Enclosing solutions of complex linear systems of equations iteratively

We consider the interval iteration [ x ] k +1 = [ A ][ x ] k + [ b ] in different interval arithmetics with the aim to enclose solutions of x = Ax + b in the case that A and b are only known to be contained in some given intervals. We give necessary and sufficient criteria for the convergence of the interval iteration for every initial interval vector [ x ] 0 to some [ x ]* = [ x ]*([ x ] 0 ) with respect to the considered interval arithmetic. Such a limit is a solution of the interval system [ x ] = [ A ][ x ] + [ b ]. If we compare the interval arithmetics with respect to the behavior of [ x ] k +1 = [ A ][ x ] k + [ b ] we come to the conclusion, that the special choice of the arithmetic has a sensitive influence on the convergence of the sequence. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)