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

8.2. Rational interpolation of uncertainty intervals 101
and the irreducible form of the generalized rational function (p/q)(t1,t2) by
rℓ,m(t1,t2). Let
Rℓ,m(Sn) = {rℓ,m(t1,t2) | rℓ,m(t (i)
1 ,t(i) 2
) ∈ Fi,q(t (i)
1 ,t(i) 2
) > 0,i = 0,... ,n}.
(8.7)
So the rational frequency response providing the data is denoted by
H(e it(i)
1 ,e it(i)
2 ) = A
B (eit(i) 1 ,e it(i)
2 )
and its rational approximant is denoted by
rℓ,m(t1,t2) = p
q (t1,t2).
Whereas in traditional rational interpolation one has ℓ + m = n, here we
envisage ℓ + m ≪ n just as in least squares approximation. For given
segments Sn and given sets N and D of respective cardinality ℓ + 1 and
m + 1, we are concerned with the problem of determining whether
The interpolation conditions
in (8.7) amount to
Rℓ,m(Sn) = ∅.
rℓ,m(t (i)
1 ,t(i)
2 ) ∈ Fi, i = 0,... ,n (8.8)
f ≤
i p(t(i) 1 ,t(i) 2 )
q(t (i)
1 ,t(i)
2 ) ≤ f i, i = 0,... ,n.
Under the assumption that q(t (i)
1 ,t(i) 2 ) > 0 for i = 0,... ,n, we obtain the
following homogeneous system of linear inequalities after linearization
−p(t (i)
1 ,t(i)
2 ) + f iq(t (i)
1 ,t(i) 2
p(t (i)
1 ,t(i) 2 ) − f i q(t(i) 1 ,t(i) 2
) ≥ 0
) ≥ 0 , i = 0,... ,n. (8.9)
As explained in Chapter 6, there is no loss of generality in assuming that
q(t1,t2) is positive in the interpolation points: the interpolation conditions
(6.7) can be linearized for arbitrary non-zero q(t (i)
1 ,t(i) 2 ), positive or negative,
without changing the nature of the problem.