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

38 3. Interpolating continued fractions
3.3 Unattainability
In the description of the algorithm unattainable points are identified in
step 3(b)i. However, there are possibly more unattainable points than detected
by the algorithm described so far. Similarly as in the Thiele case
(cf. Proposition 3.1.1), these remaining unattainable points are common
zeros of the continued fraction (3.6) [Wer79]. The following Proposition
characterizes these common zeros.
Proposition 3.3.1. The common zeros of the numerator An ′(x) and the
denominator Bn ′(x) of the n′ -th convergent of the continued fraction (3.6)
constructed from Algorithm 3.2.1 are zeros of aj(x) (j = 0,... ,n ′ − 1). If
xk is a zero of aj(x) for some j ∈ {0,... ,n ′ − 1} then
Rj+1(xk) = 0 ⇔ An ′(xk) = 0 = Bn ′(xk)
The proof follows the one of Proposition 3.1.1 closely.
Hence, including a final check to detect whether the tail Rj+1(x) vanishes
at a zero of aj(x) for j = 0,... ,n ′ − 1, guarantees the detection of all
unattainable points.
3.4 Non-normality
In order to understand how the algorithm of Werner navigates through the
table of rational interpolants, the role of the parameter rj is crucial.
Werner [Wer79] imposes rj ∈ {0,1}. Since in each step ℓj ≥ mj, clearly
the choice rj = 0 is always possible. Let us assume that no extra points are
interpolated in step 3, so that kj = γj + 1 = ℓj − mj − rj + 1. If ℓj > mj
then rj = 1 is possible. However, the choice rj = 1 then forces rj+1 = 0,
because rj = 1 implies ℓj+1 − mj+1 = mj − ℓj + kj = 0. On the other hand,
if ℓj = mj then necessarily rj = 0 so that kj = ℓj − mj + 1 = 1. Because
now ℓj+1 − mj+1 = mj − ℓj + kj = 1, the choice rj+1 = 1 is possible. Hence
subsequent values rj and rj+1 may in fact be restricted to 0 ≤ rj +rj+1 ≤ 1.
This was also pointed out by Gutknecht [Gut89].
The value of rj also affects the position in the table of rational interpolants
of the irreducible form of the j-th convergent of the resulting continued
fraction. It is not difficult to check (in analogy to Proposition 3.4.4)
that with rj = 0 for all j, the irreducible form of the convergents of the continued
fraction generated by Algorithm 3.2.1 occupy an entry in the table of
rational interpolants on the diagonal passing through (ℓ −m,0). The resulting
continued fraction is appropriately called a diagonal fraction. Whenever