  In this note, we extend the Vandermonde with Arnoldi method recently
advocated by P. D. Brubeck, Y. Nakatsukasa and L. N. Trefethen 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 P. D. Brubeck and L. N. Trefethen. The other approach
orthogonalizes the Hermite data. About which approach is preferable to another,
we made the comparison, and the result is problem dependent.
