Reliable Computing 7: 17–28, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
Automatic Computation of a Linear Interval
LUBOMIR V. KOLEV
Dept. of Theoretical Electrotechnics, Faculty of Automatica, Technical University of Soﬁa, 1756
Soﬁa, Bulgaria, e-mail: email@example.com
(Received: 30 April 1999; accepted: 17 October 1999)
Abstract. Recently, an alternative interval approximation F(X) for enclosing a factorable function
(x)inagivenboxX has been suggested. The enclosure is in the form of an afﬁne interval function
+B where only the additive term B is an interval, the coefﬁcients a
being real numbers.
The approximation is applicable to continuously differentiable, continuous and even discontinuous
In this paper, a new algorithm for determining the coefﬁcients a
and the interval B of F(X)is
proposed. It is based on the introduction of a speciﬁc generalized representation of intervals which
permits the computation of the enclosure considered to be fully automated.
Enclosing a real function
(x) in a given box X by an interval function F(X) is one
of the fundamental problems in interval analysis.
Various enclosures have been proposed over the years depending on whether
X is an interval or interval vector, on the one hand, and on the type of interval
function F(X) used, on the other. One of the most popular enclosures in the class
of ﬁrst-order (linear) interval approximations is the well-known mean-value form
(X)(X − z)
(X) is an interval extension of the ﬁrst derivative of
. Several attempts
have been made to improve this enclosure. The monotonocity test form of (1.1)
 provides a narrower extension if some of the components F
(X)ofF(X) do not
contain zero. Another way of reducing the width of F(X) by replacing some X
points in the expressions of F
) is suggested in . More speciﬁcally
In the above extensions, z is typically the midpoint m of X. An additional
narrowing of F(X) is possible ,  if m is replaced by two appropriate points
(lower and upper poles).