Skip to main content
SpringerLink
Log in

Accelerated double-step scale splitting iteration method for solving a class of complex symmetric linear systems

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

Abstract

In this paper, we introduce and study an accelerated double-step scale splitting (ADSS) iteration method for solving complex linear systems. The convergence of the ADSS iteration method is determined under suitable conditions. Also each iteration of ADSS method requires the solution of two linear systems that their coefficient matrices are real symmetric positive definite. We analytically prove that the ADSS iteration method is faster than the DSS iteration method. Moreover, to increase the convergence rate of this method, we minimize the upper bound of the spectral radius of iteration matrix. Finally, some test problems will be given and simulation results will be reported to support 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

Similar content being viewed by others

References

  1. Anderson, B.D.O., Agathoklis, P., Jury, E.I., Mansour, M.: Stability and the matrix Lyapunov equation for discrete 2–dimensional systems. IEEE Trans. Circuits Systems 33, 261–267 (1986)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Arridge, S.R.: Optical tomography in medical imaging. Inverse Probl. 15, 41–93 (1999)

    Article  MathSciNet  Google Scholar 

  4. Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithms 66, 811–841 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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  Google Scholar 

  7. 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  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, 603–626 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  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, 413–440 (2008)

    Article  MathSciNet  Google Scholar 

  11. Bai, Z.-Z., Parlett, B.N., Wang, Z.-Q.: On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102, 1–38 (2005)

    MathSciNet  MATH  Google Scholar 

  12. Bendali, A.: Numerical analysis of the exterior boundary value problem for the time–harmonic Maxwell equations by a boundary finite element method. Math. Comput. 43, 29–68 (1984)

    MathSciNet  MATH  Google Scholar 

  13. Bertaccini, D.: Efficient solvers for sequences of complex symmetric linear systems. Electr. Trans. Numer. Anal. 18, 49–64 (2004)

    MATH  Google Scholar 

  14. Christiansen, S.H.: Discrete Fredholm properties convergence estimates for the electric field integral equation. Math. Comput. 73, 143–167 (2004)

    Article  MathSciNet  Google Scholar 

  15. Clemens, M., Weiland, T.: Iterative methods for the solution of very large complex symmetric linear systems of equations in electrodynamics, Technische Hochschule Darmstadt (2002)

  16. Dehghan, M., Dehghani–Madiseh, M., Hajarian, M.: A generalized preconditioned MHSS method for a class of complex symmetric linear systems. Math. Model. Anal. 18, 561–576 (2013)

    Article  MathSciNet  Google Scholar 

  17. Demmel, J.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)

    Book  Google Scholar 

  18. Feriani, A., Perotti, F., Simoncini, V.: Iterative system solvers for the frequency analysis of linear mechanical systems. Comput. Methods Appl. Mech. Eng. 190, 1719–1739 (2000)

    Article  Google Scholar 

  19. Gambolati, G., Pini, G.: Complex solution to nonideal contaminant transport through porous media. J. Comput. Phys. 145, 538–554 (1998)

    Article  Google Scholar 

  20. Howle, V.E., Vavasis, S.A.: An iterative method for solving complex–symmetric systems arising in electrical power modeling. SIAM J. Matrix Anal. Appl. 26, 1150–1178 (2005)

    Article  MathSciNet  Google Scholar 

  21. Poirier, B.: Efficient preconditioning scheme for block partitioned matrices with structured sparsity. Numer. Linear Algebra Appl. 7, 715–726 (2000)

    Article  MathSciNet  Google Scholar 

  22. Pourbagher, M., Salkuyeh, D.K.: On the solution of a class of complex symmetric linear systems. Appl. Math. Lett. 76, 14–20 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Zheng, Z., Huang, F.-L., Peng, Y.-C.: Double–step scale splitting iteration method for a class of complex symmetric linear systems. Appl. Math. Lett. 73, 91–97 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors wish to thank three anonymous reviewers for careful reading and valuable comments and suggestions which improved the quality of the paper.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Mehdi Dehghan & Akbar Shirilord

Authors
  1. Mehdi Dehghan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akbar Shirilord
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehdi Dehghan.

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

Dehghan, M., Shirilord, A. Accelerated double-step scale splitting iteration method for solving a class of complex symmetric linear systems. Numer Algor 83, 281–304 (2020). https://doi.org/10.1007/s11075-019-00682-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00682-1

Keywords

Search

Navigation