An iterative approach to barycentric rational Hermite interpolation

Abstract

In this paper we study an iterative approach to the Hermite interpolation problem, which first constructs an interpolant of the function values at $n+1$ nodes and then successively adds m correction terms to fit the data up to the mth derivatives. In the case of polynomial interpolation, this simply reproduces the classical Hermite interpolant, but the approach is general enough to be used in other settings. In particular, we focus on the family of rational Floater–Hormann interpolants, which are based on blending local polynomial interpolants of degree d with rational blending functions. For this family, the proposed method results in rational Hermite interpolants, which depend linearly on the data, with numerator and denominator of degree at most $(m+1)(n+1)-1$ and $(m+1)(n-d)$, respectively. They converge at the rate of $O(h^{(m+1)(d+1)})$ as the mesh size h converges to 0. After deriving the barycentric form of these interpolants, we prove the convergence rate for $m=1$ and $m=2$, and show that the approximation results compare favourably with other constructions.