Sharpening Interval Computations
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(Received: 8 February 2005; accepted: 15 May 2005)
Abstract. We consider ways to use monotonicity to reduce the excess width (due to dependence) of
computed intervals. Use of monotonicity generally involves evaluation of a derivative. We show how
monotonicity can often be used without evaluating derivatives. As examples, we show how Gaussian
elimination and evaluation of slopes can be sharpened. A variable amount of extra computing is
required to obtain the sharper results.
Suppose we evaluate a function
(X)ofaninterval argument X =[x
resulting interval bounds the range of
over X.Generally, however, the bounds are
not sharp because of dependence (see ). In this paper, we consider methods for
using monotonicity to reduce the effect of dependence.
If a function is monotonic over a given interval, the exact range over the interval
can be obtained by evaluating the function at the endpoints of the interval. We
exploit this fact in two different ways.
Oneway is as follows. Suppose a given variable occurs several times in the
expression of a given function; and suppose we replace some of the occurrences
by intervals which bound the variable. If the function is monotonic as a function
of the remaining occurrences of the variable, the computed value of the function
can often be sharpened. We give a theoretical discussion of this idea in Section 4.
In Section 6, we show that the computation of slopes can be sharpened by this
To describe the other way that we use monotonicity, suppose the expression for
agiven function involves more than one occurrence of a subfunction. If the given
function is a monotonic function of the subfunction, this fact can be used to sharpen
evaluation of the given function. This is true irrespective of whether or not the
subfunction is itself monotonic. In Section 7, we show how Gaussian elimination
can be sharpened using this idea. Numerical examples are given in Section 8. Other
sections discuss subsidiary topics.
Thenew procedures provide alternative methods for evaluating a function. They
usually sharpen results; but in some cases are worse than the results using “normal”
evaluation of a function. Consequently, our methods should be used in addition to
“normal” evaluation rather than replacing it. A ﬁnal result should be obtained as