Skip to main content
SpringerLink
Log in

Preconditioned Richardson iteration for augmented linear systems

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

For solving a class of augmented linear systems, we propose a new efficient iteration method, which is called preconditioned Richardson iteration (PR). Under suitable restrictions on the iteration parameters, we prove that the iterative sequences converge to the unique solution of the augmented linear system. Moreover, the optimal iteration parameters and the corresponding optimal convergence factor are discussed in detail. Numerical results show that the PR iteration method has an advantage over several other iteration methods by computing with the preconditioned GMRES methods from the point of view of iteration steps and CPU times.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arrow, K., Hurwicz, L., Uzawa, H.: Studies in Nonlinear Programming. Stanford University Press, Stanford (1958)

    MATH  Google Scholar 

  2. Bai, Z.Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75, 791–815 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Bai, Z.Z.: Optimal parameters in the HSS-like methods for saddle-point problems. Numer. Linear Algebra Appl. 16, 447–479 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Bai, Z.Z., Benzi, M.: Regularized HSS iteration methods for saddle-point linear systems. BIT Numer. Math. 57, 287–311 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Bai, Z.Z., Golub, G.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27, 1–23 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Bai, Z.Z., Golub, G., Lu, L.Z., Yin, J.F.: Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput. 26, 844–863 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Bai, Z.Z., Golub, G., Ng, M.: On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl. 428, 413–440 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Bai, Z.Z., Golub, G., Pan, J.Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1–32 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Bai, Z.Z., Li, G.Q.: Restrictively preconditioned conjugate gradient methods for systems of linear equations. IMA J. Numer. Anal. 23, 561–580 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Bai, Z.Z., Li, G.Q., Ng, M.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Bai, Z.Z., Parlett, B., Wang, Z.Q.: Ongeneralized successive overrelaxationmethods for augmented linear systems. Numer. Math. 102, 1–38 (2005)

    MathSciNet  Google Scholar 

  12. Bai, Z.Z., Wang, Z.Q.: Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems. J. Comput. Appl. Math. 187, 202–226 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Bai, Z.Z., Wang, Z.Q.: On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl. 428, 2900–2932 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Benzi, M., Golub, G.: Apreconditioner for generalized saddle point problems. SIAM J. Matrix Anal. Appl. 26, 20–41 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Bramble, J., Pasciak, J., Vassilev, A.: Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34, 1072–1092 (1997)

    MathSciNet  MATH  Google Scholar 

  16. Cao, Y., Du, J., Niu, Q.: Shift-splitting preconditioners for saddle point problems. J. Comput. Appl. Math. 272, 239–250 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Cao, Y., Ren, Z.R., Shi, Q.: A simplified HSS preconditioner for generalized saddle point problems. BIT Numer. Math. 56, 423–439 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Chao, Z., Zhang, N.M., Lu, Y.Z.: Optimal parameters of the generalized symmetric SOR method for augmented systems. J. Comput. Appl. Math. 266, 52–60 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Darvishi, M., Hessari, P.: Symmetric SOR method for augmented systems. Appl. Math. Comput. 183, 409–415 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Darvishi, M., Hessari, P.: A modified symmetric successive overrelaxation method for augmented systems. Comput. Math. Appl. 61, 3128–3135 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Elman, H., Golub, G.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31, 1645–1661 (1994)

    MathSciNet  MATH  Google Scholar 

  22. Elman, H., Silvester, D.: Fast nonsymmetric iterations and preconditioning for NavierC-Stokes equations. SIAM J. Sci. Comput. 17, 33–46 (1996)

    MathSciNet  MATH  Google Scholar 

  23. Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press, New York (2005)

    MATH  Google Scholar 

  24. Fischer, B., Ramage, A., Silvester, D., Wathen, A.: Minimum residual methods for augmented systems. BIT Numer. Math. 38, 527–543 (1998)

    MathSciNet  MATH  Google Scholar 

  25. Golub, G., Wu, X., Yuan, J.Y.: SOR-Like methods for augmented systems. BIT Numer. Math. 41, 71–85 (2001)

    MathSciNet  MATH  Google Scholar 

  26. 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)

    MathSciNet  MATH  Google Scholar 

  27. Huang, Z.G., Wang, L.G., Xu, Z., Cui, J.J.: Improved PPHSS iterative methods for solving nonsingular and singular saddle point problems. Comput. Math. Appl. 72, 92–109 (2016)

    MathSciNet  Google Scholar 

  28. Huang, Z.G., Wang, L.G., Xu, Z., Cui, J.J.: The generalized modified shift-splitting preconditioners for nonsymmetric saddle point problems. Appl. Math. Comput. 299, 95–118 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Li, C.J., Li, B.J., Evans, D.: A generalized successive overrelaxation method for the least squares problems. BIT Numer. Math. 38, 347–356 (1998)

