Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Lopsided PMHSS iteration method for a class of complex symmetric linear systems

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

Abstract

Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, we introduce a lopsided PMHSS (LPMHSS) iteration method for solving a broad class of complex symmetric linear systems. The convergence properties of the LPMHSS method are analyzed, which show that, under a loose restriction on parameter α, the iterative sequence produced by LPMHSS method is convergent to the unique solution of the linear system for any initial guess. Furthermore, we derive an upper bound for the spectral radius of the LPMHSS iteration matrix, and the quasi-optimal parameter α ⋆ which minimizes the above upper bound is also obtained. Both theoretical and numerical results indicate that the LPMHSS method outperforms the PMHSS method when the real part of the coefficient matrix is dominant.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /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.

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(4), 197–218 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bai, Z.-Z.: Optimal parameters in the HSS-like methods for saddle-point problems. Numer. Linear Algebra Appl 16(6), 447–479 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bai, Z.-Z.: On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems. Computing 89, 171–197 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87(3–4), 93–111 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algoritm. 56(2), 297–317 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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 applications to distributed control problems. IMA J. Numer. Anal. 33, 343–369 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. 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(3), 844–863 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. 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(3), 603–626 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bai, Z.-Z., Golub, G.H., Ng, M.K.: On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numer. Linear Algebra Appl. 14(4), 319–335 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bai, Z.-Z., Golub, G.H., Ng, M.K.: On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl. 428(2–3), 413–440 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. 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), 1–32 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Benzi, M.: A generalization of the Hermitian and skew-Hermitian splitting iteration. SIAM J. Matrix Anal. Appl. 31(2), 360–374 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28(3), 598–618 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bertaccini, D.: Efficient preconditioning for sequences of parametric complex symmetric linear systems. Electron. Tran. Numer. Anal. 18, 49–64 (2004)

    MATH  MathSciNet  Google Scholar 

  15. Day, D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real formulations. SIAM J. Sci. Comput 23(2), 480–498 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Freund, R.W.: Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J. Sci. Stat. Comput. 13(1), 425–448 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Guo, X.-X., Wang, S.: Modified HSS iteration methods for a class of non-Hermitian positive-definite linear systems. Appl. Math. Comput. 218(20), 10,122–10,128 (2012)

    Article  Google Scholar 

  18. Li, L., Huang, T.-Z., Liu, X.-P.: Modified Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems. Numer. Linear Algebra Appl. 14(3), 217–235 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  20. Salkuyeh, D.K., Behnejad, S.: Modified Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems [Numer. Linear Algebra Appl. 14 (2007) 217–235]. Numer. Linear Algebra Appl. 19(5), 885–890 (2012)

  21. Yang, A.-L., An, J., Wu, Y.-J.: A generalized preconditioned HSS method for non-Hermitian positive definite linear systems. Appl. Math. Comput. 216(6), 1715–1722 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, Peoples Republic of China

    Xu Li, Ai-Li Yang & Yu-Jiang Wu

Authors
  1. Xu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ai-Li Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu-Jiang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ai-Li Yang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, X., Yang, AL. & Wu, YJ. Lopsided PMHSS iteration method for a class of complex symmetric linear systems. Numer Algor 66, 555–568 (2014). https://doi.org/10.1007/s11075-013-9748-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-013-9748-1

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation