On a theorem of Stein-Rosenberg type in interval analysis

In classical numerical analysis the asymptotic convergence factor (R 1-factor) of an iterative processx m+1=Axm+b coincides with the spectral radius of then×n iteration matrixA. Thus the famous Theorem of Stein and Rosenberg can at least be partly reformulated in terms of asymptotic convergence factor. Forn×n interval matricesA with irreducible upper bound and nonnegative lower bound we compare the asymptotic convergence factor (α T ) of the total step method in interval analysis with the factorα S of the corresponding single step method. We derive a result similar to that of the Theorem of Stein and Rosenberg. Furthermore we show thatα S can be less than the spectral radius of the real single step matrix corresponding to the total step matrix |A| where |A| is the absolute value ofA. This answers an old question in interval analysis.