  Computing mountain passes is a standard way of finding critical points. We
describe a numerical method for finding critical points that is convergent in
the nonsmooth case and locally superlinearly convergent in the smooth finite
dimensional case. We apply these techniques to describe a strategy for the
Wilkinson problem of calculating the distance of a matrix to a closest matrix
with repeated eigenvalues. Finally, we relate critical points of mountain pass
type to nonsmooth and metric critical point theory.
