On the asymptotic convergence factor of the total step method in interval computation (Q1089729)

For the total step method \(x^{m+1}=Ax^ m+b\), where A is a real \(n\times n\) interval matrix and b a real interval vector, it is known that the spectral radius \(\rho\) (\(| A|)\) of the absolute value of the matrix A is an upper bound of the asymptotic convergence factor \(\alpha\). Here \(\alpha\) is defined by \(\alpha =\sup (\lim_{m\to \infty}\sup \| q(x^ m,x)\|^{1/m})\) where q means the distance of interval vectors, \(\|, \|\) is any vector norm and x the limes, see \textit{G. Alefeld} and \textit{J. Herzberger} [Einführung in die Intervallrechnung (1974; Zbl 0333.65002)]. In this paper the open question whether this bound is sharp for every A is answered in showing that for a special class of matrices A \(\alpha\) can be less than the spectral radius \(\rho\) (\(| A|)\).