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4.2 Experimental Results
This section reports the results of the term sharing and the term combination
procedures. For the experimental tests,wehaveusedsomeknownproblems
from the numerical analysis and interval analysis communities. All the problems
were solved by a bisection algorithm with a precision of output boxes of 10
−
10
.
The experiments have been conducted on a Linux/PC Pentium III 933MHz.
The description of test problems follows. Every variable whose domain is not
explicitely mentioned lies in the interval [
−
100
,
100].
1.
Brown's almost linear problem
[11]:
x
i
+
j
=1
x
j
=
n
+1
1
i
n
−
1
j
=1
x
j
=1
In the system, the sum of variables can be shared and represented only once.
2.
Extended Wood Problem
[15] (1
i
n
):
−
200
x
i
(
x
i
+1
−
x
i
)
−
(1
−
x
i
)=0
mod(
i,
4) = 1
200(
x
i
−
x
i−
1
) + 20(
x
i
−
1) + 19
.
8(
x
i
+2
−
1) = 0
mod (
i,
4) = 2
x
i
)
−
180
x
i
(
x
i
+1
−
−
(1
−
x
i
)=0
mod(
i,
4) = 3
x
i−
1
)+20
.
2(
x
i
−
180(
x
i
−
1) + 19
.
8(
x
i−
2
−
1) = 0
mod (
i,
4) = 0
x
i−
1
) can be shared in the first two equations
andinthelasttwoconstraints.
3.
Circuit design problem
[7] (1
x
i
)and(
x
i
−
Terms (
x
i
+1
−
i
4):
x
1
x
3
=
x
2
x
4
(1
−
x
1
x
2
)
x
3
(exp(
x
5
(
a
1
i
−
a
3
i
x
7
−
a
5
i
x
8
))
−
1) =
a
5
i
−
a
4
i
x
2
a
4
i
The symbols
a
ij
are known coecients. In this system, it can be observed
that the term (1
(1
−
x
1
x
2
)
x
4
(exp(
x
6
(
a
1
i
−
a
2
i
−
a
3
i
x
7
+
a
4
i
x
9
))
−
1) =
a
5
i
x
1
−
x
1
x
2
) occurs in all but one constraint.
4.
Product problem
:
−
n
x
j
=
i,
1
i
n
j
=1
,j
=
i
Two consecutive constraints
i
and
i
+ 1 can be divided and simplified into
x
i
+1
/x
i
=
i/
(
i
+1).
5.
Extended product problem
:
n
x
i
+
x
j
=
i,
1
i
n
j
=1
The products from two consecutive constraints
i
and
i
+ 1 can be eliminated,
which corresponds to the generation of the redundant constraint
i
−
x
i
=
(
i
+1)
−
x
i
+1
.
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