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

Barycentric representation
2
In this Chapter, we consider a specific representation for the rational interpolant:
the barycentric form. This form stems from the Lagrange representation
of interpolating polynomials. Several authors [BM97, SW86] praise
its properties, both numerical and analytic. The barycentric form indeed
possesses a remarkable property which makes it attractive for implementation,
namely exact interpolation even in the presence of rounding errors
during the computation of the coefficients of the barycentric form.
First, we review the barycentric form in classical rational interpolation.
The representation turns out to be advantageous for revealing common factors
and unattainable points, although some care must be taken in correctly
interpreting this result. Besides classical rational interpolation, we also review
some other popular uses of the barycentric form, in particular a pole
free rational interpolant in barycentric form with high approximation orders.
Lastly, we examine the numerical properties of the barycentric representation
more closely. A careful analysis shows that the barycentric form is not
suited for use outside the interpolation interval. Basically, the evaluation of
the barycentric formula may suffer from catastrophic cancellation for large x.
2.1 Classical rational interpolation
Any rational function rn(x) of degree at most n in numerator and denominator,
interpolating the n+1 points in Xn can be written in the barycentric
form [BBM05]
n
rn(x) =
i=0
n
i=0
15
ui
x − xi
ui
x − xi
fi
. (2.1)