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Reliable Computing (2005) 11: 165-190 @ Springer 2005 Enclosing Solutions of Singular Interval Systems
 

Summary: Reliable Computing (2005) 11: 165-190 @ Springer 2005
Enclosing Solutions of Singular Interval Systems
Iteratively
Dedicated to Professor G. Maeß on the occasion of his 65th birthday.
GÖTZ ALEFELD
Institutfür Angewandte Mathematik, Universität Karlsruhe, D-76128 Karlsruhe, Germany,
e-mail: goetz.alefeld@math.uni-karlsruhe.de
and
GÜNTER MAYER
Institutfür Mathematik, Universität Rostock, D-18051 Rostock, Germany,
e-mail: guenter.mayer@uni-rostock.de
(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].

  

Source: Alefeld, Götz - Institut für Angewandte und Numerische Mathematik & Fakultät für Mathematik, Universität Karlsruhe

 

Collections: Mathematics