逼近最近的结构化奇异矩阵多项式
Approximating the closest structured singular matrix polynomial
AI总结：
本文提出一种基于梯度系统数值积分的迭代方法，利用 Frobenius 范数距离逼近最近的结构化奇异矩阵多项式，支持多种结构约束与扰动限制。
摘要：
考虑矩阵多项式 $P(\lambda)= A_0 + \lambda A_1 + \ldots + \lambda^d A_d$，其中 $A_0,\ldots, A_d$ 为具有特定结构的复（或实）矩阵。本文讨论了一种迭代方法，利用 Frobenius 范数诱导的距离，数值逼近最近的结构化奇异矩阵多项式 $\widetilde P(\lambda)$。该方法的一个重要特点是能够包含不同类型的结构约束。该方法还允许将扰动限制在少数矩阵上，并包含额外的结构，例如保留一个或多个矩阵 $A_i$ 的稀疏模式，以及类似回文结构的集体属性。该迭代方法基于与一个合适的泛函相关的梯度系统的数值积分，该泛函量化了矩阵多项式到奇异的距离。
英文摘要：
Consider a matrix polynomial $P \left( λ\right)= A_0 + λA_1 + \ldots + λ^d A_d$, 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 $\widetilde P\left( λ\right)$, 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.
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