Computing the closest singular matrix polynomial
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Given a matrix polynomial P ( λ)= A_0 + λ A_1 + … + λ^d A_d, with A_0,…, A_d complex (or real) matrices, in this paper we discuss an iterative method to compute the closest singular matrix polynomial P( λ), using the distance induced by the Frobenius norm. An important peculiarity of the approach we propose is the possibility to limit the perturbations to just a few matrices (we recall that in many examples some of the matrices may have a topological/combinatorial function which does not allow to change them) and also to include structural constraints, as the preservation of the sparsity pattern of one or more matrices A_i, as well as collective-like properties, like a palindromic structure. The iterative method is based on the numerical integration of the gradient system associated with a suitable functional which quantifies the distance to singularity of a matrix polynomial.
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