Numerische Mathematik

, Volume 85, Issue 1, pp 109–127 | Cite as

Semiconvergence of nonnegative splittings for singular matrices

  • Yongzhong Song
Original article


In this paper, we discuss semiconvergence of the matrix splitting methods for solving singular linear systems. The concepts that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting is quasi-regular or quasi-nonnegative. The equivalent conditions for the semiconvergence are proved. Comparison theorem on convergence factors for two different quasi-nonnegative splittings is presented. As an application, the semiconvergence of the power method for solving the Markov chain is derived. The monotone convergence of the quasi-nonnegative splittings is proved. That is, for some initial guess, the iterative sequence generated by the iterative method introduced by a quasi-nonnegative splitting converges towards a solution of the system from below or from above.

Mathematics Subject Classification (1991):65F10 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Yongzhong Song
    • 1
  1. 1.Department of Mathematics, Nanjing Normal University, Nanjing 210097, People's Republic of China; e-mail: CN

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