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## 2012.107: The probability that a slightly perturbed numerical analysis problem is difficult

2012.107: Peter Burgisser, Felipe Cucker and Martin Lotz (2008) The probability that a slightly perturbed numerical analysis problem is difficult. Mathematics of Computation, 77. pp. 1559-1583. ISSN 1088-6842

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## Abstract

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 $\sigma$. Besides $\varepsilon$ and $\sigma$, this bound depends only on the dimension of the sphere and on the degree of the defining equations.

Item Type: Article MSC 2000 > 65 Numerical analysis 2012.107 Dr. Martin Lotz 23 October 2012