On Stein-Rosenberg type theorems for nonnegative and Perron-Frobenius splittings (Q947663)

The author discusses the iterative solution of linear systems by extending and generalizing the Stein-Rosenberg theorem. \textit{P. Stein} and \textit{R. L. Rosenberg} stated and proved an important theorem which compares the spectral radius of the Jacobi and Gauss-Seidel iterative methods [see J. Lond. Math. Soc. 23, 111--118 (1948; Zbl 0036.36501)]. Since 1948 the theorem of Stein and Rosenberg was generalized in many different ways. In this paper Stein-Rosenberg type theorems for a class of nonnegative splittings of the matrix of the linear system are discussed. Also, the class of Perron-Frobenius splittings are considered. Two types of Stein-Rosenberg theorems for each class of splittings are stated and proved. A comparison of these theorems to characterize which theorem is stronger than the other is presented. Characteristic examples to confirm the theoretical results are presented.