Nonnegative splitting theory

  • Zbigniew I. Woźnicki


In this paper the nonnegative splitting theory, playing a fundamental role in the convergence analysis of iterative methods for solving large linear equation systems with monotone matrices and representing a broad class of physical and engineering problems, is formulated. As the main result of this theory, it is possible to make the comparison of spectral radii of iteration matrices in particular iterative methods.

Key words

linear equation systems iterative methods monotone matrices eigenvalues eigenvectors regular nonnegative weak nonnegative and weak splittings comparison theorems 


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  1. [1]
    R.S. Varga, Matrix Iterative Analysis. Prentice Hall, Englewood Cliffs, N.J., 1962.Google Scholar
  2. [2]
    Z.I. Woźnicki, Two-sweep iterative methods for solving large linear systems and their application to the numerical solution of multi-group, multi-dimensional neutron diffusion equations. Doctoral, Dissertation, Rep. No. 1447-CYFRONET-PM-A, Inst. Nuclear Res., Swierk-Otwock, Poland, 1973.Google Scholar
  3. [3]
    Z.I. Woźnicki, AGA two-sweep iterative method and their application in critical reactor calculations. Nukleonika,9 (1978), 941–968.Google Scholar
  4. [4]
    Z.I. Woźnicki, AGA two-sweep iterative methods and their application for the solution of linear equation systems. Proc. International Conference on Linear Algebra and Applications., Valencia, Spain, Sept. 28–30, 1987 (published in Linear Algebra Appl.,121 (1989), 702–710.Google Scholar
  5. [5]
    Z.I. Woźnicki, Estimation of the optimum relaxation factors in the partial factorization iterative methods. Proc. International Conference on the Physics of Reactors: Operation, Design and Computations, Marseille, France, April 23–27, 1990, pp. P-IV-173-186. (To appear at beginning 1993 in SIAM J. Matrix Anal. Appl.)Google Scholar
  6. [6]
    J.M. Ortega and W. Rheinboldt, Monotone iterations for nonlinear equations with applications to Gauss-Seidel methods. SIAM J. Numer. Anal.,4 (1967), 171–190.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    G. Csordas and R.S. Varga, Comparison of regular splittings of matrices. Numer. Math.,44 (1984), 23–35.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    G. Alefeld and P. Volkmann, Regular splittings and monotone iteration functions. Numer. Math.,46 (1985), 213–228.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    V.A. Miller and M. Neumann, A note on comparison theorems for nonnegative matrices. Numer. Math.,47 (1985), 427–434.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    L. Elsner, Comparisons of weak regular splittings and multisplitting methods. Numer. Math.,56 (1989), 283–289.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    I. Marek and D.B. Szyld, Comparison theorems for weak splittings of bounded operators. Numer. Math.,58 (1990), 389–397.CrossRefMathSciNetGoogle Scholar
  12. [12]
    Z.I. Woźnicki, HEXAGA-II-120, −60, −30 Two-dimensional multi-group neutron diffusion programmes for a uniform triangular mesh with arbitrary group scattering. Report KfK-2789, 1979.Google Scholar
  13. [13]
    Z.I. Woźnicki, HEXAGA-III-120, −30 Three-dimensional multi-group neutron diffusion programmes for a uniform triangular mesh with arbitrary group scattering. Report KfK-3572, 1983.Google Scholar
  14. [14]
    Z.I. Woźnicki, Two- and three-dimensional benchmark calculations for triangular geometry by means of HEXAGA programmes. Proc. International Meeting on Advances in Nuclear Engineering Compuational Methods, Knoxville, Tennessee, April 9–11, 1985, pp. 147–156.Google Scholar
  15. [15]
    R. Beauwens, Factorization iterative methods, M-operators and H-operators. Numer. Math.,31 (1979), 335–357.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    H.C. Elman and G.H. Golub, Line iterative methods for cyclically reduced discrete convection-diffusion problems. SIAM J. Sci. Statist. Comput.,13 (1992), 339–363.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Z.I. Woźnicki, The graphic representation of the algorithims of the AGA two-sweep iterative method (under preparation).Google Scholar
  18. [18]
    Z.I. Woźnicki, On numerical analysis of conjugate gradient method. Japan J. Indust. Appl. Math.,10 (1993), 487–519.MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    Z.I. Woźnicki, The Sigma-SOR algorithm and the optimal strategy for the utilization of the SOR iterative method. Math. Comp.,62 206 (1994), 619–644.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© JJIAM Publishing Committee 1994

Authors and Affiliations

  • Zbigniew I. Woźnicki
    • 1
  1. 1.Institute of Atomic EnergyOtwock-SwierkPoland

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