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

6 1. Classical rational interpolation
always exists. Indeed, when expanding p(x) and q(x) in terms of basis
functions, such as the monomials,
p(x) =
ℓ
αkx k , q(x) =
k=0
m
k=0
βkx k
then (1.2) reduces to a homogeneous system of linear equations in the unknown
coefficients of p(x) and q(x). Since this system has one more unknown
than its total number of linear equations, the linearized problem (1.2) always
has a non-trivial solution. Moreover, for any two non-trivial solutions
(p1(x),q1(x)) and (p2(x),q2(x)) of (1.2) in R(ℓ,m), the polynomial
(p1q2 − p2q1)(x) of degree at most ℓ + m = n vanishes at n + 1 distinct
points:
[(fq2 − p2)q1 − (fq1 − p1)q2] (xi) = 0 if f(xi) is finite
(p1q2 − p2q1)(xi) = 0 if f(xi) is ∞
i = 0,... ,n.
Therefore it must vanish identically and we have p1(x)q2(x) = p2(x)q1(x).
Hence all solutions of (1.2) in R(ℓ,m) are equivalent and, up to a normalization,
have the same irreducible form rℓ,m(x), which is called the rational
interpolant of order (ℓ,m) for f(x). Let p0(x) and q0(x) denote the numerator
and denominator of rℓ,m(x). For the notation not to become overloaded,
the explicit dependence of ℓ and m in p0(x) and q0(x) is omitted.
1.2 Unattainability
It is well-known that the interpolating polynomial of degree at most n,
interpolating n+1 points Xn always exists and is unique. The condition ℓ+
m = n is imposed in order to obtain this exact same analogy. It is true that
the linearized rational interpolation problem (1.2) maintains the analogy
to some extent, because it always has a solution and delivers the unique
representation rℓ,m(x). However, in contrast with polynomial interpolation,
the rational interpolation problem (1.1) is not always soluble and may give
rise to unattainable points. An interpolation point is called unattainable for
a non-trivial solution (p(x),q(x)) ∈ R(ℓ,m) of (1.2) if and only if
q(xi) = 0 = p(xi)
and, after cancellation of (x − xi) in p(x)/q(x), in addition
p(xi)
= f(xi).
q(xi)