Geography Reference
In-Depth Information
This defines the general problem of homogeneous polynomial series inversion. It reduces to con-
struct the matrices
{
B
11
,
B
12
, ...,
B
1
n−
1
,
B
1
n
}
from the given matrices
{
A
11
,
A
12
, ...,
A
1
n−
1
,
A
1
n
}
by the GKS algorithm as outlined in Box
B.3
.
Box B.3 (Algorithm for the construction of a multivariate homogeneous polynomial of degree
n
).
1st polynomial:
n
A
1
k
2
x
[
k
2
]
[
k
1
]
n
+
β
1
n
+1
.
x
=
B
1
k
1
(B.61)
k
1
=1
k
2
=1
2nd polynomial:
n
A
1
k
2
x
[
k
2
]
[
k
1
]
n
n
x
[2]
=
B
2
k
y
[
k
]
+
β
2
n
+1
=
+
β
2
n
+1
.
B
2
k
1
(B.62)
k
=2
k
1=2
k
2
=1
n
th polynomial:
x
[
n
]
=B
nn
y
[
n
]
+
β
nn
+1
=B
nn
A
nn
x
[
n
]
+
β
nn
+1
.
(B.63)
Again taking advantage of the basic product rule between Cayley and Kronecker-Zehfuss
multiplication, i.e. (
AC
)
⊗
(
BD
)=(
A ⊗ B
)(
C ⊗ D
), we arrive at the following results.
Forward substitution:
⎡
⎤
⎡
⎤
x
x
[2]
·
x
[
n−
1]
x
[
n−
1]
x
x
[2]
·
x
[
n−
1]
x
[
n−
1]
⎣
⎦
⎣
⎦
=
=
.
(B.64)
B
A
(Note that
A
is given by Box
B.4
.
)
Backward substitution:
B
=I
⇒
A
(B.65)
see first row B
1
n
given in Box
B.5
.
}
,
where the columm array powers are to be understood with respect to the Kronecker-Zehfuss
product. Again, advantage is taken from the basic product rule (
AC
)
⊗
(
BD
)=(
A⊗B
)(
C ⊗D
),
also called “reduction of the Kronecker-Zehfuss product to the Cayley product”, by replacing such
an identity into the power series equations, the upper triangular matrix
First, in the forward substitution, we construct a set of equations for
{
x
,
x
[2]
, ...,
x
[
n−
1]
,
x
[2]
as well as its inverse
A
−
1
B
:=
A
. Second, while Box
B.4
collects the input matrices A
mn
, which built up
A
,by
means of Box
B.5
, we compute its inverse
B
, namely its first row
{
B
11
,
B
12
, ...,
B
1
n−
1
,
B
1
n
}
,
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