Reliable Computing 3: 83–93, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Use of Interval Slopes for the Irrational Part of
LUBOMIR V. KOLEV
Dept. of Theoretical Electrotechnics, Faculty of Automatica, Technical University of Soﬁa, 1756
Soﬁa, Bulgaria, e-mail: LKOLEV@vmei.acad.bg
(Received: 2 February 1995; accepted: 21 February 1996)
Abstract.Interval slopes are known to provide sharper enclosures for the range of factorable functions
in comparison to interval derivatives. Presently, only the rational part of the functions is, however,
treated by way of interval slopes while interval derivatives are used for the irrational components.
In this paper, it is suggested to use, whenever appropriate, ﬁrst- or second-order slopes for
the irrational components of the factorable functions also. Theoretical considerations as well as
illustrative examples show that the new approach leads to enclosures for the range that are narrower
in comparison with those obtained by the traditional scheme.
In many applications of Interval Analysis such as solving systems of nonlinear
equations and global analysis it is extremely important to determine, without too
much computational effort, relatively sharp enclosures for the range of a given
→ R over an interval X in R
. If the continuously differentiable
(D)) is a rational function then interval extensions of
relatively small excess can be obtained using interval slopes , , –. Intro-
duced originally for the case of functions in a single variable  slope arithmetic
was extended for the multivariate case in . Later it has been generalized to fac-
torable functions , , – containing irrational components. Presently, only
the rational part of the function is, however, treated by way of ﬁrst- or second-order
 interval slopes, while ﬁrst- or second-order interval derivatives are used for the
In this paper, the approach based on interval slopes is extended to cover the
irrational components of factorable functions. For the case where the irrational
elementary function is either convex or concave an optimal interval extension
providing the range of the ﬁrst-order slopes is suggested. If the function’s ﬁrst
derivative has the same property an optimal formula of similar type is presented for
the second-order interval slopes too.
Theoretical considerations and illustrative examples show that the enclosures
for the range of factorable functions thus obtained are narrower in comparison with
those obtained by the traditional scheme.