The probability that a small perturbation of a numerical analysis problem is difficult

Details The probability that a small perturbation of a numerical analysis problem is difficult The probability that a small perturbation of a numerical analysis problem is difficult

2006-10-09

arxiv.org/abs/math/0610270v1

Math. Comp. 77 (2008), 1559-1583

Mathematics

Numerical Analysis

30 pages, 4 figures

Scientific paper

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 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.

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