Smoothed analysis of componentwise condition numbers for sparse matrices
نویسندگان
چکیده
منابع مشابه
Smoothed Analysis of Condition Numbers
The running time of many iterative numerical algorithms is dominated by the condition number of the input, a quantity measuring the sensitivity of the solution with regard to small perturbations of the input. Examples are iterative methods of linear algebra, interior-point methods of linear and convex optimization, as well as homotopy methods for solving systems of polynomial equations. Thus a ...
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Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain smoothed analysis estimates for elements in this class depending only on geometric invariants of the corresponding sets of ill-posed inputs. These estimates are for...
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We provide estimates on the volume of tubular neighborhoods around a subvariety Σ of real projective space, intersected with a disk of radius σ . The bounds are in terms of σ , the dimension of the ambient space, and the degree of equations defining Σ . We use these bounds to obtain smoothed analysis estimates for some conic condition numbers. To cite this article: P. Bürgisser et al., C. R. Ac...
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Let A be any matrix and let A be a slight random perturbation of A. We prove that it is unlikely that A has large condition number. Using this result, we prove it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we bound the smoothed precision needed by Gaussian elimination without pivoting. Our results improve the average-case ...
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denote its Euclidean operator norm (often called the 2-norm). If is nonsingular, then its condition number () is defined by () = kk°°−1°° = 1() () where 1 ≥ 1 ≥ ≥ ≥ 0 are the singular values of . The s constitute lengths of the semi-axes of the hyperellipsoid = { : kk = 1} in -dimensional space; thus measures elongation of at its extreme [1]. The role that ...
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ژورنال
عنوان ژورنال: IMA Journal of Numerical Analysis
سال: 2014
ISSN: 0272-4979,1464-3642
DOI: 10.1093/imanum/drt068