  We propose a new approach to the theory of conditioning for numerical
analysis problems for which both classical and stochastic perturbation theory
fail to predict the observed accuracy of computed solutions. To motivate our
ideas, we present examples of problems that are discontinuous at a given input
and have infinite classical and stochastic condition number, but where the
solution is still computed to machine precision without relying on structured
algorithms. Stimulated by the failure of classical and stochastic perturbation
theory in capturing such phenomena, we define and analyse a weak worst-case and
a weak stochastic condition number. This new theory is a more powerful
predictor of the accuracy of computations than existing tools, especially when
the worst-case and the expected sensitivity of a problem to perturbations of
the input is not finite. We apply our analysis to the computation of simple
eigenvalues of matrix polynomials, including the more difficult case of
singular matrix polynomials. In addition, we show how the weak condition
numbers can be estimated in practice.
