The Inverse of a Multivariate Homogeneous Polynomial - Map Projections: Cartographic Information Systems - page 688

+B 13 3

x 1
x 2

[ k 1 ]

3

x 1
x 2

[ k 2 ]

A 1 k 1 ⊗ A 1 k 2

k 1 =1 k 2 =1

3

[ k 3 ] =

x 1
x 2

⊗ A 1 k 3 (B.51)

k 3 =1

=B 11 A 11 x 1

+(B 11 A 12 +B 12 A 11 ⊗ A 11 ) x 1 [2]

+

x 2 x 2

+[B 11 A 13 +B 12 (A 11 ⊗ A 12 +A 12 ⊗ A 11 )+B 13 (A 11 ⊗ A 11 ⊗ A 11 )]

x 1
x 2

[3]

+ β 14 .

2nd polynomial:

x 1
x 2

x 1
x 2

= x 1 [2]

=B 22 y 1

[2]

+B 23 y 1

[3]

+ β 24 = ⊗

x 2 y 2 y 2

(B.52)

A 11 ) x 1

[2]

=B 22 (A 11 ⊗ +[B 22 (A 11 ⊗ A 12 +A 12 ⊗ A 11 )

x 2

A 11 )] x 1

[3]

+ β 24 .

+B 23 (A 11 ⊗ A 11 ⊗

x 2

3rd polynomial:

x 1
x 2

x 1
x 2

x 1
x 2

= x 1 [3]

A 11 ) x 1

[3]

+ β 34 . ⊗ ⊗ =B 33 (A 11 ⊗ A 11 ⊗ (B.53)

x 2 x 2

Forward substitution:

⎡ ⎤ ⎡ ⎤

x 1
x 2

[1] x 1

x 2

[1]

⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x 1
x 2

[2] x 1

x 2

[2]

β 14
β 24
β 34

B 11 B 12 B 13
OB 22 B 23
00B 33

A 11 A 12 A 13
0A 22 A 23
00A 33

⎣ ⎦ ⎣ ⎦ = ⎣ ⎦ ,

= + (B.54)

x 1
x 2

[3] x 1

x 2

[3]

subject to

A 22 =A 11 ⊗ A 11 , A 23 =A 11 ⊗ A 12 +A 12 ⊗ A 11 , A 33 =A 11 ⊗ A 11 ⊗ A 11 . (B.55)

Both the matrices A :=

A and B := B are upper triangular such that

− 1

B =I 14

⇔

B = A .

(B.56)

A