Graphics Reference
In-Depth Information
Figure 18.1.
Component intervals.
the components are sufficiently separated. The dashed rectangles in Figure 18.1 are
some sample rectangles returned by the algorithm.
Finally, [Snyd92] points out that interval Newton methods can be exploited to
improve the bounds on the set of solutions to the constraint problem. Furthermore,
such methods provide very robust methods for solving for zeros of sets of equations.
18.5
An Application: Implicit Curve Approximations
The goal of this section is to clarify Algorithm 18.4.1 by applying it to a concrete
problem. Before describing the general implicit curve algorithm we work through an
example.
18.5.1
Example.
Let
(
)
=-
2
fxy
,
y x
.
The problem is to find an approximation to the part of the implicit curve in the plane
defined by
(
)
= 0
fxy
,
that lies in the unit square A = [0,1] ¥ [0,1]. See Figure 18.2(a).
Solution.
The generic inclusion function F for f is
[
]
2
[
]
¥
[
]
)
=
[
]
-
[
]
2
2
(
Fab
,
cd
,
cd
,
ab
,
=-
c b d a
,
-
.
As solution acceptance set constraint function we choose
Œ
()
R
2
()
=
(
()
<
)
H B
w B
d
,
for
B
I
and some fixed
d
>
0
.
Figures 18.2(b)-(e) show several iterations of Algorithm 18.4.1. The shaded rectangles
are the rectangles that the algorithm generates and which are subdivided in the next
iteration. For example, as we move from Figure 18.2(d) to Figure 18.2(e) we lose
rectangle [1/2,3/4] ¥ [3/4,1] because the rectangle
(
[
]
¥
[
]
)
=
[
]
F1234
,
341
,
316716
,
