  Rational Krylov subspaces have become a reference tool in dimension reduction
procedures for several application problems. When data matrices are symmetric,
a short-term recurrence can be used to generate an associated orthonormal
basis. In the past this procedure was abandoned because it requires twice the
number of linear system solves per iteration than with the classical long-term
method. We propose an implementation that allows one to obtain key rational
subspace matrices without explicitly storing the whole orthonormal basis, with
a moderate computational overhead associated with sparse system solves. Several
applications are discussed to illustrate the advantages of the proposed
procedure.
