Confluent Vandermonde with Arnoldi

In this note, we extend the Vandermonde with Arnoldi method recently advocated by Brubeck et al. (2021) to dealing with the confluent Vandermonde matrix. To apply the Arnoldi process, it is critical to find a Krylov subspace which generates the column space of the confluent Vandermonde matrix. A theorem is established for such Krylov subspaces for any order derivatives. This enables us to compute the derivatives of high degree polynomials to high precision. It also makes many applications involving derivatives possible, as illustrated by numerical examples. We note that one of the approaches orthogonalizes only the function values and is equivalent to the formula given by Brubeck and Trefethen (2022). The other approach orthogonalizes the Hermite data. About which approach is preferable to another, we made the comparison, and the result is problem dependent.