Computational and Applied Mathematics

, Volume 36, Issue 2, pp 877–883 | Cite as

On the preconditioned AOR iterative method for Z-matrices

  • Davod Khojasteh Salkuyeh
  • Mohsen Hasani
  • Fatemeh Panjeh Ali Beik


In this paper, considering a general class of preconditioners \(P(\alpha )\), we study the convergence properties of the preconditioned AOR (PAOR) iterative methods for solving linear system of equations. It is shown that the spectral radius of the iteration matrix of the PAOR method has a monotonically decreasing property when the value of \(\alpha \) increases.


Linear system of equations Preconditioner AOR iterative method Z-matrix 

Mathematics Subject Classification

65F10 65F50 



The authors would like to express their heartfelt gratitude to the anonymous referees for their valuable recommendations and useful comments which have improved the quality of the paper. The work of the first author is partially supported by University of Guilan.


  1. Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences, Academic Press, New York, 1979. SIAM, Philadelphia, PAGoogle Scholar
  2. Evans DJ, Martins MM, Trigo ME (2001) The AOR iterative method for new preconditioned linear systems. J Comput Appl Math 132:461–466MathSciNetCrossRefMATHGoogle Scholar
  3. Gunawardena AD, Jain SK, Snyder L (1991) Modified iterative methods for consistent linear systems. Linear Algebra Appl 154–156:123–143Google Scholar
  4. Hadjidimos A (1978) Accelerated overrelaxation method. Math Comput 32:149–157MathSciNetCrossRefMATHGoogle Scholar
  5. Huang TZ, Cheng GH, Evans DJ, Cheng XY (2005) AOR type iterations for solving preconditioned linear systems. Int J Comput Math 82:969–976MathSciNetCrossRefMATHGoogle Scholar
  6. Li W (2005) A note on the preconditioned Gauss-Seidel (GS) method for linear systems. J Comput Appl Math 182:81–90MathSciNetCrossRefMATHGoogle Scholar
  7. Li W (2003) The convergence of the modified Gauss-Seidel methods for consistent linear systems. J Comput Appl Math 154:97–105MathSciNetCrossRefMATHGoogle Scholar
  8. Li W, Sun W (2000) Modified Gauss-Seidel type methods and Jacobi type methods for Z-matrices. Linear Algebra Appl 317:227–240MathSciNetCrossRefMATHGoogle Scholar
  9. Saad Y (1995) Iterative methods for sparse linear systems. PWS Press, New YorkGoogle Scholar
  10. Sun LY (2005) A comparison theorem for the SOR iterative method. J Comput Appl Math 181:336–341MathSciNetCrossRefMATHGoogle Scholar
  11. Varga RS (1962) Matrix iterative analysis. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  12. Wang L, Song YZ (2009) Preconditioned AOR iterative method for M-matrices. J Comput Appl Math 226:114–124MathSciNetCrossRefMATHGoogle Scholar
  13. Woznicki ZI (2001) Basic comparison theorems for weak and weaker matrix splitting. Electron J Linear Algebra 8:53–59MathSciNetCrossRefMATHGoogle Scholar
  14. Woznicki ZI (1994) Nonnegative splitting theory. Jpn J Ind Appl Math 11:289–342MathSciNetCrossRefMATHGoogle Scholar
  15. Wu M, Wang L, Song Y (2007) Preconditioned AOR iterative method for linear systems. Appl Numer Math 57:672–685MathSciNetCrossRefMATHGoogle Scholar
  16. Zheng Y, Huang TZ (2005) Modified iterative method for nonnegative matrices and M-matrices linear systems. Comput Math Appl 50:1587–1602MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015

Authors and Affiliations

  • Davod Khojasteh Salkuyeh
    • 1
  • Mohsen Hasani
    • 2
  • Fatemeh Panjeh Ali Beik
    • 3
  1. 1.Faculty of Mathematical SciencesUniversity of GuilanRashtIran
  2. 2.Faculty of Science, Department of MathematicsIslamic Azad UniversityShahroodIran
  3. 3.Department of MathematicsVali-e-Asr University of RafsanjanRafsanjanIran

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