Practical Rational Interpolation of Exact and Inexact Data Theory ...

A.2. Rational interpolation of vertical segments 123
need to define a rule for the case that
d
ki =
i=1
We restrict ourselves to lexicographical orderings. We have chosen the backwards
lexicographical ordering (i.e. ordering by kd first) by default, but this
is merely a matter of taste. Priority can be assigned to any ki. In fact, any
lexicographical ordering can be described in a simple inductive way once a
preference is fixed upon the ki. Let
⎧
⎨
Vn =
⎩ (k1,... ,kd) ∈ N d ⎫
d ⎬
| kj = n
⎭ .
A lexicographical preference fixes the order of the elements of Vn.
Example A.2.2. For d = 3, the lexicographical preference (1,2,3) means
that first is sorted on k1, then on k2 and finally on k3. Hence the ordering
of V1 is (0,0,1) < (0,1,0) < (1,0,0). The preference (1,3,2) means that
first is sorted on k1, then on k3 and finally on k2. The ordering of V1 is then
(0,1,0) < (0,0,1) < (1,0,0).
Recall that it is our goal to enumerate over N d , not simply sorting elements.
A first idea is to use the fact that for each n > 0, the elements
of Vn can be decomposed in a unique way as the sum of n elements out of
V1. Here addition is understood component wise. This gives rise to a d-ary
tree, which can be traversed efficiently in a breadth-first fashion. Provided
that the tree is composed in the right way, breadth-first traversal of the tree
corresponds exactly to the discussed enumeration of N d . For a lexicographical
ordering at each level (Vn), its construction is as follows. The origin is
always the root element, and has d children: the elements of V1. For ease of
notation, denote the elements of V1 by
d
i=1
k ′ i .
j=1
ei = (0,... ,0,1,0,... ,0)
where the position of 1 is on the i-th location. Each arrow connects a node
(k1,...,kd) ∈ Vn−1 to a node (k ′ 1 ,...,k′ d ) ∈ Vn and has a label 1 ≤ i ≤ d. A
label i means that
(k ′ 1 ,...,k′ d ) = (k1,... ,kd) + ei
The way that children are generated depends on the chosen lexicographical
preference. If an arrow with label i enters a node, then arrows with