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A modified GHSS method for non-Hermitian positive definite linear systems

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Abstract

This paper is concerned with a modified version of the generalization of Hermitian and skew-Hermitian splitting iteration (MGHSS) to solve the non-Hermitian positive definite linear systems. The corresponding inexact MGHSS (IMGHSS) method is developed by employing some Krylov subspace methods as its inner process. Numerical examples are reported to confirm the efficiency of the proposed methods.

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References

  1. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979). Reprinted by SIAM, Philadelphia (1994)

  2. Varga R.S.: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs (1962)

    Google Scholar 

  3. Saad Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston (1996)

    MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai Z.-Z., Golub G.H., Ng M.K.: On successive overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numer. Linear Algebra Appl. 14, 319–335 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bai Z.-Z., Golub G.H., Li C.-K.: Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM J. Sci. Comput. 28, 583–603 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Bai Z.-Z., Golub G.H., Li C.-K.: Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math. Comput. 76, 287–298 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bai Z.-Z.: On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems. Computing 89, 171–197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Benzi M., Bertaccini D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bai Z.-Z.: Splitting iteration methods for non-Hermitian positive definite systems of linear equations. Hokkaido Math. J. 36, 801–814 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Benzi M.: A generalization of the Hermitian and skew-Hermitian splitting iteration. SIAM J. Matrix Anal. Appl. 31, 360–374 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Marchuk G.I.: Methods of Numerical Mathematics. Springer, New York (1984)

    Google Scholar 

  18. Benzi M., Gander M.J., Golub G.H.: Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems. BIT 43, 881–900 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Greif C., Varah J.: Iterative solution of cyclically reduced systems arising from discretization of the three-dimensional convection-diffusion equation. SIAM J. Sci. Comput. 19, 1918–1940 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Greif C., Varah J.: Block stationary methods for nonsymmetric cyclically reduced systems arising from three-dimensional elliptic equation. SIAM J. Matrix Anal. Appl. 20, 1038–1059 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cheung W.M., Ng M.K.: Block-circulant preconditioners for systems arising from discretization of the three-dimensional convection-diffusion equation. J. Comput. Appl. Math. 140, 143–158 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Anyang Normal University, Anyang, 455000, People’s Republic of China

    Cui-Xia Li & Shi-Liang Wu

  2. College of Mathematics, Chengdu University of Information Technology, Chengdu, 610255, People’s Republic of China

    Shi-Liang Wu

Authors
  1. Cui-Xia Li
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  2. Shi-Liang Wu
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Corresponding author

Correspondence to Shi-Liang Wu.

Additional information

This research of this author was supported by NSFC Tianyuan Mathematics Youth Fund (11026040).

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Li, CX., Wu, SL. A modified GHSS method for non-Hermitian positive definite linear systems. Japan J. Indust. Appl. Math. 29, 253–268 (2012). https://doi.org/10.1007/s13160-012-0059-z

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  • DOI: https://doi.org/10.1007/s13160-012-0059-z

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