B-2 Inversion of a bivariate homogeneous polynomial of degree n 509
B-2 Inversion of a bivariate homogeneous polynomial of degree n
Assume the bivariate homogeneous polynomial of degree n, namely y(x) or {y 1 (x 1 , x 2 ), y 2 (x 1 , x 2 )} of
Box B.2, to be given and find the inverse bivariate homogeneous polynomial of degree n, namely x(y)
or {x 1 (y 1 , y 2 ), x 2 (y 1 , y 2 )}, i. e. from the set of coefficients {A 11 , A 12 , . . . , A 1n−1 , A 1n }, by the algorithm
that is outlined in Box B.2, find the set of coefficients {B 11 , B 12 , . . . , B 1n−1 , B 1n }.
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
x 2
#
+ A 12
"
x 1
x 2
#
⊗
"
x 1
x 2
#
+ · · · + A 1n−1
"
x 1
x 2
#
⊗ · · · ⊗
| {z }
n − 1 times
"
x 1
x 2
#
+ A 1n
"
x 1
x 2
#
⊗ · · · ⊗
| {z }
n times
"
x 1
x 2
#
=
=
n
X
k=1
A 1k
"
x 1
x 2
# [k]
,
(B.21)
x(y) :=
"
x 1
x 2
#
=
= B 11
"
y 1
y 2
#
+ B 12
"
y 1
y 2
#
⊗
"
y 1
y 2
#
+ · · · + B 1n−1
"
y 1
y 2
#
⊗ · · · ⊗
| {z }
n − 1 times
"
y 1
y 2
#
+ B 1n
"
y 1
y 2
#
⊗ · · · ⊗
| {z }
n times
"
y 1
y 2
#
=
=
n
X
k=1
B 1k
"
y 1
y 2
# [k]
.
(B.22)
GKS algorithm: given {A 11 , A 12 , . . . , A 1n−1 , A 1n }, find {B 11 , B 12 , . . . , B 1n−1 , B 1n }.
1st polynomial:
"
x 1
x 2
#
= B 11
n
X
k=1
A 1k
"
x 1
x 2
# [k]
+ B 12
0
@
n
X
k 1 =1
A 1k 1
"
x 1
x 2
# [k 1 ]
1
A ⊗
0
@
n
X
k 2 =1
A 1k 2
"
x 1
x 2
# [k 2 ]
1
A + · · · +
+B 1n−1
0
@
n
X
k 1 =1
A 1k 1
"
x 1
x 2
# [k 1 ]
1
A ⊗ · · · ⊗
0
@
n
X
k n−1 =1
A 1k n−1
"
x 1
x 2
# [k n−1 ]
1
A +
+B 1n
0
@
n
X
k 1 =1
A 1k 1
"
x 1
x 2
# [k 1 ]
1
A ⊗ · · · ⊗
0
@
n
X
k n =1
A 1k n
"
x 1
x 2
# [k n ]
1
A =
= 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 1n−1 + · · · + B 1n−1 A n−1n−1 )
"
x 1
x 2
# [n−1]
+
+ (B 11 A 1n + · · · + B 1n A nn )
"
x 1
x 2
# [n]
+
+β
1n+1 .
(B.23)