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⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
x
x
2
x
3
b
11
b
12
b
13
0
b
22
b
23
00
b
33
a
11
a
12
a
13
0
a
22
a
23
00
a
33
x
x
2
x
3
β
14
β
24
β
34
⎣
⎦
=
⎣
⎦
⎣
⎦
⎣
⎦
+
⎣
⎦
,
(B.16)
subject to
a
22
=
a
11
,a
23
=2
a
11
a
12
,
and
a
33
=
a
11
.
Both the matrices A :=
A
and B :=
B
are upper triangular such that
−
1
B
=I
3
⇔
A
.
B
=
(B.17)
A
3rd step (backward substitution):
⎡
⎤
⎡
⎤
a
−
1
a
−
3
11
a
12
a
−
4
11
(2
a
−
1
11
a
12
−
a
11
a
12
a
13
0
a
11
2
a
11
a
12
00
a
11
−
a
13
)
11
⎣
⎦
=
⎣
⎦
⇒
−
1
a
−
2
−
2
a
−
4
0
11
a
12
B
=
=
(B.18)
11
A
a
−
3
0
0
11
b
11
=
a
−
1
a
−
3
11
a
12
,b
13
=
a
−
4
11
(2
a
−
1
11
a
12
−
⇒
11
,b
12
=
−
a
13
)
,
or
b
33
a
33
=
b
33
a
11
=1
b
33
=
a
−
1
33
=
a
−
3
11
,b
22
a
22
=
b
22
a
11
=1
⇒
⇒
b
22
=
a
−
1
22
=
a
−
2
11
,b
22
a
23
+
b
23
a
33
=2
a
−
1
11
a
12
+
b
23
a
11
=0
⇒ b
23
2
a
−
4
=
−
11
a
12
,
(B.19)
b
11
=
a
−
1
11
,b
11
a
12
+
b
12
a
22
=
a
−
1
11
a
12
+
b
12
a
11
=0
b
11
a
11
=1
⇒
⇒
b
12
a
−
3
11
a
12
,b
11
a
13
+
b
12
a
23
+
b
13
a
33
=
a
−
1
a
−
3
11
a
12
a
23
+
b
13
a
11
=0
=
−
11
a
13
−
⇒
b
13
=
a
−
4
11
(2
a
−
1
11
a
12
− a
13
)
,
x
(
y
)=
a
−
1
a
−
3
11
a
12
y
2
+
a
−
4
11
(2
a
−
1
11
a
12
−
a
13
)
y
3
.
11
y
−
(B.20)
End of Example.
B-2 Inversion of a Bivariate Homogeneous Polynomial of Degree
n
}
of Box
B.2
, to be given and find the inverse bivariate homogeneous polynomial of degree
n
,namely
x
(
y
)or
Assume the bivariate homogeneous polynomial of degree
n
,namely
y
(
x
)or
{
y
1
(
x
1
,x
2
)
,y
2
(
x
1
,x
2
)
{
x
1
(
y
1
,y
2
)
,x
2
(
y
1
,y
2
)
}
, i.e. from the set of coecients
{
A
11
,
A
12
, ...,
A
1
n−
1
,
A
1
n
}
,by
the algorithm that is outlined in Box
B.2
, find the set of coecients
{
B
11
,
B
12
, ...,
B
1
n−
1
,
B
1
n
}
.
Box B.2 (Algorithm for the construction of an inverse bivariate homogeneous polynomial of
degree
n
).
y
(
x
):=
y
1
=
y
2
=A
11
x
1
+A
12
x
1
x
1
x
2
+
+A
1
n−
1
x
1
x
1
x
2
⊗
···
⊗···⊗
n
x
2
x
2
x
2
−
1times
+A
1
n
x
1
x
1
x
2
=
⊗···⊗
(B.21)
n
times
x
2
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