A note on comparison theorems for nonnegative matrices (Q761767)

In a recent paper, \textit{G. Csordas} and \textit{R. S. Varga} [Comparisons of regular splittings and matrices, ibid. 44, 23-25 (1984)] unified and extended earlier theorems of \textit{R. S. Varga} [Boundary Probl. in diff. Equations, Proc. Sympos., Madison, April 20-22, 1959, 121-142 (1960; Zbl 0100.125)] and \textit{Z. Woźnicki} [Two-sweep iterative methods for solving large linear systems and their application to the numerical solution of multi-group multi-dimensional neutron diffusion equation, Diss., Inst. Nucl. Research, Swicrk/Otwocka, Poland (1973)] on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings. The main purpose of this note is to show a connection between the Csordas-Varga paper and a paper by \textit{R. Beauwens} [Numer. Math. 31, 335-357 (1979; Zbl 0431.65012)], in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular. Monotonic norms already used in Beauwens play an important role in our work here.