Enclosing Solutions of Singular Interval Systems Iteratively

摘要

Richardson splitting applied to a consistent system of linear equations C_x = b with a singular matrix C yields to an iterative method x~(k+1) = Ax~k + b where A has the eigenvalue one. It is known that each sequence of iterates is convergent to a vector x~* = x~*(x~0) if and only if is semi-convergent. In order to enclose such vectors we consider the corresponding interval iteration [x]~(k+1) = [A][x]~k + [b] with ρ(∣[A]∣) = 1 where ∣[A]∣ denotes the absolute value of the interval matrix [A]. If [A] is irreducible we derive a necessary and sufficient criterion for the existence of a limit [x]~* = [x]~*([x]~0) of each sequence of interval iterates. We describe the shape of [x]~* and give a connection between the convergence of ([x]~k) and the convergence of the powers [A]~k of [A].