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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold

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Volume 24, Issue 1


A note on the parallel GSAOR method for block diagonally dominant matrices

Xuezhong Wang / Cuixia Li
Published Online: 2016-03-18 | DOI: https://doi.org/10.1515/jnma-2014-1001


Recently, Liu et al. [Q. B. Liu, G. L. Chen, J.Huang, On the parallel GSAOR method for block diagonallydominant matrices, Appl. Math. Comput. 215(2009), 707–715] study the convergence of the parallel multisplitting generalized SAOR iterative methods based on the generalized AOR iterative method for solving a linear system whose coefficient matrix is a block diagonally dominant matrix or a generalized block diagonal dominant matrix. In this paper, we extend the domain of convergence from 1≤ωi< 2/(1+μ1(PMQ)) and 1≤ωi < 2/(1+μ2(PMQ)) to 0 < ωi < 2/(1 + μ1(PMQ)) and 0 < ωi < 2/(1 + μ2(PMQ)) for the parallel multisplitting generalized SAOR iterative methods.

Keywords: convergence; multisplitting; generalized SAOR method; block diagonally dominant matrices; generalized block diagonally dominant matrices

MSC 2010: 65F10; 65F50


  • [1]

    Q. B. Liu, G. L. Chen, and J. Huang, On the parallel GSAOR method for block diagonally dominant matrices, Appl. Math. Comput., 215(2009), 707–715.Web of ScienceGoogle Scholar

  • [2]

    K. R. James, Convergence of matrix iterations subject to diagonal dominance, SIAM J. Numer. Anal., 10(1986), 117–132.Google Scholar

  • [3]

    A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994.Google Scholar

  • [4]

    D. P. O’Leary and R. E. White, Multi-splittings of matrices and parallel solution of linear systems, SIAM J. Alg. Disc. Math., 6(1985), 630–640.Google Scholar

  • [5]

    Z. Z. Bai, Parallel matrix multisplitting block relaxation iteration methods, Math. Numer. Sinica, 17(1995), 238–252.Google Scholar

  • [6]

    Z. Z. Bai, A class of asynchronous parallel multisplitting blockwise relaxation methods, Parallel Computing, 25(1999), 681–701.Google Scholar

  • [7]

    M. Neumann and R. J. Plemmons, Convergence of parallel multisplitting iterative methods for M-matrices, Lin. Alg. Appl., 88-89(1987), 559–573.Google Scholar

  • [8]

    L. T. Zhang, T. Z. Huang, T. X. Gu, and X. L. Guo, Convergence of relaxed multisplitting USAOR methods for H-matrices linear systems, Appl. Math. Comput., 202(2008), 121–132.Google Scholar

  • [9]

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

  • [10]

    A. Frommer and G.Mayer, Convergence of relaxed parallel multisplitting methods, Lin. Alg. Appl., 119(1989), 141–152.Google Scholar

  • [11]

    R. E. White, Multisplitting of a symmetric positive deRnite matrix, SIAM J. Matrix Anal. Appl., 11(1990), 69–82.Google Scholar

  • [12]

    R. E. White, Multisplitting with di_erent weighting schemes, SIAM J. Matrix Anal. Appl., 10(1989), 481–493.Google Scholar

  • [13]

    W. Li, Comparison results for parallel multisplitting methods with applications to AOR methods, Lin. Alg. Appl., 206(2008), 738–747.Google Scholar

  • [14]

    D. R.Wang, On the convergence of the parallel multisplitting AOR algorithm, Lin. Alg. Appl., 154-156(1991), 473–486.Google Scholar

  • [15]

    D. G. Feingold and R. S. Varga, Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem, Paci_c J. Math., 12(1962), 1241–1249.Google Scholar

  • [16]

    B. Polman, Incomplete blockwise factorizations of (block) H-matrices, Lin. Alg. Appl., 90(1987), 119–132.Google Scholar

  • [17]

    R. Bru, V. Migallon, J. Penadés, and D. B. Szyld, Parallel synchronous and asynchronous two-stage multisplitting methods, Elec. Tran. Num. Anal., 3(1995), 24–38.Google Scholar

  • [18]

    D. B. Szyld, Synchronous and asynchronous two-stage multisplitting methods, in: Proc. 5th SIAM Conference on Applied Linear Algebra, (Ed.J. G. Lewis), SIAM, Philadelphia, 1994.Google Scholar

  • [19]

    J. Mos and D. B. Szyld, Nonstationary Parallel Relaxed Multisplitting Methods, Lin. Alg. Appl., 241-243(1996), 733–747.Google Scholar

  • [20]

    Y. Z. Song, On the convergence of GAOR methods, Math. Numer. Sinica, 4(1989), 405–412 (in Chinese).Google Scholar

About the article

Received: 2014-01-15

Accepted: 2014-05-28

Published Online: 2016-03-18

Published in Print: 2016-03-01

Citation Information: Journal of Numerical Mathematics, Volume 24, Issue 1, Pages 35–44, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnma-2014-1001.

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© 2016 by Walter de Gruyter Berlin/Boston.Get Permission

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