Advertisement

Numerical Algorithms

, Volume 80, Issue 2, pp 337–354 | Cite as

Efficient parameterized rotated shift-splitting preconditioner for a class of complex symmetric linear systems

  • Cheng-Liang Li
  • Chang-Feng MaEmail author
Original Paper

Abstract

By utilizing the equivalent real block two-by-two linear systems and the shift-splitting techniques, we establish an efficient parameterized rotated shift-splitting (PRSS) preconditioner for solving a class of complex symmetric linear systems. The proposed preconditioner is extracted from a stationary iteration method which is unconditionally convergent. Moreover, some spectral properties of the corresponding preconditioned matrix are studied in detail. Finally, numerical results are presented to show the feasibility and effectiveness of the proposed preconditioner.

Keywords

Complex symmetric linear systems Parameterized rotated shift-splitting preconditioner Spectral properties 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their detailed comments which greatly improve the presentation. This research is supported by National Science Foundation of China (41725017, 41590864), National Basic Research Program of China under grant number 2014CB845906. It is also partially supported by the CAS/CAFEA international partnership Program for creative research teams (No. KZZD-EW-TZ-19 and KZZD-EW-TZ-15), Strategic Priority Research Program of the Chinese Academy of Sciences (XDB18010202) and Fujian Natural Science Foundation (2016J01005).

References

  1. 1.
    Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arridge, S.R.: Optical tomography in medical imaging. Inverse Prob. 15, 41–93 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertaccini, D.: Efficient solvers for sequences of complex symmetric linear systems. Electron. Trans. Numer. Anal. 18, 49–64 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bertaccini, D.: Efficient preconditioning for sequences of parametric complex symmetric linear systems. Electron. Tran. Numer. Anal. 18, 49–64 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benzi, M., Simoncini, V.: On the eigenvalues of a class of saddle point matrices. Numer. Math. 103, 173–196 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75, 791–815 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56, 297–317 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bai, Z.-Z.: Rotated block triangular preconditioning based on PMHSS. Sci. China Math. 56, 2523–2538 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93, 41–60 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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 64, 655–675 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bai, Z.-Z.: Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91, 379–395 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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, 1–32 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bai, Z.-Z., Yin, J.-F., Su, Y.-F.: A shift-splitting preconditioner for non-Hermitian positive definite matrices. J. Comput. Math. 24, 539–552 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Bai, Z.-Z.: Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl. Math. Comput. 109, 273–285 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Bai, Z.-Z.: Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math. 283, 71–78 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chen, C.-R., Ma, C.-F.: A generalized shift-splitting preconditioner for saddle point problems. Appl. Math. Lett. 43, 49–55 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen, C.-R., Ma, C.-F.: A generalized shift-splitting preconditioner for singular saddle point problems. Appl. Math. Comput. 269, 947–955 (2015)MathSciNetGoogle Scholar
  22. 22.
    Chen, C.-R., Ma, C.-F.: AOR-Uzawa iterative method for a class of complex symmetric linear system of equations. Comput. Math. Appl. 72, 2462–2472 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cao, Y., Du, J., Niu, Q.: Shift-splitting preconditioners for saddle point problems. J. Comput. Appl. Math. 272, 239–250 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Cao, Y., Miao, S.-X.: On semi-convergence of the generalized shift-splitting iteration method for singular nonsymmetric saddle point problems. Comput. Math. Appl. 71, 1503–1511 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    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
  26. 26.
    Frommer, A., Lippert, T., Medeke, B., Schilling, K.: Numerical challenges in lattice quantum chromodynamics. Lect. Notes Comput. Sci. Eng. 15, 66–83 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Freund, R.W.: Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J. Sci. Stat. Comput. 13, 425–448 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hezari, D., Edalatpour, V., Salkuyeh, D.K.: Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations. Numer. Linear Algebra Appl. 22, 761–776 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lang, C., Ren, Z.-R.: Inexact rotated block triangular preconditioners for a class of block two-by-two matrices. J. Eng. Math. 93, 87–98 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ling, S.-T., Liu, Q.-B.: New local generalized shift-splitting preconditioners for saddle point problems. Appl. Math. Comput. 302, 58–67 (2017)MathSciNetGoogle Scholar
  31. 31.
    Li, X., Yang, A.-L., Wu, Y.-J.: Lopsided PMHSS iteration method for a class of complex symmetric linear systems. Numer. Algorithms 66, 555–568 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Yan, H.-Y., Huang, Y.-M.: Splitting-based block preconditioning methods for block two-by-two matrices of real square blocks. Appl. Math. Comput. 243, 825–837 (2014)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Zhang, F.-Z.: Matrix Theory. Springer, New York (2011)CrossRefGoogle Scholar
  36. 36.
    Zeng, M.-L., Ma, C.-F.: A parameterized SHSS iteration method for a class of complex symmetric system of linear equations. Comput. Math. Appl. 71, 2124–2131 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Zhang, J.-H., Dai, H.: Inexact splitting-based block preconditioners for block two-by-two linear systems. Appl. Math. Lett. 60, 89–95 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zeng, M.-L., Zhang, G.-F.: Parameterized rotated block preconditioning techniques for block two-by-two systems with application to complex linear systems. Comput. Math. Appl. 70, 2946–2957 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zheng, Q.-Q., Lu, L.-Z.: A shift-splitting preconditioner for a class of block two-by-two linear systems. Appl. Math. Lett. 66, 54–60 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and Informatics & FJKLMAAFujian Normal UniversityFuzhouPeople’s Republic of China

Personalised recommendations