Generalized Stein-Rosenberg theorems for the regular splittings and convergence of some generalized iterative methods

AbstractWe study the convergence properties of the AOR and GAOR iterative methods for the solution of the linear system Ax = b, and we give more general forms of the Stein-Rosenberg theorem under the assumptions that (1) L, U ⩾ 0, in A = I − B = I − L − U, are not necessarily strictly lower and strictly upper triangular matrices respectively, and (2) ϱ(L) < 1. We derive new results regarding the spectral radii of the GAOR (or AOR) and the GJ (or Jacobi) iteration matrices, and give, in the (γ,ω) plane, sufficient conditions (domains) for the GAOR (and AOR) method to converge