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

34 3. Interpolating continued fractions
convergent of R0(x) = U0(x)/V0(x) is irreducible except for factors of the
form (x − xj) with xj ∈ Xn−1 unattainable. These factors are characterized
by
Rj+1(xj) = 0 ⇔ xj is unattainable .
Proof. We denote by
Ri(x) =
Ui(x)
, i = 0,... ,n
Vi(x)
the tails Ri(x) in vector form. Then (3.3) readily becomes
bj
Rj(x) =
1
x − xj Uj+1(x)
= Tj(x)Rj+1(x),
0 Vj+1(x)
j = 0,... ,n − 1.
Assuming that bi = ∞ (i = 0,... ,n), we first show that the only common
factors of R0(x) are of the form (x − xj), with xj ∈ Xn−1.
Assume that R0(x) has a common factor (x − α) with α /∈ Xn−1. For
α /∈ Xn−1, every Tj(α) is not singular, so that we may write
Rn(α) =
bn
1
= T −1
n−1 · · ·T−1 0 R0(α) =
which is a contradiction. Hence α ∈ Xn−1. This means that R0(x) is
irreducible up to common factors of the form (x − xj) and by construction
interpolates every xj for which Rj+1(xj) = 0.
If R0(x) contains the factor (x−xj) in numerator and denominator, then
and also
Rj(xj) = T −1
j−1 · · ·T−1 0 R0(xj) =
Rj(xj) = TjRj+1(xj) =
0
0
0
0
bjUj+1(xj)
=
Uj+1(xj)
,
0
.
0
Hence the numerator Uj+1(x) of Rj+1(x) vanishes at xj. Because
bn
Rn(xj) = = T
1
−1
n−1 · · ·T−1
j+1Rj+1(xj)
0
= ,
0
it cannot be the case that also Vj+1(xj) = 0, such that Rj+1(xj) = 0.
Analogously, if Rj+1(xj) = 0 then
R0(x) = T0 · · ·Tj−1Rj(xj) =
0
.
0