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Real root finding for determinants of linear matrices

Didier Henrion 1, 2 Simone Naldi 1, 3 Mohab Safey El Din 3
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
3 PolSys - Polynomial Systems
LIP6 - Laboratoire d'Informatique de Paris 6, Inria de Paris
Abstract : Let A_0 , A_1 , . . . , A_n be given square matrices of size m with rational coefficients. The paper focuses on the exact computation of one point in each connected component of the real determinantal variety {x ∈ R^n : det(A_0 + x_1 A_1 + · · · + x_n A_n) = 0}. Such a problem finds applications in many areas such as control theory, computational geometry, optimization, etc. Using standard complexity results this problem can be solved using m^{O(n)} arithmetic operations. Under some genericity assumptions on the coefficients of the matrices, we provide an algorithm solving this problem whose runtime is essentially quadratic in {{n+m}\choose{n}}^{3} . We also report on experiments with a computer implementation of this algorithm. Its practical performance illustrates the complexity estimates. In particular, we emphasize that for subfamilies of this problem where m is fixed, the complexity is polynomial in n.
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Didier Henrion, Simone Naldi, Mohab Safey El Din. Real root finding for determinants of linear matrices. Journal of Symbolic Computation, Elsevier, 2016, 74, pp.205-238. ⟨10.1016/j.jsc.2015.06.010⟩. ⟨hal-01077888v3⟩

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