Graphics Reference
In-Depth Information
18.7
Conclusions
Interval analysis has many advantages. It can be implemented on a computer in hard-
ware or software. The software only has to ensure that any rounding that takes place
goes outward from the interior of an interval. It can produce robust algorithms. In
fact, it has lead to new results that are not just extensions of the corresponding real
number result. A good example of this is the iterative Newton-Raphson method for
finding the zeros of a function f. If F is an inclusion function for f, then the interval
analysis Newton-Raphson method takes the form
()
FA
F mid A
n
(
-
A
=
mid A
.
n
+
1
n
(
()
)
ยข
n
This form is shown in [Moor66] to produce much better results than the usual
Newton-Raphson method.
Some disadvantages of interval analysis are:
(1) Interval arithmetic computations are slower than floating point operations,
roughly by a factor of three, although there are problems that are solved faster
when implemented using interval arithmetic.
(2) There are no additive or multiplicative inverses for intervals.
(3) We do not have a strict distributive law of multiplication over addition, only
subdistributivity (see Lemma 18.2.3(6)). One consequence of this is that gen-
eric inclusion functions do not give as tight a bound as would be desirable.
18.8
E
XERCISES
Prove any or all of the unproved facts in Section 18.2.
