The probability that a slightly perturbed numerical analysis problem is difficult

Peter Bürgisser, Felipe Cucker, Martin Lotz
2008 Mathematics of Computation  
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
more » ... lgebraic subvariety of a sphere, intersected with a disk of radius \sigma. Besides \epsilon and \sigma, this bound depends only the dimension of the sphere and on the degree of the defining equations.
doi:10.1090/s0025-5718-08-02060-7 fatcat:evzix4if55ax5ft6hf3354z2w4