Structured distance to singularity as a nonlinear system of equations

In this article we study the structured distance to singularity for a nonsingular matrix $A\in\mathbb{C}^{n\times n}$, with a prescribed linear structure $\mathcal{S}$ (for instance, a sparsity pattern, or a real Toeplitz structure), i.e., the norm of the smallest perturbation $\Delta \in \mathcal{S}$, such that $A + \Delta$ is singular. This is an example of structured matrix nearness problem: a family of problems that arise in control and systems theory and in numerical analysis, when characterizing the robustness of a certain property of a system with respect to perturbations that are constrained to a certain structure (for example the structure of the nominal system). We start by highlighting the parallelism between two main tools which have been proposed in the literature: a gradient system approach for a functional in the eigenvalues, which requires the solution of certain low-rank matrix differential equations (see [Guglielmi, Lubich, Sicilia, SINUM 2023]), and a two-level optimization approach in which the inner linear least-squares problem is solved explicitly (see [Usevich, Markovsky, JCAM 2014] and [Gnazzo, Noferini, Nyman, Poloni, FoCM 2025]). In particular, these articles underline the remarkable property that $\Delta$ is (at least generically) the orthogonal projection onto the structure $\mathcal{S}$ of a rank-1 matrix $uv^*$. This property and the parallelism suggest a new reformulation of the problem into a system of nonlinear equations in the two vector unknowns $u,v \in\mathbb{C}^n$. We study this new formulation, and propose an algorithm to solve these nonlinear equations directly with the multivariate Newton's method. We discuss how to avoid the singularity of such system of nonlinear equations, and how to ensure monotonic convergence. The resulting algorithm is faster than the existing ones for large matrices, and maintains comparable accuracy.