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, 55:8 | Cite as

Two-parameter TSCSP method for solving complex symmetric system of linear equations

  • Davod Khojasteh Salkuyeh
  • Tahereh Salimi Siahkolaei
Article

Abstract

We introduce a two-parameter version of the two-step scale-splitting iteration method, called TTSCSP, for solving a broad class of complex symmetric system of linear equations. We present some conditions for the convergence of the method. An upper bound for the spectral radius of the method is presented and optimal parameters which minimize this bound are given. Inexact version of the TTSCSP iteration method (ITTSCSP) is also presented. Some numerical experiments are reported to verify the effectiveness of the TTSCSP iteration method and the numerical results are compared with those of the TSCSP, the SCSP and the PMHSS iteration methods. Numerical comparison of the ITTSCSP method with the inexact version of TSCSP, SCSP and PMHSS are presented. We also compare the numerical results of the BiCGSTAB method in conjunction with the TTSCSP and the ILU preconditioners.

Keywords

Complex linear systems Symmetric positive definite MHSS PMHSS GSOR SCSP TSCSP 

Mathematics Subject Classification

65F10 65F50 65F08 

Notes

Acknowledgements

The work of Davod Khojasteh Salkuyeh is partially supported by University of Guilan. The authors would like to thank Prof. M. Benzi and anonymous referees for their valuable comments and suggestions which greatly improved the quality of the paper.

References

  1. 1.
    Arridge, S.R.: Optical tomography in medical imaging. Inverse Prob. 15, 41–93 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56, 297–317 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bai, Z.-Z., Chen, F., Wang, Z.-Q.: Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices. Numer. Algorithms 62, 655–675 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bai, Z.-Z., Parlett, B.N., Wang, Z.-Q.: On generalized successive overrelaxation methods for augmented linear systems. Numer. Math. 102, 1–38 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bertaccini, D.: Efficient solvers for sequences of complex symmetric linear system. Electron. Trans. Numer. Anal. 18, 49–64 (2004)MathSciNetMATHGoogle Scholar
  12. 12.
    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)CrossRefMATHGoogle Scholar
  13. 13.
    Frommer, A., Lippert, T., Medeke, B., Schilling, K.: Numerical Challenges in Lattice Quantum Chromodynamics. Lecture Notes in Computational Science and Engineering, vol. 15, pp. 1719–1739. Springer, Berlin (2000)CrossRefMATHGoogle Scholar
  14. 14.
    Hezari, D., Salkuyeh, D.K., Edalatpour, V.: A new iterative method for solving a class of complex symmetric system of linear equathions. Numer. Algorithms 73, 927–955 (2016)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Moro, G., Freed, J.H.: Calculation of ESR spectra and related Fokker–Planck forms by the use of the Lanczos algorithm. J. Chem. Phys. 74, 3757–3773 (1981)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Poirier, B.: Effecient preconditioning scheme for block partitioned matrices with structured sparsity. Numer. Linear Algebra Appl. 7, 715–726 (2000)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Press, New York (1995)Google Scholar
  18. 18.
    Salkuyeh, D.K., Hezari, D., Edalatpour, V.: Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations. Int. J. Comput. Math. 92, 802–815 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Salkuyeh, D.K.: Two-step scale-splitting method for solving complex symmetric system of linear equations. arXiv:1705.02468
  20. 20.
    Schmitt, D., Steffen, B., Weiland, T.: 2D and 3D computations of lossy eigenvalue problems. IEEE Trans. Magn. 30, 3578–3581 (1994)CrossRefGoogle Scholar
  21. 21.
    Van der Vorst, H.A.: BiCGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 10, 631–644 (1992)CrossRefMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesUniversity of GuilanRashtIran

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