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Some comparison theorems for nonnegative splittings of matrices

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In a recent paper the author has proposed some theorems on the comparison of the asymptotic rates of convergence of two nonnegative splittings. They extended the corresponding result of Miller and Neumann and implied the earlier theorems of Varga, Beauwens, Csordas and Varga. An open question by Miller and Neumann, which additional and appropriate conditions should be imposed to obtain strict inequality, was also answered. This article continues to investigate the comparison theorems for nonnegative splittings. The new results extend and imply the known theorems by the author, Miller and Neumann.

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References

  1. Beauwens, R. (1979): Factorization iteration methods, M-operators and H-operators. Numer. Math.31, 335–357

    Google Scholar 

  2. Beauwens, R., Bouzid, M. Ben (1987): On sparse block factorization iterative methods. SIAM J. Numer. Anal.24, 1066–1076

    Google Scholar 

  3. Berman, A., Plemmons, R.J. (1979): Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York

    Google Scholar 

  4. Csordas, G., Varga, R.S. (1984): Comparisons of regular splittings of matrices. Numer. Math.44, 23–35

    Google Scholar 

  5. Elsner, L. (1989): Comparisons of weak regular splittings and multisplitting methods. Numer. Math.56, 283–289

    Google Scholar 

  6. Miller, V.A., Neumann, M. (1985): A note on comparison theorems for nonnegative matrices. Numer. Math.47, 427–434

    Google Scholar 

  7. Rheinholdt, W.C., Vandergraft, J.S. (1973): A simple approach to the Perron-Frobenius Theory for positive operators on general partially-ordered finite-dimensional linear spaces. Math. Comput.27, 139–145

    Google Scholar 

  8. Song, Y. (1989): The monotone convergence of splitting of matrices. Math. Appl.2, 31–36

    Google Scholar 

  9. Song, Y. (1991): Comparisons of nonnegative splittings of matrices. Lin. Alg. Appl.154/156, 433–455

    Google Scholar 

  10. Varga, R.S. (1960): Factorization and normalized iterative methods. In: R.E. Langer, ed., Boundary problems in differential equations, pp. 121–142. The University of Wisconsin Press, Madison

    Google Scholar 

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

    Google Scholar 

  12. Woźnicki, Z. (1973): Two-sweep iterative methods for solving large linear systems and their application to the numerical solution of multi-group multi-dimensional neutron diffusion equation. Doctoral Dissertation, Institute of Nuclear Research, Swicrk k/Otwocda, Poland

    Google Scholar 

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

    Google Scholar 

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The Project Supported by the Natural Science Foundation of Jiangsu Province Education Commission

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Song, Y. Some comparison theorems for nonnegative splittings of matrices. Numer. Math. 65, 245–252 (1993). https://doi.org/10.1007/BF01385750

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  • DOI: https://doi.org/10.1007/BF01385750

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