A generalized modified SOR-like method for the singular saddle point problems

  • Hong-Tao Fan
  • Xin-Yun Zhu
  • Bing ZhengEmail author
Original Research


Recently, Guo et al. proposed a modified SOR-like (MSOR-like) iteration method for solving the nonsingular saddle point problem. In this paper, we further prove the semi-convergence of this method when it is applied to solve the singular saddle point problems under suitable conditions on the involved iteration parameters. Moreover, the optimal iteration parameters and the corresponding optimal semi-convergence factor for the MSOR-like method are determined. In addition, numerical experiments are used to show the feasibility and effectiveness of the MSOR-like method for solving singular saddle point problems, arising from the incompressible flow problems.


Singular saddle point problems Modified SOR-like method Semi-convergence Semi-convergence factor Optimal iteration parameters 

Mathematics Subject Classification

65F10 65F50 



This work is supported by the National Natural Science Foundation of China (Nos.11571004, 11171371.)


  1. 1.
    Bai, Z.Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75, 791–815 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Bai, Z.Z.: On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems. Computing 89, 171–197 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadephia, PA (1994)CrossRefzbMATHGoogle Scholar
  5. 5.
    Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Philadelphia, PA (2001)zbMATHGoogle Scholar
  6. 6.
    Chao, Z., Zhang, N.M.: A generalized preconditioned HSS method for singular saddle point problems. Numer. Algor. 66, 203–221 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chao, Z., Chen, G.L.: Semi-convergence analysis of the Uzawa-SOR methods for singular saddle point problems. Appl. Math. Lett. 35, 52–57 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Y., Zhang, N.M.: A note on the generalization of parameterized inexact Uzawa method for singular saddle point problems. Appl. Math. Comput. 235, 318–322 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Performance and analysis of saddle point preconditioners for the discrete steady-state Navier–Stokes equations. Numer. Math. 90, 665–688 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31, 1645–1661 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible fluid dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2005). xiv+400 ppzbMATHGoogle Scholar
  12. 12.
    Elman, H.C.: Preconditioners for saddle point problems arising in computational fluid dynamics. 19th Dundee Biennial Conference on Numerical Analysis (2001). Applied Numerical Mathematics, vol. 43, pp. 75–89. (2002)Google Scholar
  13. 13.
    Fan, H.T., Zheng, B.: A preconditioned GLHSS iteration method for non-Hermitian singular saddle point problems. Comput. Math. Appl. 67, 614–626 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guo, P., Li, C.X., Wu, S.L.: A modified SOR-like method for the augmented systems. J. Comput. Appl. Math. 274, 58–69 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Li, W., Liu, Y.P., Peng, X.F.: The generalized HSS method for solving singular linear systems. J. Comput. Appl. Math. 236, 2338–2353 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Perugia, I., Simoncini, V.: Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Preconditioning techniques for large sparse matrix problems in industrial applications, vol. 7, pp. 585–616. Numerical Linear Algebra, Minneapolis (1999)Google Scholar
  17. 17.
    Wang, L., Bai, Z.Z.: Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts. BIT Numer. Math. 44, 363–386 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Yang, A.L., Dou, Y., Wu, Y.J., Li, X.: On generalized parameterized inexact methods for singular saddle-point problems. Numer. Algor. 69, 579–593 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zhang, N.M., Lu, T.T., Wei, Y.M.: Semi-convergence analysis of Uzawa methods for singular saddle point problems. J. Comput. Appl. Math. 255, 334–345 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhang, N.M., Wei, Y.M.: On the convergence of general stationary iterative methods for range-Hermitian singular linear systems. Numer. Linear Algebra Appl. 17, 139–154 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zheng, B., Bai, Z.Z., Yang, X.: On semi-convergence of parameterized Uzawa methods for singular saddle point problems. Linear Algebra Appl. 431, 808–817 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouChina
  2. 2.Department of MathematicsUniversity of Texas of the Permian BasinOdessaUSA

Personalised recommendations