Geography Reference
In-Depth Information
Box B.1 (Algorithm for the construction of an inverse univariate homogeneous polynomial
of degree
n
).
y
(
x
)=
a
11
x
+
a
12
x
2
+
···
+
a
1
n−
1
x
n−
1
+
a
1
n
x
n
,
x
(
y
)=
b
11
x
+
b
12
x
2
+
+
b
1
n−
1
x
n−
1
+
b
1
n
x
n
.
···
(B.1)
GKS algorithm: given
{
a
11
,a
12
..., a
1
n−
1
,α
1
n
}
, find
{
b
11
,b
12
, ..., b
1
n−
1
,b
1
n
}
.
Forward substitution:
x
=
b
11
y
+
b
12
y
2
+
+
b
1
n−
1
y
n−
1
+
b
1
n
y
n
+
β
1
n
+1
,
···
x
2
=
b
22
y
2
+
b
23
y
3
+
+
b
2
n−
1
y
n−
1
+
b
2
n
y
n
+
β
2
n
+1
,
···
(B.2)
x
n−
1
=
b
n−
1
n−
1
y
n−
1
+
b
n−
1
n
y
n
+
β
n−
1
n
+1
,
x
n
=
b
nn
y
n
+
β
nn
+1
.
(B.3)
⎡
⎤
⎡
⎤
⎡
⎤
y
y
2
y
n
a
11
a
12
···
a
1
n
x
x
2
x
n
⎣
⎦
⎣
⎦
⎣
⎦
0
a
22
a
2
n
· · ··· ·
00
···
+
α
n
,
=
(B.4)
···
a
nn
subject to
a
22
=
a
11
,
a
23
=2
a
11
a
12
,
a
24
=2
α
11
a
13
+
a
12
,
a
25
=2
a
11
a
14
+2
a
12
a
13
,
etc
.
a
33
=
a
11
,
a
34
=3
a
11
a
12
,
a
35
=3
a
11
a
13
+3
a
11
a
12
,
etc
.
a
44
=
a
11
,
a
45
=4
a
11
a
12
,
etc
.
(B.5)
⎡
⎤
a
11
a
12
a
13
···
a
1
n
⎣
⎦
0
a
11
2
a
11
a
12
···
a
2
n
A
=
·
00
... a
n−
1
n−
1
a
n−
1
n
00
...
·
·
...
...
,
(B.6)
0
a
nn
⎡
⎤
b
11
b
12
... b
1
n−
1
b
1
n
⎣
⎦
0
b
22
... b
2
n−
1
b
2
n
B
=
· ·
00
...b
n−
1
n−
1
b
n−
1
n
00
...
·
·
...
.
(B.7)
0
b
nn
Consult Box
B.4
for the general representation of
a
mn
.
Backward substitution:
B
=
I
A
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