Geography Reference
In-Depth Information
⇔
(i)
b
11
a
11
=1
,
⇒
b
11
=
a
−
1
11
(ii)
b
11
a
12
+
b
12
a
22
=0
⇒
b
11
a
12
a
−
1
a
−
3
b
12
=
−
22
=
−
11
a
12
,
(B.8)
(iii)
b
11
a
13
+
b
12
a
23
+
b
13
a
33
=0
⇒
b
13
=
−
(
b
11
a
13
+
b
12
a
23
)
a
−
1
33
=
a
−
1
11
(
a
12
a
−
1
22
a
23
− a
13
)
a
−
1
33
,
(iv)
b
11
a
14
+
b
12
a
24
+
b
13
a
34
+
b
14
a
44
=0
⇒
b
14
=
−
(
b
11
a
14
+
b
12
a
24
+
b
13
a
34
)
a
−
1
44
=
a
−
1
11
[
a
12
a
−
1
22
(
a
24
− a
23
a
−
1
33
a
34
)
+
a
13
a
−
1
a
14
]
a
−
1
33
a
34
−
44
.
Consult Box
B.5
for the general representation of
b
1
n
.
Notable for the GKS algorithm is the following. In the first step or the forward substitution,
a set of equations
x, x
2
, ..., x
n−
1
,x
n
is constructed by substituting
x
(
y
)intothepowers
x, x
2
, ...,x
n−
1
,x
n
, finally written into a matrix equation. The upper triangular matrix
{
}
A
is
gained by a multinomial expansion as indicated. In contrast, the second step or the backward
substitution is based upon the upper triangular matrix
−
1
B
:=
A
, the inversion of
A
.Its
first row contains the unknown coecients we are looking for:
{
b
11
,b
12
,..., b
1
n−
1
,b
1
n
}
The
−
1
construction of
A
can be based on symbolic computer manipulation. The
algebraic manipulation becomes more concrete when we pay attention to Examples
B.1
and
B.2
.
The first example aims at the inversion of an univariate homogeneous polynomial of degree
n
=2,
namely
y
(
x
)=
a
11
x
+
a
12
x
2
A
as well as
x
(
y
)=
b
11
y
+
b
12
y
2
. The GKS algorithm determines the set of
→
coecients
from the two given coecients
a
11
and
a
12
. In contrast, the second example
focuses on the inversion of an univariate homogeneous polynomial of degree
n
=3,namely
y
(
x
)=
a
11
x
+
a
12
x
2
+
a
13
x
3
{
b
11
,b
12
}
→ x
(
y
)=
b
11
y
+
b
12
y
2
+
b
13
y
3
. The GKS algorithm determines the
set of coecients
{b
11
,b
12
,b
13
}
from the three given coecients
a
11
,a
12
,and
a
13
.
Example B.1 (Inversion of an univariate homogeneous polynomial of degree
n
=2).
Assume the univariate homogeneous polynomial
y
(
x
)=
a
11
x
+
a
12
x
2
to be given and find the
inverse univariate homogeneous quadratic polynomial
x
(
y
)=
b
11
y
+
b
12
y
2
by the GKS algorithm.
1st step:
x
(
y
)=
b
11
y
+
b
12
y
2
=
b
11
a
11
x
+(
b
11
a
12
+
b
12
a
11
)
x
2
+
β
13
,
(B.9)
x
2
(
y
)=
b
22
y
2
+
β
23
=
b
22
a
11
x
2
+
β
23
.
Search WWH ::
Custom Search