On Stein–Rosenberg type theorems for nonnegative and Perron–Frobenius splittings

AbstractThe Stein–Rosenberg theorem is extended and generalized to the class of nonnegative splittings A=M1-N1=M2-N2, as well as to the most generalized class of Perron–Frobenius splittings. Two types of the Stein–Rosenberg theorem are stated and proved for both classes. These theorems allow us to obtain comparison results for the rate of convergence of the associated iterative methods. Specific assumptions are given under which the inequalities of the spectral radii become equalities or strict inequalities. The theoretical results are confirmed by numerical examples