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

36 3. Interpolating continued fractions
3.2 Werner’s algorithm
Werner [Wer79] suggests a recursive method, based on successive continued
fraction reductions and reordering of the data, such that the intermediate
computations always remain bounded and such that insolubility of the problem
can be detected. For the description of the algorithm it is assumed that
1. ℓ ≥ m. If ℓ ≤ m, then we may swap the roles of ℓ and m and consider
rm,ℓ(x) which interpolates 1/f(xi), xi ∈ Xn.
2. f(xi) = 0, ∞ for xi ∈ Xn. We come back to the case f(xi) = 0, ∞ in
Section 3.5.
The algorithm recursively constructs a continued fraction representation of
a rational function R0(x)
R0(x) = b0(x) + a0(x)
b1(x)
a1(x)
+
b2(x) + ... + an ′ −1(x)
bn ′(x)
, n ′ ≤ n (3.6)
by applying continued fraction reductions of the form
Rj(x) = bj(x) + aj(x)
Rj+1(x)
j = 0... ,n ′ − 1,
where the polynomials aj(x) and bj(x) are determined by the interpolation
conditions. Algorithm 3.2.1 clarifies the roles of bj(x) and aj(x).
The algorithm has two different stopping criteria: in step 3a when all
interpolation data have been processed, or in step 3(b)i in which case there
are unattainable points. The second stopping criterion is motivated by the
following Proposition.
Proposition 3.2.1 (Werner [Wer79]). Let b(x) be the polynomial of degree
∂b ≤ γ = ℓ − m that interpolates x0,...,xγ. If there exists an integer
k ≥ ℓ + 1 such that
b(xi) = 0 for i = γ + 1,... ,k − 1
b(xi) = 0 for i = k,... ,ℓ + m,
then the polynomials p ∗ (x) and q ∗ (x) given by
q ∗ (x) =
(x − xk)... (x − xℓ+m) if k − 1 < ℓ + m
1 if k − 1 = ℓ + m
p ∗ (x) = q ∗ (x) · b(x)
solve the linearized rational interpolation problem (1.2).
We remark that if the algorithm starts with f(xi) = 0, ∞, then by
construction and (3.7), also Rj(xi) = 0, ∞ for xi ∈ Sj.