Convergence conditions for splitting iteration methods for non-Hermitian linear systems

Necessary and sufficient convergence conditions are studied for splitting iteration methods for non-Hermitian system of linear equations when the coefficient matrix is nonsingular. When this theory is specialized to the generalized saddle-point problem, we obtain convergence theorem for a class of modified accelerated overrelaxation iteration methods, which include the Uzawa and the inexact Uzawa methods as special cases. Moreover, we apply this theory to the two-stage iteration methods for non-Hermitian positive definite linear systems, and obtain sufficient conditions for guaranteeing the convergence of these methods.