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

2.1. Classical rational interpolation 17
the barycentric form does not readily reflect the actual numerator and denominator
degrees, these methods basically determine a denominator q(x)
of degree ∂q ≤ m using a basis other than the Lagrange basis and restore
the connection with the Lagrangian basis afterward. For example Schneider
and Werner [SW86] first compute a denominator in a Newton basis with
respect to Xm ⊆ Xn. For the resulting linear system of which the coefficient
matrix is a divided-difference matrix (or Löwner matrix) a stable solver is
proposed in [Gem96]. Only in a second phase the barycentric weights with
respect to Xn are computed from these Newton coefficients using an algorithm
from [Wer84]. A more direct approach is due to Berrut et al. [BM97].
Using monomial basis functions, conditions for the weights are derived such
that rn(x) = p(x)/q(x) satisfies ∂p ≤ ℓ and ∂q ≤ m. A very similar approach
is due to Zhu et al. [ZZ02] who directly determine function values q0,... ,qn
of a denominator of degree ∂q ≤ m from the same degree conditions, but
now by considering the Newton representation.
A slightly different, but related approach is due to Antoulas and Anderson
[AA86]. They start from the following implicit form of (2.1)
n
i=0
rn(x) − fi
ui
x − xi
= 0 (2.4)
and investigate the algebraic structure of the problem of parameterizing all
interpolating rational functions of minimal complexity. Here, the complexity
of a rational function rn(x) = p(x)/q(x) is understood as max{∂p,∂q}. This
problem formulation is further considered in Section 5.2.
Instead of the full barycentric form (2.4), Antoulas and Anderson [AA86]
start from a lower complexity barycentric representation 1 [Ber00] rn ′(x) of
rn(x)
n ′
i=0
rn ′(x) − fi
ui
x − xi
= 0 (2.5)
where n ′ < n. When specifying rn ′(x), it is implicitly assumed that Xn is
ordered such that the first n ′ + 1 points are those appearing in (2.5). For
rn ′(x) to interpolate also in the remaining points Xn \ Xn ′, a non-trivial
weight vector u = (u0,... ,un ′)T must exist in the kernel of the (n − n ′ ) ×
(n ′ + 1) Löwner matrix L with entries
Lj,i = fj − fi
xj − xi
i = 0,... ,n ′ ;j = n ′ + 1,... ,n.
1 This nomenclature is not used explicitly in [AA86].