F.: Matrix computations, 3rd edn. The Johns Hopkins University Press , (1996)

  17. Golub, G.H., Greif, C.: On solving block-structured indefinite linear systems. SIAM J. Sci. Comput. 24, 2076–2092 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lang, C., Ren, Z.-R.: Inexact rotated block triangular preconditioners for a class of block two-by-two matrices. J. Eng. Math. 93, 87–98 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, X., Yang, A.-L., Wu, Y.-J.: Lopsided PMHSS iteration method for a class of complex symmetric linear systems. Numer. Algor. 66, 555–568 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pan, J.-Y., Ng, M.K., Bai, Z.-Z.: New preconditioners for saddle point problems. Appl. Math. Comput. 72, 762–771 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Pennacchio, M., Simoncini, V.: Fast structured AMG preconditioning for the bidomain model in electrocardiology. SIAM J. Sci. Comput. 33, 721–745 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  24. Wu, S.-L.: Several variants of the Hermitian and skew-Hermitian splittingmethod for a class of complex symmetric linear systems. Numer. Linear Algebra Appl. 22, 338–356 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, J.-L., Gu, C.-Q., Zhang, K.: A relaxed positive-definite and skew-Hermitian splitting preconditioner for saddle point problems. Appl. Math. Comput. 249, 468–479 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Zhang, J.-L., Gu, C.-Q.: A variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems. BIT Numer. Math. 56, 587–604 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, J.-H., Dai, H.: A new block preconditioner for complex symmetric indefinite linear systems. Numer. Algor. doi:10.1007/s11075-016-0175-y

  28. Zhang, J.-H., Dai, H.: A new splitting preconditioner for the iterative solution of complex symmetric indefinite linear systems. Appl. Math. Lett. 49, 100–106 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zheng, Q.-Q, Ma, C.-F.: Accelerated PMHSS iteration methods for complex symmetric linear systems. Numer. Algor. doi:10.1007/s11075-016-0105-z

Download references

Author information

Authors and Affiliations

  1. School of Fundamental Studies, Shanghai University of Engineering Science, Shanghai, 201620, People’s Republic of China

    Ju-Li Zhang

  2. Department of Mathematics, Zhejiang A&F University, Zhejiang, 311300, People’s Republic of China

    Ju-Li Zhang

  3. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu, 730000, People’s Republic of China

    Hong-Tao Fan

  4. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Chuan-Qing Gu

Authors
  1. Ju-Li Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hong-Tao Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chuan-Qing Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hong-Tao Fan.

Additional information

The work is partially supported by National Natural Science Foundation (No. 11371243), (No. 11571004), (No. 11471211), Foundation of Zhe-jiang Educational Committee (Y201431769).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, JL., Fan, HT. & Gu, CQ. An improved block splitting preconditioner for complex symmetric indefinite linear systems. Numer Algor 77, 451–478 (2018). https://doi.org/10.1007/s11075-017-0323-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0323-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation