510 B The inverse of a multivariate homogeneous polynomial
Continuation of Box.
2nd polynomial:
"
x 1
x 2
# [2]
= B 22
"
y 1
y 2
# [2]
+ B 23
"
y 1
y 2
# [3]
+ · · · + B 2n−1
"
y 1
y 2
# [n−1]
+ B 2n
"
y 1
y 2
# [n]
+ β
2n+1 =
= B 22 (A 11 ⊗ A 11 )
"
x 1
x 2
# [2]
+ (B 22 A 23 + B 23 A 33 )
"
x 1
x 2
# [3]
+ · · · + (B 22 A 2n + · · · + B 2n A nn )
"
x 1
x 2
# [n]
+
+β
2n+1 .
(B.24)
nth polynomial:
"
x 1
x 2
# [n]
= B nn A nn
"
x 1
x 2
# [n]
+ β
nn+1 .
(B.25)
(According to E. Grafarend and B. Schaffrin (1993) or W. H. Steeb (1991).)
Forward substitution:
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
"
x 1
x 2
# [1]
"
x 1
x 2
# [2]
.
"
x 1
x 2
# [n]
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
4
B 11 B 12 . . . B 1n
0 B 22 . . . B 2n
.
. ... .
0
0 . . . B nn
3
7
7
7
7
5
2
6
6
6
6
4
A 11 A 12 . . . A 1n
0 A 22 . . . A 2n
.
. ... .
0
0 . . . A nn
3
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
"
x 1
x 2
# [1]
"
x 1
x 2
# [2]
.
"
x 1
x 2
# [n]
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
+
2
6
6
6
6
6
4
β
1n+1
β
2n+1
.
β
nn+1
3
7
7
7
7
7
5
,
(B.26)
subject to
A 22 = A 11 ⊗ A 11 , A 23 = A 11 ⊗ A 12 + A 12 ⊗ A 11 ,
A 2n =
n−1 X
i=1
A 1i ⊗ A 1n−i ;
A 33 = A
[3]
11 ,
A 3n =
n−2 X
i=1
A 1i ⊗
n−i−1 X
j=1
A 1j ⊗ A 1n−i−j ;
A 44 = A
[4]
11 ,
A 4n =
n−3 X
i=1
A 1i ⊗
" n−i−2 X
j=1
A 1j ⊗
n−i−j−1
X
k=1
A 1k ⊗ A 1n−i−j−k
!#
;
A 5n =
n−4 X
i=1
A 1i ⊗
" n−i−3 X
j=1
A 1j ⊗
n−i−j−2
X
k=1
A 1k ⊗
" n−i−j−k−1 X
l=1
A 1l ⊗ A 1n−i−j−k−l
#!#
.
(B.27)
(Consult Box B.4 for the general representation of A mn .)