Summary: Reliable Computing (2005) 11: 165-190 @ Springer 2005
Enclosing Solutions of Singular Interval Systems
Dedicated to Professor G. Maeß on the occasion of his 65th birthday.
Institutfür Angewandte Mathematik, Universität Karlsruhe, D-76128 Karlsruhe, Germany,
Institutfür Mathematik, Universität Rostock, D-18051 Rostock, Germany,
(Received: 4 May 2004; accepted: 16 August 2004)
Abstract. 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].