508 B The inverse of a multivariate homogeneous polynomial
Example B.2 (Inversion of an univariate homogeneous polynomial of degree n = 3).
Assume the univariate homogeneous polynomial y(x) = a 11 x + a 12 x
2 + a 13 x
3 to be given and find
the inverse univariate homogeneous quadratic polynomial x(y) = b 11 y + b 12 y
2 + b 13 y
3 by the GKS
algorithm.
1st step:
x(y) = b 11 y + b 12 y
2 + b 13 y
3
= b 11 a 11 x +
b 11 a 12 + b 12 a
2
11
x
2 +
b 11 a 13 + 2b 12 a 11 a 12 + b 13 a
3
11
x
3 + β
14 ,
x
2 (y) = b 22 y
2 + b 23 y
3 + β
24
= b 22 a
2
11 x
2 +
2b 12 a 11 a 12 + b 23 a
3
11
x
3 + β
24 ,
x
3 (y) = b 33 y
3 + β
34
= b 33 a
3
11 x
3 + β
34 .
(B.15)
2nd step (forward substitution):
⎡
⎢
⎢
⎣
x
x
2
x
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
x
2
x
3
⎤
⎥
⎥
⎦ +
⎡
⎢
⎢
⎣
β
14
β
24
β
34
⎤
⎥
⎥
⎦ ,
(B.16)
subject to a 22 = a
2
11 , a 23 = 2a 11 a 12 , and a 33 = a
3
11 .
Both the matrices A := A and B := B are upper triangular such that
A B = I 3 ⇔ B =
−1
A .
(B.17)
3rd step (backward substitution):
B =
−1
A =
⎡
⎢
⎢
⎣
a 11 a 12
a 13
0 a
2
11 2a 11 a 12
0 0
a
3
11
⎤
⎥
⎥
⎦
−1
=
⎡
⎢
⎢
⎣
a
−1
11 −a
−3
11 a 12 a
−4
11 (2a
−1
11 a
2
12 − a 13 )
0
a
−2
11
−2a
−4
11 a 12
0
0
a
−3
11
⎤
⎥
⎥
⎦ ⇒
⇒ b 11 = a
−1
11 , b 12 = −a
−3
11 a 12 , b 13 = a
−4
11 (2a
−1
11 a
2
12 − a 13 ) ,
(B.18)
or
b 33 a 33 = b 33 a
3
11 = 1 ⇒ b 33 = a
−1
33 = a
−3
11 , b 22 a 22 = b 22 a
2
11 = 1 ⇒ b 22 = a
−1
22 = a
−2
11 ,
b 22 a 23 + b 23 a 33 = 2a
−1
11 a 12 + b 23 a
3
11 = 0 ⇒ b 23 = −2a
−4
11 a 12 ,
b 11 a 11 = 1 ⇒ b 11 = a
−1
11 , b 11 a 12 + b 12 a 22 = a
−1
11 a 12 + b 12 a
2
11 = 0 ⇒ b 12 = −a
−3
11 a 12 ,
b 11 a 13 + b 12 a 23 + b 13 a 33 = a
−1
11 a 13 − a
−3
11 a 12 a 23 + b 13 a
3
11 = 0 ⇒ b 13 = a
−4
11 (2a
−1
11 a
2
12 − a 13 ) ,
(B.19)
x(y) = a
−1
11 y − a
−3
11 a 12 y
2 + a
−4
11 (2a
−1
11 a
2
12 − a 13 )y
3 .
(B.20)
End of Example.