The Inverse of a Multivariate Homogeneous Polynomial - Map Projections: Cartographic Information Systems - page 683

+B 2 n y 1

[ n ]

+ β 2 n +1 =

y 2

A 11 ) x 1

[2]

+(B 22 A 23 +B 23 A 33 ) x 1

[3]

=B 22 (A 11 ⊗ + ··· +

x 2 x 2

(B 22 A 2 n + ··· +B 2 n A nn ) x 1

[ n ]

+ (B.24)

x 2

+ β 2 n +1 .
n th polynomial:

x 1
x 2

[ n ]

=B nn A nn x 1

[ n ]

+ β nn +1 .

(B.25)

x 2

(According to Grafarend and Schaffrin 1993 or Steeb 1991 .)

Forward substitution:

⎡ ⎤ ⎡ ⎤

x 1
x 2

[1] x 1

x 2

[1]

⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤

x 1
x 2

[2] x 1

x 2

[2]

B 11 B 12 ...B 1 n
0 B 22 ...B 2 n
·

A 11 A 12 ...A 1 n
0 A 22 ...A 2 n
·

⎣ ⎦ ⎣ ⎦

= =

·
00 ...B nn

· ... ·

00 ...A nn

· ...

· ·

x 1
x 2

[ n ] x 1

x 2

[ n ]

⎡ ⎤

β 1 n +1
β 2 n +1
·
β nn +1

⎣ ⎦

+ , (B.26)

subject to

A 22 =A 11 ⊗ A 11 , A 23 =A 11 ⊗ A 12 +A 12 ⊗ A 11 ,

n− 1

A 2 n = A 1 i ⊗ A 1 n−i ;

i =1

A 33 =A [3]

11 ,

n− 2 n−i− 1

A 3 n = A 1 i ⊗ A 1 j ⊗ A 1 n−i−j ; (B.27)

i =1 j =1

A 44 =A [4]

11 ,

n−i− 2

A 1 n−i−j−k ;

n − 3 n−i−j− 1

A 4 n = A 1 i ⊗ A 1 j ⊗ A 1 k ⊗

i =1 j =1 k =1

n−i− 3

n − i − j − 2

n− 4

A 5 n = A 1 i ⊗ A 1 j ⊗ A 1 k ⊗

i =1 j =1

k =1