A parameterized splitting iteration method for complex symmetric linear systems

  • Guo-Feng ZhangEmail author
  • Zhong Zheng
Original Paper Area 2


In this paper, we propose a parameterized splitting (PS) iteration method for solving complex symmetric linear systems. The convergence theory of the method is established and the spectral properties of the corresponding iteration matrix are analyzed. The explicit expression for the spectral radius of the iteration matrix is given. In addition, the optimal choice of the iteration parameter is discussed. It is shown that the eigenvalues of the preconditioned matrix are cluster at 1. Numerical experiments illustrate the theoretical results and also examine the numerical effectiveness of the new parameterized splitting iteration method served either as a preconditioner or as a solver.


Complex symmetric linear systems PMHSS iteration method GMRES Spectral properties Preconditioning 

Mathematics Subject Classification (2000)

65F08 65F10 65F50 65N22 


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  1. 1.
    Arridge SR: Optical tomography in medical imaging. Inverse Probl. 15, 41–93 (1999)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Axelsson O, Kucherov A: Real valued iteration methods for solving complex symmetric linear syetems. Numer. Linear Algebra Appl. 7, 197–218 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bai Z-Z: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput 75, 791–815 (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bai Z-Z, Benzi M, Chen F: Modified HSS iteration methods for complex symmetric linear systems. Computing 87, 93–111 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bai Z-Z, Benzi M, Chen F: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algor. 56, 297–317 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bai Z-Z, Golub GH, Ng MK: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    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(1):343–369 (2013)Google Scholar
  8. 8.
    Benzi M, Golub GH, Liesen J: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Benzi M, Bertaccini D: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Bertaccini D: Efficient solvers for sequences of complex symmetric linear systems. Electr. Trans. Numer. Anal. 18, 49–64 (2004)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Betts, J.T.: Practical Methods For Optimal Control Using Nonlinear Programming. SIAM, Philadelphia, (2001)Google 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)CrossRefzbMATHGoogle Scholar
  13. 13.
    Frommer, A., Lippert, T., Medeke, B., Schilling, K.: Numerical challenges in lattice quantum chromodynamics, Lecture notes in computational science and engineering. Springer, Heidelberg, 15 (2000)Google Scholar
  14. 14.
    Poirier B: Efficient preconditioning scheme for block partitioned matrices with structured sparsity. Numer. Linear Algebra Appl. 7, 715–726 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    van Dijk, W., Toyama, F.M.: Accurate numerical solutions of the time-dependent Schrodinger equation. Phys. Rev. E, 75, 036707 (2007)Google Scholar
  16. 16.
    Young, D.M.: Iterative Solution of Large Linear Systems. Academic Press, New York, (1971)Google Scholar
  17. 17.
    Zhou Y-Y, Zhang G-F: A generalization of parameterized inexact Uzawa method for generalized saddle point problems. Appl. Math. Comput. 215, 599–607 (2009)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China

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