Barycentric Hermite Interpolation

The Hermite interpolation problem—defined in the article text—is more complicated than the Lagrange interpolation problem—also defined there—and occurs less frequently in practice. But it does occur, and solving it is occasionally useful. Solutions have been reinvented many times since the problem was first posed and solved in 1878 by Charles Hermite. This article shows how the barycentric forms of the solution, invented about a hundred years after Hermite, work. All one needs to do is to compute a partial fraction expansion by a numerically stable method, and this gives us numerically stable and efficient forms to evaluate the Hermite interpolational polynomial. I describe the Maple program BHIP and some of its ancillary routines (available for download from the Maple Cloud, by clicking on the link below the link to the article PDF, to the right of this abstract), and mention the equivalent Matlab versions genbarywts and hermiteeval. I also compare to some less numerically stable and less efficient approaches. I also show how to find the roots of a Hermite interpolational polynomial by constructing a companion matrix pencil with the routine CMP, which does not change the polynomial basis, and then using standard software to compute the generalized eigenvalues of the pencil, which then give us the roots of the polynomial.