Consider 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 with a certain structure. In this paper we discuss an iterative method to numerically approximate the closest structured 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 include different types of structural constraints. The method also allows us to limit the perturbations to just a few matrices and also to include additional structures, such as the preservation of the sparsity pattern of one or more matrices $A_i$, and also 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.

翻译：考虑一个矩阵多项式 $P \left( \lambda \right)= A_0 + \lambda A_1 + \ldots + \lambda^d A_d$，其中 $A_0,\ldots, A_d$ 是具有特定结构的复（或实）矩阵。本文讨论一种迭代方法，用于数值近似在 Frobenius 范数诱导的距离下最近的结构奇异矩阵多项式 $\widetilde P\left( \lambda \right)$。我们所提方法的一个重要特点是能够包含不同类型的结构约束。该方法还允许我们将扰动限制在少数几个矩阵上，并能包含额外的结构，例如保持一个或多个矩阵 $A_i$ 的稀疏模式，以及集体性性质，如回文结构。该迭代方法基于对与特定泛函相关的梯度系统的数值积分，该泛函用于量化矩阵多项式到奇异的距离。