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- Title
A randomized algorithm for testing nonsingularity of structured matrices with an application to asserting nondefectivity of Segre varieties.
- Authors
Vannieuwenhoven, Nick; Vandebril, Raf; Meerbergen, Karl
- Abstract
Tensors admitting an expression as the sum of at most s rank 1 tensors can be considered points of the sth order secant variety of a Segre variety, yet the most basic invariant of this variety—its dimension—is not yet fully understood. A conjecture was nevertheless proposed by Abo et al. (2009, Induction for secant varieties of Segre varieties. Trans. Amer. Math. Soc., 361, 767–792), which was proved to be correct for s≤6. We propose a numerical randomized algorithm for testing whether a mathematically exact and structured matrix is singular, requiring only the availability of an approximate matrix–vector product. Using this method, we test whether the aforementioned conjecture is true. The proposed method requires several orders of magnitude less memory than approaches based on symbolic arithmetic, thus greatly increasing the number of varieties that can be handled. Our experiments establish that the Segre varieties embedded in with expected generic rank s≤55 obey the conjecture; the probability that a defective variety is incorrectly classified as nondefective is less than <10−55.
- Subjects
ALGORITHMS; MATRICES (Mathematics); VECTOR algebra; MATHEMATICAL analysis; NUMERICAL analysis
- Publication
IMA Journal of Numerical Analysis, 2015, Vol 35, Issue 1, p289
- ISSN
0272-4979
- Publication type
Article
- DOI
10.1093/imanum/drt069