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.Keywords condition number · stochastic perturbation theory · weak condition number · polynomial eigenvalue problem · singular matrix polynomial Mathematics Subject Classification (2010) 15A15 · 15A18 · 15B52 · 60H99 · 65F15 · 65F35
IntroductionThe condition number of a computational problem f : V → W measures the sensitivity of f with respect to perturbations in the input. If the problem is differentiable, then the condition number is the norm of the derivative of f . In the case of solving systems of linear equations, the idea of conditioning dates back at least to the work of von Neumann and Goldstine [43] and Turing [41], who coined the term. For an algorithm computing f in finite precision arithmetic, the importance of the condition number κ stems from the "rule of thumb" popularized by N. J. Higham [27, §1.6], forward error κ · (backward error).The backward error is small if the algorithm computes the exact value of f at a nearby input, and a small condition number would certify that this is enough to get a small overall error. Higham's rule of thumb comes from a first order expansion, and in practice it often holds as an approximate equality and is valuable for practitioners who wish to predict the accuracy of numerical computations. Suppose that a solution is computed with, say, a backward error equal to 10 −16 . If κ = 10 2 then one would trust the computed value to have (at least) 14 meaningful decimal digits.The condition number can formally still be defined when f is not differentiable, though it may not be finite. In particular, if f is discontinuous at an input, then the condition number is +∞; a situation