B-2 Inversion of a bivariate homogeneous polynomial of degree n 515
1st polynomial:
x 1
x 2
= B 11
3
k=1
A 1k
x 1
x 2
[k]
+
+B 12
⎛
⎝
3
k 1 =1
A 1k 1
x 1
x 2
[k 1 ]
⎞
⎠ ⊗
⎛
⎝
3
k 2 =1
A 1k 2
x 1
x 2
[k 2 ]
⎞
⎠ +
+B 13
⎛
⎝
3
k 1 =1
A 1k 1
x 1
x 2
[k 1 ]
⎞
⎠ ⊗
⎛
⎝
3
k 2 =1
A 1k 2
x 1
x 2
[k 2 ]
⎞
⎠ ⊗
⎛
⎝
3
k 3 =1
A 1k 3
x 1
x 2
[k 3 ]
⎞
⎠ =
= B 11 A 11
x 1
x 2
+ (B 11 A 12 + B 12 A 11 ⊗ A 11 )
x 1
x 2
[2]
+
+ [B 11 A 13 + B 12 (A 11 ⊗ A 12 + A 12 ⊗ A 11 ) + B 13 (A 11 ⊗ A 11 ⊗ A 11 )]
x 1
x 2
[3]
+ β
14 .
(B.51)
2nd polynomial:
x 1
x 2
⊗
x 1
x 2
=
x 1
x 2
[2]
= B 22
y 1
y 2
[2]
+ B 23
y 1
y 2
[3]
+ β
24 =
= B 22 (A 11 ⊗ A 11 )
x 1
x 2
[2]
+ [B 22 (A 11 ⊗ A 12 + A 12 ⊗ A 11 ) + B 23 (A 11 ⊗ A 11 ⊗ A 11 )]
x 1
x 2
[3]
+ β
24 .
(B.52)
3rd polynomial:
x 1
x 2
⊗
x 1
x 2
⊗
x 1
x 2
=
x 1
x 2
[3]
= B 33 (A 11 ⊗ A 11 ⊗ A 11 )
x 1
x 2
[3]
+ β
34 .
(B.53)
Forward substitution:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
x 1
x 2
[1]
x 1
x 2
[2]
x 1
x 2
[3]
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎣
B 11 B 12 B 13
0 B 22 B 23
0
0 B 33
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
A 11 A 12 A 13
0 A 22 A 23
0
0 A 33
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
x 1
x 2
[1]
x 1
x 2
[2]
x 1
x 2
[3]
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎣
β
14
β
24
β
34
⎤
⎥
⎥
⎥
⎦
,
(B.54)
subject to
A 22 = A 11 ⊗ A 11 , A 23 = A 11 ⊗ A 12 + A 12 ⊗ A 11 , A 33 = A 11 ⊗ A 11 ⊗ A 11 .
(B.55)
Both the matrices A := A and B := B are upper triangular such that
A B = I 14 ⇔ B =
−1
A .
(B.56)