Jim Wilkinson discovered that the computation of zeros of polynomials is ill
conditioned when the polynomial is given by its coefficients. For many
problems we need to compute zeros of polynomials, but we do not necessarily
need to represent the polynomial with its coefficients. We develop
algorithms that avoid the coefficients. They turn out to be stable, however,
the drawback is often heavily increased computational effort. Modern
processors on the other hand are mostly idle and wait for crunching numbers
so it may pay to accept more computations in order to increase stability and
also to exploit parallelism. We apply the method for nonlinear eigenvalue
problems.