The probability that a slightly perturbed numerical analysis problem is difficult

We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of epsilon-tubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a spherical disk of radius sigma. Besides epsilon and sigma, this bound depends only on the dimension of the sphere and on the degree of the defining equations.