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

48 4. Asymptotic behavior
In the sequel, we let ℓ −m = k and consider rational interpolants rm+k,m(x)
with m + k ≥ 0. Hence, we are interested in determining an irreducible
rational function rm+k,m(x) for which an interpolation condition (4.1) at
infinity and interpolation conditions (1.1) at finite points X2m+k−1 are given.
It may be assumed that m > 0, i.e. rm+k,m(x) is truly rational. Otherwise
rk,0(x) is a polynomial and the problem is trivial. Note that if m > 0 and
m + k ≥ 0, then X2m+k−1 = ∅, i.e. there is at least one finite interpolation
point. For practical considerations, as before, we may also assume that
m + k ≥ m, hence k ∈ N. If m + k < m then we consider rm,m+k(x) =
1/rm+k,m(x) which interpolates
rm,m+k(xi) = 1
f(xi) , xi ∈ X2m+k−1 ∪ {∞}.
4.2 Unattainability and non-normality
A non-trivial polynomial couple (p(x),q(x)) ∈ R(m+k,m) that satisfies the
linearized finite interpolation conditions (1.2) for X2m+k−1 and also
αm+k − ξβm = 0 (4.2)
always exists, because there are only 2m+k+1 conditions for the 2m+k+2
unknowns α0,... ,αm+k and β0,...,βm. In addition, for any two non-trivial
polynomial couples (p1(x),q1(x)) and (p2(x),q2(x)) in R(m + k,m) that
satisfy (4.2)–(1.2), the polynomial (p1q2 − p2q1)(x) is of degree at most
2m + k − 1 — due to (4.2), the coefficient of the term x 2m+k is zero — and
vanishes at 2m + k points X2m+k−1. Therefore it must vanish identically
and we have p1(x)q2(x) = p2(x)q1(x). Hence all solutions of (4.2)–(1.2) are
equivalent and have the same irreducible form rm+k,m(x). As in the case
of rational interpolation involving only finite data, the original non-linear
problem (4.1)–(1.1) need not have a solution. We say that the point at
infinity is unattainable if and only if rm+k,m(∞) = f(∞). Because there
is no such thing as a deficiency polynomial including the point at infinity,
the complete block structure for rational interpolation problems involving
conditions at infinity is more complex than the one given in Chapter 1 and
is fully described by Gutknecht [Gut93a]. Here we assume that the ordering
of the data is such that all finite interpolation data X2m+k−1 come first and
that the point at infinity is always last. With this consideration we can say
the following concerning two subsequent entries on a staircase in the table of
rational interpolants through the diagonals starting at (k,0) and (k + 1,0).