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

5.2. Parameterizations 61
(s1,s2)-degree ≥ 0 such that the linearized rational interpolation conditions
f(xi)d(xi) − n(xi) = 0, i = 0,... ,j (5.2)
are satisfied.
We already briefly touched this kind of problem setting in Section 2.1,
where the barycentric form was used by Antoulas and Anderson [AA86] for
obtaining a solution for (5.2) of minimal (0,0)-degree. In the discussion here
we restrict ourselves to one particular, so-called single step method, where
one is interested to compute a sequence of neighboring entries in the table of
rational interpolants (or in case of blocks, a maximal subsequence) [BL97].
In each step one more interpolation condition is added, hence each time 0 ≤ j
is increased by one. It is shown in [VBB92, VBB90] that all polynomial
couples (n(x),d(x)) of a certain (s1,s2)-degree ≤ α that satisfy (5.2) can
be parameterized by two polynomial couples vj(x) = (nj,v(x),dj,v(x)) and
wj(x) = (nj,w(x),dj,w(x)). Therefore (vj(x),wj(x)) is called a basic pair.
Denote by Rivj and Riwj the linearized residual at xi of the polynomial
couple vj(x) and wj(x) respectively
Rivj = f(xi)dj,v(xi) − nj,v(xi)
Riwj = f(xi)dj,w(xi) − nj,w(xi).
The steps to obtain a possible choice for such a basic pair are given in
Algorithm 5.2.1. For each 0 ≤ j, a basic pair (vj(x),wj(x)) is constructed
from a basic pair (vj−1(x),wj−1(x)) of the previous level. In each step,
both vj(x) and wj(x) solve the linearized rational interpolation problem for
or equivalently
{xi} j
i=0
Rivj = 0, i = 0,... ,j
Riwj = 0, i = 0,... ,j.
The algorithm is such that the polynomial couple vj(x) is always of minimal
(s1,s2)-degree ≥ 0. In fact vj(x) is a minimal degree solution for some
rational interpolant in the table of rational interpolants. Hence, vj(x) need
not be irreducible, but if vj(x) contains common factors, they are unattainable
points. It is also shown in [VBB92, VBB90] that all solutions of the
(non-linear) rational interpolation problem of minimal (s1,s2)-degree are
also parameterized by a linear combination of vj(x) and wj(x).
Example 5.2.1. Let us consider (s1,s2) = (0,0) and the case of a normal
table of rational interpolants, i.e. without blocks. Figure 5.2 illustrates