Skip to main content
Log in

Two lopsided TSCSP (LTSCSP) iteration methods for solution of complex symmetric positive definite linear systems

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In this paper we make two new lopsided methods (LTSCSP1 and LTSCSP2) based on the two-scale-splitting (TSCSP) method and show that the convergence speed of TSCSP iteration method can be increased under some conditions without adding any additional parameter. The convergence analysis of the new methods in detail is given. Then we will obtain the quasi-optimal parameter to minimize the spectral radius of iteration matrix for the new methods. The inexact version of these methods is derived. To illustrate the effectiveness of the proposed framework, several numerical examples are given.

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

Access this article

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

Similar content being viewed by others

References

  1. Anderson BDO, Agathoklis P, Jury EI, Mansour M (1986) Stability and the matrix Lyapunov equation for discrete 2-dimensional systems. IEEE Trans. Circ Syst 33:261–267

    Article  MathSciNet  Google Scholar 

  2. Axelsson O, Farouq S, Neytcheva M (2016) Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Poisson and convection-diffusion control. Numer Algor 73:631–663

    Article  Google Scholar 

  3. Axelsson O, Salkuyeh DK (2019) A new version of a preconditioning method for certain two-by-two block matrices with square blocks. BIT Numer Math 59:321–342

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Bai Z-Z, Golub GH, Ng MK (2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl 24:603–626

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Bai Z-Z, Golub GH, Ng MK (2008) On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl 428:413–440

    Article  MathSciNet  Google Scholar 

  12. Bendali A (1984) 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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  15. Christiansen SH (2004) Discrete Fredholm properties convergence estimates for the electric field integral equation. Math Comput 73:143–167

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  17. Dehghan M, Shirilord A (2020) Accelerated double-step scale splitting iteration method for solving a class of complex symmetric linear systems. Numer Algorithms 83:281–304

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Dehghan M, Mohammadi-Arani R (2017) Generalized product-type methods based on Bi-conjugate gradient(GPBiCG) for solving shifted linear systems. Comput Appl Math 36(4):1591–1606

    MathSciNet  MATH  Google Scholar 

  20. Dehghan M, Hajarian M (2014) Modified AOR iterative methods to solve linear systems. J Vib Control 20(5):661–669

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Howle VE, Vavasis SA (2005) An iterative method for solving complex-symmetric systems arising in electrical power modeling. SIAM J Matrix Anal Appl 26:1150–1178

    Article  MathSciNet  Google Scholar 

  25. Frommer A, Lippert T, Medeke B, Schilling K (2000) Numerical Challenges in Lattice Quantum Chromodynamics. Lecture Notes in Computational Science and Engineering, vol 15. Springer, Berlin

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Salkuyeh, D.K.: Two-step scale-splitting method for solving complex symmetric system of linear equations, arXiv:1705.02468

  30. Salkuyeh DK, Siahkolaei TS (2018) Two-parameter TSCSP method for solving complex symmetric system of linear equations. Calcolo 55:1–22

    Article  MathSciNet  Google Scholar 

  31. Sommerfeld A (1949) Partial differential equations. Academic Press, New York

    MATH  Google Scholar 

  32. Wang T, Zheng Q, Lu L (2017) A new iteration method for a class of complex symmetric linear systems. J Comput Appl Math 325:188–197

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

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. Two lopsided TSCSP (LTSCSP) iteration methods for solution of complex symmetric positive definite linear systems. Engineering with Computers 38, 1867–1881 (2022). https://doi.org/10.1007/s00366-020-01126-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-020-01126-4

Keywords

AMS subject classification

Navigation