Graphics Reference
In-Depth Information
The inclusion function F is called
isotonic
or
inclusion monotonic
if
()
Õ
()
AB
Õ
implies that
F A
FB
.
It is said to be
convergent
if for each sequence of intervals A
1
, A
2
,..., in I(
X
)
()
=
(
()
)
=
lim
w A
0
implies that
lim
w F A
0
.
i
i
i
Æ•
i
Æ•
Since induced functions are continuous by Theorem 18.2.8, they are certainly con-
vergent, because this only requires continuity at 0. An arbitrary inclusion function
may not be convergent however.
Inclusion functions allow us leeway in specifying accuracy. The induced functions
clearly have the tightest possible bounds. For that reason they are also often called
ideal functions
. It is often interesting to know how far an inclusion function deviates
from the ideal one.
Definition.
Using the notation of the previous definition, define the
excess width
of
the inclusion function F at A ΠI(
R
n
) to be
(
()
)
-
(
()
)
.
wFA
wf A
I
The inclusion function F is said to be of
order k
if
(
)
k
(
()
)
-
(
()
)
=
()
wFA
wf A
OwA
I
for all A ΠI(
R
n
).
Clearly, the higher the order is for an inclusion function for a function f, the tighter
it matches f.
Definition.
Let
X
Õ
R
m
and f :
X
Æ
R
n
. We shall say that f satisfies a
Lipschitz con-
dition
if there exists an inclusion function F for f and a constant c > 0 so that
(
()
)
£
()
Œ
()
wFA
cwA
,
for all A
I
X
.
Let
X
Õ
R
m
and f, g :
X
Æ
R
n
. Let F and G be inclusion functions for f and g,
respectively. Let
*
Œ {+, -, ◊, /}. It is easy to show that F
*
G defined by
(
)(
)
=
() ()
FG A
FA GA
(18.1)
*
*
is an inclusion function for f
*
g. Although theoretically correct, this may not actually be
true on an actual computer because of round-off errors. It can be made to work on com-
puters that support the IEEE floating point standard though by using special rounding
modes called “round-to--•” and “round-to-+•.” See [Snyd92] for more details.
At any rate, equation (18.1) means that once we have inclusion functions on some
primitive functions we can compute inclusion functions for a great many other
functions either by applying arithmetic operators directly or by using recursion. We
