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Convergence of Generalized SOR, Jacobi and Gauss–Seidel Methods for Linear Systems

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Abstract

In this paper, we study the convergence of generalized Jacobi and generalized Gauss–Seidel methods for solving linear systems with symmetric positive definite matrix, L-matrix and H-matrix as co-efficient matrix. A generalization of successive overrelaxation (SOR) method for solving linear systems is proposed and convergence of the proposed method is presented for linear systems with strictly diagonally dominant matrices, symmetric positive definite matrices, M-matrices, L-matrices and for H-matrices. Finally, numerical experiments are carried out to establish the advantages of generalized SOR method over generalized Jacobi, generalized Gauss–Seidel, and SOR methods.

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References

  1. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Science. SIAM, Philadelphia (1994)

    Book  Google Scholar 

  2. Salkuyeh, D.K.: Generalized Jacobi and Gauss–Seidel methods for solving linear system of equations. Numer. Math. J. Chin. Univ. (English Ser.) 16(2), 164–170 (2007)

    MathSciNet  Google Scholar 

  3. Varga, R.S.: Matrix Iteraive Analysis. Springer, Berlin (2000)

    Book  Google Scholar 

  4. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)

    Book  Google Scholar 

  5. Young, D.M.: Iterative Solution of Large Linear Systems. Elsevier, Amsterdam (2014)

    Google Scholar 

  6. Hackbusch, W.: Iteraive Solution of Large Sparse Systems of Equations. Springer, Berlin (1994)

    Book  Google Scholar 

  7. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  8. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  9. Hadjidimos, A.: Successive overrelaxation (SOR) and related methods. J. Comput. Appl. Math. 123(1–2), 177–199 (2000)

    Article  MathSciNet  Google Scholar 

  10. Bouni, J., Varga, R.S.: Theorems of Stein–Rosenburg type. Numer. Math. 49, 65–75 (1979)

    Google Scholar 

  11. Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.: Block Triangular and Skew Hermitian Splitting Methods for Positive Definite Linear Systems. SIAM, Philadelphia (2005)

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, NIT Meghalaya, Shillong, India

    Manideepa Saha & Jahnabi Chakravarty

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  1. Manideepa Saha
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  2. Jahnabi Chakravarty
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Correspondence to Manideepa Saha.

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The work was supported by DST-SERB (Grant No. ECR/2017/002116).

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Saha, M., Chakravarty, J. Convergence of Generalized SOR, Jacobi and Gauss–Seidel Methods for Linear Systems. Int. J. Appl. Comput. Math 6, 77 (2020). https://doi.org/10.1007/s40819-020-00830-5

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  • DOI: https://doi.org/10.1007/s40819-020-00830-5

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