The Inverse of a Multivariate Homogeneous Polynomial - Map Projections: Cartographic Information Systems - page 682

A 1 k x 1

[ k ] n

= ,

x 2

k =1

x ( y ):= x 1

=

x 2

=B 11 y 1

+B 12 y 1 y 1

y 2

+

+B 1 n− 1 y 1

y 1
y 2

⊗ ··· ⊗···⊗

n− 1times

y 2 y 2 y 2

+B 1 n y 1

y 1
y 2

=

⊗···⊗ (B.22)

n times

y 2

B 1 k y 1

[ k ]

n

= .

y 2

k =1

GKS algorithm: given { A 11 , A 12 , ..., A 1 n− 1 , A 1 n } ,

find { B 11 , B 12 , ..., B 1 n− 1 , B 1 n } .

1st polynomial:

+B 12 n

[ k 1 ] x 1

x 2

=B 11

A 1 k x 1

[ k ] x 1

x 2

n

A 1 k 1

x 2

k =1 k 1 =1

n

[ k 2 ] + x 1

x 2

⊗ A 1 k 2 ··· +

k 2 =1

⎛

[ k n− 1 ] ⎞
⎠ +

+B 1 n− 1 n

[ k 1 ] x 1

x 2

x 1
x 2

n

⎝

A 1 k 1 ⊗···⊗ A 1 k n− 1

k 1 =1 k n− 1 =1

+B 1 n n

[ k 1 ]

n

[ k n ] = x 1

x 2

x 1
x 2

⊗···⊗

A 1 k 1 A 1 k n

k 1 =1 k n =1

=B 11 A 11 x 1

+(B 11 A 12 +B 12 A 11 ⊗

A 11 ) x 1

[2]

+ ··· + (B.23)

x 2 x 2

+B 1 n− 1 A n− 1 n− 1 ) x 1

[n − 1]

+(B 11 A 1 n− 1 + ··· +

x 2

+B 1 n A nn ) x 1

[ n ]

+(B 11 A 1 n + ··· +

x 2

+ β 1 n +1 .

2nd polynomial:

x 1
x 2

[2]

=B 22 y 1

[2]

+B 23 y 1

[3]

+ ··· +B 2 n− 1 y 1

[ n− 1]

y 2 y 2 y 2