Information Technology Reference
In-Depth Information
zω
4
+
ω
1
=(2
z
−
2
+
4
z
−
3
+6
z
−
4
+10
z
−
5
+2
z
−
6
,
2
z
−
4
+4
z
−
5
+
z
−
6
,z
−
6
+
z
−
7
,z
−
6
)
t
,
ω
1
Since
θ
(
ω
4
)=
−
θ
(
ω
1
)and
v
(
ω
4
)
≤
v
(
ω
1
), we have
ξ
←
←
ω
4
,
ω
4
←
ξ
and the others are not changed.
r
=3.
I
=
.
Since
θ
(
ω
3
)=
θ
(
ω
1
)+
θ
(
ω
2
)and
v
(
ω
3
)
{
3
,
4
}
z
2
ω
3
+
z
2
ω
1
+
≤
v
(
ω
2
), we have
ξ
=
−
ω
2
=(
z
−
1
+2
z
−
2
+3
z
−
3
+9
z
−
4
,z
−
3
+3
z
−
4
,
z
−
4
+
z
−
5
,z
−
5
)
t
,
ω
2
←
−
ω
3
,
ω
3
←
ξ
and the others are unchanged.
16
z
−
6
,
Since
θ
(
ω
4
)=2
θ
(
ω
1
)and
v
(
ω
4
)=
v
(
ω
1
), we have
ω
4
←
ω
4
−
2
ω
1
=(
−
5
z
−
6
,
z
−
6
z
−
7
,z
−
6
2
z
−
7
)
t
, and the others are not changed.
−
−
−
r
=4.
I
=
.
Since
θ
(
ω
3
)=
θ
(
ω
1
)and
v
(
ω
3
)
<v
(
ω
1
), we have
ω
3
←
{
3
}
zω
1
=(4
z
−
4
ω
3
−
−
9
z
−
5
,z
−
4
3
z
−
5
,
z
−
4
+
z
−
5
z
−
6
,z
−
5
z
−
6
)
t
,
ω
4
←
16
z
−
6
,
5
z
−
6
,z
−
6
−
−
−
−
(
−
−
−
z
−
7
,z
−
6
2
z
−
7
)
t
,
ω
2
←
(
z
−
2
+
z
−
3
+
z
−
4
+2
z
−
5
,
z
−
2
+
z
−
4
+
z
−
5
,z
−
6
,
0)
t
and
−
−
(
z
−
2
+2
z
−
3
+3
z
−
4
+5
z
−
5
+9
z
−
6
,z
−
4
+2
z
−
5
+3
z
−
6
,z
−
7
,z
−
7
)
t
.
At this time they become normal and so
M
6
(
z
)=
ω
1
←
.
z
3
+
z
2
−
−
zz
−
1
z
2
−
zz
−
2
6
Conclusions
In this paper we extend a minimal partial realization algorithm for vector se-
quences to matrix sequences by means of the lattice basis reduction. The main
idea of the algorithm lies in finding the reduction relation of the basis elements
such that their valuations become smaller and smaller till they become reduced.
Therefore different reduction ways lead to different realization algorithms. As
we see, Algorithm 4.2 is similar to the minimal partial realization algorithms
in [5, 18].
In Algorithm 4.1, if the matrix (
θ
(
ω
1
)
... θ
(
ω
p
)) is kept lower-triangular, we
have a special method to solve the equation
θ
(
ω
p
+
i
)=
j
=1
a
j
θ
(
ω
j
), that is,
eliminating the nonzero
j
th component, 1
p
,of
θ
(
ω
p
+
i
) with the corre-
sponding
θ
(
ω
j
) step by step. Similar to the method of deriving Algorithm 4.2
from Algorithm 4.1, we can deduce the algorithm in [13].
Therefore our algorithm provides a greater insight into the minimal partial
realization problem and gives a unified algorithm for this problem.
≤
j
≤
Acknowledgment
The research is supported by the National Natural Science Foundation of China
(Grant No. 60773141 and No. 60503010) and the National 863 Project of China
(Grant No. 2006AA 01Z420). The author also would like to thank the reviewers
for their helpful suggestions and comments.
