Approximating the closest structured singular matrix polynomial

TL;DR

This paper introduces an iterative numerical method to find the closest structured singular matrix polynomial, accommodating various structural constraints and preserving specific matrix properties, using gradient system integration.

Contribution

It presents a novel iterative approach for approximating the nearest singular structured matrix polynomial with flexible structural constraints.

Findings

The method effectively handles different structural constraints.

It can limit perturbations to selected matrices.

The approach preserves additional structures like sparsity and palindromic forms.

Abstract

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.