  The problem of computing recurrence coefficients of sequences of rational
functions orthogonal with respect to a discrete inner product is formulated as
an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two
procedures are proposed to solve this inverse eigenvalue problem, via the
rational Arnoldi iteration and via an updating procedure using unitary
similarity transformations. The latter is shown to be numerically stable. This
problem and both procedures are generalized by considering biorthogonal
rational functions with respect to a bilinear form. This leads to an inverse
eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil
implies short recurrence relations for the biorthogonal rational functions,
which is more efficient than the orthogonal case. However the procedures
solving this problem must rely on nonunitary operations and might not be
numerically stable.
