Computing the closest singular matrix polynomial

Given a matrix polynomial $P \left( \lambda \right)= A_0 + \lambda A_1 + \ldots + \lambda^d A_d$, with $A_0,\ldots, A_d$ complex (or real) matrices, in this paper we discuss an iterative method to compute the closest singular matrix polynomial $\widetilde P\left( \lambda \right)$, 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.