Geography Reference
In-Depth Information
2nd step (forward substitution):
x
x
2
=
b
11
b
12
a
11
a
12
0
a
22
x
x
2
+
β
13
=
b
11
a
11
b
11
a
12
+
b
12
a
22
β
23
0
b
22
0
b
22
a
22
x
x
2
+
β
13
,
(B.10)
β
23
subject to
a
22
=
a
11
.
Both the matrices A :=
and B :=
are upper triangular such that
A
B
−
1
B
=I
2
⇔
B
=
A
.
(B.11)
A
3rd step (backward substitution):
=(
a
11
a
22
)
−
1
a
22
−a
12
=
a
−
1
−a
−
3
11
a
12
−
1
⇒ b
11
=
a
−
1
11
11
,b
12
B
=
a
−
2
0
a
11
0
A
11
a
−
3
=
−
11
a
12
,
(B.12)
or
b
22
a
22
=
b
22
a
11
=1
b
22
=
a
−
1
22
=
a
−
2
b
11
=
a
−
1
⇒
11
,
11
a
11
=1
⇒
11
,
(B.13)
a
−
1
a
−
3
b
11
a
12
+
b
12
a
22
=0
⇒
b
12
=
−
22
a
12
b
11
=
−
11
a
12
,
x
(
y
)=
a
−
1
a
−
3
11
a
12
y
2
.
11
y
−
(B.14)
End of Example.
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
11
)
x
2
+(
b
11
a
13
+2
b
12
a
11
a
12
+
b
13
a
11
)
x
3
+
β
14
,
x
2
(
y
)=
b
22
y
2
+
b
23
y
3
+
β
24
(B.15)
=
b
22
a
11
x
2
+(2
b
12
a
11
a
12
+
b
23
a
11
)
x
3
+
β
24
,
x
3
(
y
)=
b
33
y
3
+
β
34
=
b
33
a
11
x
3
+
β
34
.
2nd step (forward substitution):
Search WWH ::
Custom Search