Enclosing Solutions of Singular Interval Systems Iteratively

Richardson splitting applied to a consistent system of linear equations Cx = b with a singular matrix C yieJds to an iterative method ..1+1 = Axk + b where A has the eigenvalue one. It is known that each sequence of iterates is convergent to a vector x* = x*(xO)if and onJy if A is semi-convergent. In order to enclose such vectors we consider the corresponding interval iteration [X]k+1 = [A][x]k + [b] with p(I[AJI) = I where I[A]I denotes the absolute value 01' the interval matrix [A]. If I[A]I is irreducible we derive a necessary and sufficient criterion for the existence 01'a limit [x]* = [x]*([x]o) of each sequence 01' interval iterates. We describe the shape 01'[x]* and give a connection between the convergence of ([x]k) and the convergence 01'the powers [At of [A].