The Inverse of a Multivariate Homogeneous Polynomial - Map Projections: Cartographic Information Systems - page 691

the one being only needed. A symbolic computer manipulation software package of Box B.5 is
available from the author, in particular, for the backward substitution step.

Box B.4 (Inversion of a multivariate homogeneous polynomial of degree n : the upper trian-
gular matrix A :=

A ).

Recurrence relation:

n− ( m− 1)

A mn = A 1 i ⊗ A m− 1 n−i ∀ m ≤ n. (B.66)

i =1

Inversion relations (A 11 , A 1 n given) :

n− 1

A 22 =A [2]

11 , A 2 n =

A 1 i ⊗ A 1 n−i ,

i =1

n− 2 n−i− 1

A 33 =A [3]

11 , A 3 n =

A 1 i ⊗ A 1 j ⊗ A 1 n−i−j ,

i =1 j =1

n − i − j − 1

n− 3 n−i− 2

A 44 =A [4]

11 , A 4 n =

A 1 i ⊗ A 1 j ⊗ A 1 k ⊗ A 1 n−i−j−k ,

i =1 j =1 k =1

n − 4

n−i− 3 n−i−j− 2 n−i−j−k− 1

A 55 =A [5]

11 , A 5 n =

A 1 i ⊗ A 1 j ⊗ A 1 k ⊗

i =1 j =1 κ =1

l =1

A 1 l ⊗ A 1 n−i−j−k−l , (B.67)

n − ( m − 1) n − k 1 − ( m − 2) n − k 1 − k 2 − ( m − 3)

A mn = A 1 k 1 ⊗ A 1 k 2 ⊗ A 1 k 3 ⊗···⊗

k 1 =1 k 2 =1 k 3 =1

n−k 1 −k 2 −···−k m− 2

⊗···⊗Φ A 1 k m− 1 ⊗ A 1 n−k 1 −k 2 −···−k m− 2 −k m− 1 .

k m− 1 =1

Box B.5 (Inversion of a multivariate homogeneous polynomial of degree n :1strowofthe
triangular matrix B := ).

B

Recurrence relation:

B 1 i A in A − 1

nn = A [ n ] 11 − 1

n− 1

2 , subject to A − 1

B 1 n = − ∀ n ≥

nn

i =1

= A − 1

11 [ n ] .

(B.68)

Inversion relations (A 11 , A 1 n given) :

B 11 =+A − 1

11 , B 12 = − A − 1 11 A 12 A − 1 22 , B 13 =+A − 1 11 [A 12 A − 1 22 A 23 − A 13 ]A − 1 33 ,