The probability that a slightly perturbed numerical analysis problem is difficult

Burgisser, Peter and Cucker, Felipe and Lotz, Martin (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
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Dr. Martin Lotz
Date Deposited:	23 Oct 2012
Last Modified:	20 Oct 2017 14:13
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1903

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