Advertisement

Numerical Algorithms

, Volume 71, Issue 3, pp 655–671 | Cite as

On SSOR iteration method for a class of block two-by-two linear systems

  • Zhao-Zheng Liang
  • Guo-Feng Zhang
Original Paper

Abstract

In this paper, the optimal iteration parameters of the symmetric successive overrelaxation (SSOR) method for a class of block two-by-two linear systems are obtained, which result in optimal convergence factor. An accelerated variant of the SSOR (ASSOR) method is presented, which significantly improves the convergence rate of the SSOR method. Furthermore, a more practical way to choose iteration parameters for the ASSOR method has also been proposed. Numerical experiments demonstrate the efficiency of the SSOR and ASSOR methods for solving a class of block two-by-two linear systems with the optimal parameters.

Keywords

Block two-by-two matrices Complex symmetric linear systems Symmetric SOR method Convergence Optimal parameter 

Mathematics Subject Classification (2010)

65F10 65F50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithm 66(4), 811–841 (2014)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75(254), 791–815 (2006)CrossRefMATHGoogle Scholar
  4. 4.
    Bai, Z.-Z.: Rotated block triangular preconditioning based on PMHSS. Sci. China Math. 56, 2523–2538 (2013)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. doi: 10.1007/s10665-013-9670-5 (2015)
  6. 6.
    Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric liinear systems. Numer. Algorithm 56, 297–317 (2011)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric liinear systems. Computing 87, 93–111 (2014)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: Preconditioned MHSS iteration method for a class of block two-by-two linear systems with applications to distributed control problems. IMA J. Numer. Anal. 33, 343–369 (2013)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    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. Algorithm 62, 655–675 (2013)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    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(3), 603–626 (2003)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Bai, Z.-Z., Parlett, B.N., Wang, Z.-Q.: On generalized successive overrelaxation methods for augmented linear systems. Numer. Math. 102(1), 1–38 (2005)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Bertaccini., D.: Efficient solvers for sequences of complex symmetric linear systems. Electr. Trans. Numer. Anal. 18, 49–64 (2004)MathSciNetMATHGoogle Scholar
  15. 15.
    Betts, J.T.: Practical methods foroptimal controlusing nonlinear programming. SIAM, Philadelphia (2001)Google Scholar
  16. 16.
    Bunse, G.M., Stöver, R.: On a conjugate gradient-type method for solving complex symmetric linear systems. Linear Algebra Appl. 287(1), 105–123 (1999)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Day, D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real formulations. SIAM J. Sci. Comput. 23(2), 480–498 (2001)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Dehghan, M., Dehghani, M. M., Hajarian, M.: A generalized preconditioned MHSS method for a class of complex symmetric linear systems. J. Math. Model. Anal. 18(4), 561–576 (2013)CrossRefMATHGoogle Scholar
  19. 19.
    Feriani, A., Perotti, F., Simoncini, V.: Iterative system solvers for the frequency analysis of linear mechanical systems. Methods Appl. Mech. Eng. 190, 1719–1739 (2000)CrossRefMATHGoogle Scholar
  20. 20.
    Frommer, A., Lippert, T., Medeke, B., Schilling, K.: Numerical challenges in lattice quantum chromodynamics. Lecture notes in computational science and engineering, vol. 15. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  21. 21.
    Guo, X.-X., Wang, S.: Modified HSS iteration method for a class of non-symmetric positive definite linear systems. Appl. Math. Comput. 218, 10122–10128 (2012)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Golub, G.H., Wu, X., Yuan, J.-Y.: SOR-like methods for augmented systems. BIT Numer. Math. 41(1), 71–85 (2001)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Hezari, D., Edalatpour, V., Salkuyeh, D.K.: Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations. Numer. Linear Algebera Appl. doi: 10.1002/nla.1987 (2015)
  24. 24.
    Lang, C., Ren, Z.-R.: Inexact rotated block triangular preconditioners for a class of block two-by-two matrices. J. Eng. Math. doi: 10.1007//s13160-014-0140-x (2013)
  25. 25.
    Lass, O., Vallejos, M., Borzì, A., Douglas, C.C.: Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems. Computing 84, 27–48 (2009)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadephia (2003)CrossRefMATHGoogle Scholar
  27. 27.
    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(4), 802–815 (2015)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Sommerfeld, A.: Partial Differential Equations. Academic Press, New York (1949)MATHGoogle Scholar
  29. 29.
    Young, D.M.: Iterative Solution for Large Linear Systems. Academic Press, New York (1971)Google Scholar
  30. 30.
    Wu, S.-L.: Several variants of the Hermitian and skew-Hermitian splitting method for a class of complex symmetric linear systems. Numer. Linear Algebera Appl. 22 (2), 338–356 (2015)CrossRefGoogle Scholar
  31. 31.
    Zhang, G.-F., Zheng, Z.: A parameterized splitting iteration methods for complex symmetric linear systems. Jpn. J. Indust. Appl. Math. 31, 265–278 (2014)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

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

Personalised recommendations