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The probability that a slightly perturbed numerical analysis problem is difficult
Zitatschlüssel BCL-The-Probability-That-A-Slightly-Perturbed-Numerical-Analysis-Problem-Is-Difficult
Autor Peter Bürgisser and Felipe Cucker and Martin Lotz
Seiten 1559-1583
Jahr 2008
Journal Math. Comp.
Jahrgang 77
Notiz Warning: unfortunately, due to an error in the production of the paper, binomial coefficients have been replaced by fractions at several places in the journal version! For a correct version see arXiv math/0610270
Zusammenfassung 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 $ \varepsilon$-tubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a spherical disk of radius σ. Besides 𝜀 and σ, this bound depends only on the dimension of the sphere and on the degree of the defining equations.
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