Reliable Computing 9: 339–347, 2003.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
Extended Interval Power Function
at GH Wuppertal, Fachbereich 7 (Mathematik), Gaußstraße 20, D–42097
Wuppertal, Germany, e-mail: firstname.lastname@example.org
URGEN WOLFF VON GUDENBERG
ur Informatik 2, Am Hubland, D–97074 W
(Received: 24 July 2002; accepted: 24 March 2003)
Abstract. The containment set as an extension of the range of a function has been introduced in a
series of white papers; see e.g. Walster, G. W.: Closed Interval Systems, Sun Microsystems, 2002,
and Walster, G. W. et al.: Extended Real Intervals and the Topological Closure of Extended Real
Relations, Sun Microsystems, 2002. The containment evaluation provides an exception free evaluation
of functions over an arbitrary range.
In this paper we discuss alternative existing implementations (C++ Interval Arithmetic Program-
ming Reference, Sun Microsystems, 2000, and Hofschuster, W. et al.: The Interval Library ﬁ
2.0, Design, Features and Sample Programs,Universit
at Wuppertal, 2001) of the power function,
introduce a new version, develop containment sets and discuss algorithms for the implementation.
1. Containment Sets
The simplest method to enclose the range of a function
⊆ R → R
compute the interval extension f :
IR → IR
that is obtained by replacing every
occurrence of the variable x by the interval variable x and by replacing every
operator by its interval arithmetic counterpart and every elementary function by an
enclosure of its range.
Usually, we have
(x) | x
However, an interval extension of a function may not be deﬁned, even if the
function is continuous everywhere.
To overcome the difﬁculties with partially deﬁned functions throwing excep-
tions, a second mode of interval arithmetic, the “extended” mode, has been intro-
duced by Hansen and Walster, see , . Here, usually no exceptions are raised,
but the domains of interval functions and ranges of interval results are consistent-
ly extended, so that they include all accumulation points on the boundary of the
Instead of the range we now deﬁne the containment set according to :