  In this paper, we consider the problem of efficiently computing the
eigenvalues of limited-memory quasi-Newton matrices that exhibit a compact
formulation. In addition, we produce a compact formula for quasi-Newton
matrices generated by any member of the Broyden convex class of updates. Our
proposed method makes use of efficient updates to the QR factorization that
substantially reduces the cost of computing the eigenvalues after the
quasi-Newton matrix is updated. Numerical experiments suggest that the proposed
method is able to compute eigenvalues to high accuracy. Applications for this
work include modified quasi-Newton methods and trust-region methods for
large-scale optimization, the efficient computation of condition numbers and
singular values, and sensitivity analysis.
