Barycentric Hermite Interpolation

Abstract

Let $z_{1},\ldots,z_{K}$ be distinct grid points. If $f_{k,0}$ is the prescribed value of a function at the grid point $z_{k}$ and $f_{k,r}$ the prescribed value of the $r$th derivative, for $1\leq r\leq n_{k}-1$, the Hermite interpolant is the unique polynomial of degree $N-1$ ($N=n_{1}+\cdots+n_{K}$) which interpolates the prescribed function values and function derivatives. We obtain another derivation of a method for Hermite interpolation recently proposed by Butcher et al. [Numer. Algorithms, 56 (2011), pp. 319--347]. One advantage of our derivation is that it leads to an efficient method for updating the barycentric weights. If an additional derivative is prescribed at one of the interpolation points, we show how to update the barycentric coefficients using only $\mathcal{O}(N)$ operations. Even in the context of confluent Newton series, a comparably efficient and general method to update the coefficients appears not to be known. If the method is properly implemented, it computes the barycentric weights with fewer operations than other methods and has very good numerical stability even when derivatives of high order are involved. We give a partial explanation of its numerical stability.