  The problem of barycentric Hermite interpolation is highly susceptible to
overflows or underflows. In this paper, based on Sturm-Liouville equations for
Jacobi orthogonal polynomials, we consider the fast implementation on the
second barycentric formula for higher order Hermite-Fej\'{e}r interpolation at
Gauss-Jacobi or Jacobi-Gauss-Lobatto pointsystems, where the barycentric
weights can be efficiently evaluated and cost linear operations corresponding
to the number of grids totally. Furthermore, due to the division of the second
barycentric form, the exponentially increasing common factor in the barycentric
weights can be canceled, which yields a superiorly stable method for computing
the simplified barycentric weights, and leads to a fast implementation of the
higher order Hermite-Fej\'{e}r interpolation with linear operations on the
number of grids. In addition, the convergence rates are derived for
Hermite-Fej\'{e}r interpolation at Gauss-Jacobi pointsystems.
