On the numerical stability of barycentric rational interpolation
Chiara Fuda  1@  , Kai Hormann  2  , Rosanna Campagna  3  
1 : Universitá della Svizzera Italiana
2 : Universitá della Svizzera Italiana
3 : Università degli studi della Campania "Luigi Vanvitelli"

The barycentric forms of polynomial and rational interpolation have recently gained popularity, because they can be computed with simple, efficient, and numerically stable algorithms. In this talk, we show more generally that the evaluation of any function that can be expressed as $r(x)=\sum_{i=0}^n a_i(x) f_i\big/\sum_{j=0}^m b_j(x)$ in terms of data values $f_i$ and some functions $a_i$ and $b_j$ for $i=0,\dots,n$ and $j=0,\dots,m$ with a simple algorithm that first sums up the terms in the numerator and the denominator, followed by a final division, is forward and backward stable under certain assumptions. This result includes the two barycentric forms of rational interpolation as special cases. Our analysis further reveals that the stability of the second barycentric form depends on the Lebesgue constant associated with the interpolation nodes, which typically grows with $n$, whereas the stability of the first barycentric form depends on a similar, but different quantity, that can be bounded in terms of the mesh ratio, regardless of $n$. We support our theoretical results with numerical experiments, which indicate that the first barycentric form is stable even in situations where the second barycentric form is completely unstable.

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