    MathSciNet  MATH  Google Scholar 

  30. Li, C.J., Li, Z., Evans, D., Zhang, T.: A note on an SOR-like method for augmented systems. IMA J. Numer. Anal. 23, 581–592 (2003)

    MathSciNet  MATH  Google Scholar 

  31. Li, J., Zhang, N.M.: A triple-parameter modified SSOR method for solving singular saddle point problems. BIT Numer. Math. 56, 501–521 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Liang, Z.Z., Zhang, G.F.: PU-STS Method for non-Hermitian saddle-point problems. Appl. Math. Lett. 46, 1–6 (2015)

    MathSciNet  MATH  Google Scholar 

  33. Liang, Z.Z., Zhang, G.F.: Convergence behavior of generalized parameterized Uzawa method for singular saddle-point problems. J. Comput. Appl. Math. 311, 293–305 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Liu Q.H., Liu A.J.: Block SOR methods for the solution of indefinite least squares problems. Calcolo 51, 367–379 (2014)

    MathSciNet  Google Scholar 

  35. Ma, C.F., Zheng, Q.Q.: The corrected Uzawa method for solving saddle point problems. Numer. Linear Algebra Appl. 22, 717–730 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Njeru, P., Guo, X.P.: Accelerated SOR-like method for augmented linear systems. BIT Numer. Math. 56, 557–571 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Salkuyeh, D., Hezari, D., Edalatpour, V.: Generalized SOR iterative method for a class of complex symmetric linear system of equations. Int. J. Comp. Math. 92, 802–815 (2015)

    MATH  Google Scholar 

  38. Santos, C., Sival, B., Yuan, J.Y.: Block SOR method for rank deficient least squares problems. Comput. Appl. Math. 100, 1–9 (1998)

    MathSciNet  Google Scholar 

  39. Wang, X., Li, W.W., Mao, L.Z.: On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation AX + XB = C. Comput. Math. Appl. 66, 2352–2361 (2013)

    MathSciNet  MATH  Google Scholar 

  40. Wang, X., Li, Y., Dai, L.: On Hermitian and skew-Hermitian splitting iteration methods for the linear matrix equation AXB = C. Comput. Math. Appl. 65, 657–664 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Wright, S.: Stability of augmented systems factorizations in interior-point methods. SIAM Matrix Anal. Appl. 18, 191–222 (1997)

    MathSciNet  MATH  Google Scholar 

  42. Xiao, X.Y., Wang, X.: A new single-step iteration method for solving complex symmetric linear systems. Numer. Algorithms 78, 643–660 (2018)

    MathSciNet  MATH  Google Scholar 

  43. Yin, J.F., Bai, Z.Z.: The restrictively preconditioned conjugate gradient methods on normal residual for block two-by-two linear systems. J. Comput. Math. 26, 240–249 (2008)

    MathSciNet  MATH  Google Scholar 

  44. Young, D.: Iterative Solutions of Large Linear Systems. Academic Press, New York (1971)

    Google Scholar 

  45. Zhang C.H., Wang X., Tang X.B.: Generalized AOR method for solving a class of generalized saddle point problems. J. Comput. Appl. Math. 350, 69–79 (2019)

    MathSciNet  Google Scholar 

  46. Zhang, G.F., Lu, Q.H.: On generalized symmetric SOR method for augmented systems. J. Comput. Appl. Math. 219, 51–58 (2008)

    MathSciNet  MATH  Google Scholar 

  47. Zhang, J.J., Shang, J.J.: A class of uzawa-SOR methods for saddle point problems. Appl. Math. Comput. 216, 2163–2168 (2010)

    MathSciNet  MATH  Google Scholar 

  48. Zhou, R., Wang, X., Tang, X.B.: Preconditioned positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equations AX + XB = C. East Asian. J. Appl. Math. 7, 55–69 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Sciences, Nanchang University, Nanchang, 330031, China

    X. Y. Xiao, X. Wang & H. W. Yin

  2. Numerical Simulation and High-Performance Computing Laboratory, School of Sciences, Nanchang University, Nanchang, 330031, China

    X. Y. Xiao, X. Wang & H. W. Yin

Authors
  1. X. Y. Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. X. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. W. Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X. Y. Xiao.

Additional information

This work is supported by NNSF with Nos. 11461046 and 61563033; NSF of Jiangxi Province with Nos. 20181ACB20001, and 20161ACB21005.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiao, X.Y., Wang, X. & Yin, H.W. Preconditioned Richardson iteration for augmented linear systems. Numer Algor 82, 843–867 (2019). https://doi.org/10.1007/s11075-018-0629-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-018-0629-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation