Floater–Hormann barycentric interpolant

Let $n\ge 1$, let $x_0<\cdots <x_n$ be distinct real nodes, let $f_0,\dots ,f_n\in \mathbb{R}$ be data, and let $d$ be an integer with $0\le d<n$. For $j=0,\dots ,n$, define $w_j=(-1{)}^{j-d}{\sum }_{i=\max (0,j-d)}^{\min (j,n-d)}{\prod }_{\begin{array}{c}i\le k\le i+d\\ k\mathrm{\ne }j\end{array}}\mathrm{\mid }{x}_j-x_k{\mathrm{\mid }}^{-1}\mathrm{.}$ The index range is nonempty and every product is positive, so every $w_j$ is nonzero. The Floater–Hormann interpolant of degree parameter $d$ is the barycentric rational quotient of these nodes, data, and weights from Barycentric rational quotient with support values and reduced poles. Away from the nodes, exact evaluation uses the quotient with this denominator; a zero of the reduced denominator is reported as pole status. The no-real-pole theorem below proves that this status never occurs for admissible exact real inputs.