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

8 1. Classical rational interpolation
valuable computational tool for three reasons. First of all, the condition
number of Vandermonde-like matrices grows exponentially with n [GI87],
such that the obtained coefficients of p(x) and q(x) may be very inaccurate.
Changing to orthogonal basis polynomials is necessary for improving
the conditioning of the problem. Second, there is the normalization. Usually,
an a priori normalization is agreed upon for one of the coefficients in
the numerator or denominator polynomial. Although an appropriate normalization
can always be found, choosing an inappropriate normalization
for the problem at hand may cause the linear system to be inconsistent.
Third, if the homogeneous system is not of full rank and the nullspace is
therefore multidimensional, finding a representation for the minimal degree
solution from a basis for the nullspace is not obvious. If the obtained couple
(p(x),q(x)) is not the minimum degree solution, then it may happen that
both p(xi) = q(xi) = 0 for some xi ∈ Xn, while after cancellation of (x −xi)
nonetheless p(xi)/q(xi) = fi. This complicates the practical detection of
unattainable points.
Example 1.3.1. As an example, consider the data xi = i+1 and fi = 1/xi
for i = 0,... ,4 and let ℓ = m = 2. The rational interpolant for these data
is rℓ,m(x) = 1/x. Let us consider a solution of (1.2) in classical powerform.
A popular normalization is β0 = 1. With this normalization, the linear
system (1.2) is inconsistent, although the rational interpolation problem has
a solution. Without normalization, the system (1.2) for these data is rank
deficient by one, and its kernel (or nullspace) is therefore two dimensional.
All solutions of (1.2) are parameterized as follows α2 = 0 = β0, α1 = β2,
α0 = β1. Hence all solutions of (1.2) are polynomials of the form p(x) =
α0 + α1x and q(x) = x(α0 + α1x). If we choose for example α0 = −α1 = 0,
then both p(x) and q(x) contain the factor (x − 1) = (x − x0). Although x0
is not unattainable, both p(x0) and q(x0) vanish.
Due to such difficulties, solving (1.2) directly is not recommended. A
more convenient alternative are recursive algorithms that build up consecutive
interpolants along a path in the table of rational interpolants, which is
defined below.
In a strictly formal sense, consider the infinite, fixed sequence {xn} ∞ n=0 .
For different values of ℓ + m = n, the rational interpolants of order (ℓ,m)