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Approximating the closest structured singular matrix polynomial

Miryam Gnazzo, Nicola Guglielmi

TL;DR

The paper develops a gradient-flow framework to compute the structured distance to singularity for regular matrix polynomials $P(\lambda)=\sum_{i=0}^d\lambda^i A_i$, under the Frobenius norm. It introduces a two-level scheme: an inner constrained gradient flow on a structure subspace $\mathcal{S}$ to minimize a smooth objective $G_{\varepsilon}$, and an outer Newton-bisection step to find the minimal perturbation magnitude $\varepsilon^*$ that renders $P+\Delta P$ singular. The approach accommodates a variety of structures (per-coefficient linear subspaces, fixed coefficients, sparsity, and palindromic relations) and also extends to the problem of finding the nearest polynomial with a common kernel, all while providing a posteriori bounds via established algebraic techniques. Numerical experiments demonstrate tight upper bounds on the distance to singularity under different structural constraints and offer practical insights into the method’s robustness and applicability to DAEs and other structured systems.

Abstract

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.

Approximating the closest structured singular matrix polynomial

TL;DR

The paper develops a gradient-flow framework to compute the structured distance to singularity for regular matrix polynomials , under the Frobenius norm. It introduces a two-level scheme: an inner constrained gradient flow on a structure subspace to minimize a smooth objective , and an outer Newton-bisection step to find the minimal perturbation magnitude that renders singular. The approach accommodates a variety of structures (per-coefficient linear subspaces, fixed coefficients, sparsity, and palindromic relations) and also extends to the problem of finding the nearest polynomial with a common kernel, all while providing a posteriori bounds via established algebraic techniques. Numerical experiments demonstrate tight upper bounds on the distance to singularity under different structural constraints and offer practical insights into the method’s robustness and applicability to DAEs and other structured systems.

Abstract

Consider a matrix polynomial , with 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 , 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 , 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.
Paper Structure (15 sections, 11 theorems, 112 equations, 1 table)

This paper contains 15 sections, 11 theorems, 112 equations, 1 table.

Table of Contents

  1. Introduction
  2. An associated optimization problem
  3. Structured distance to singularity
  4. Methodology
  5. The inner iteration
  6. The outer iteration
  7. Fixed coefficients
  8. Linear structure
  9. Structure on the whole polynomial
  10. Computational approach and numerical examples
  11. Numerical examples
  12. Computing the nearest singular polynomial with common kernel
  13. Numerical examples
  14. A posteriori bounds to singularity
  15. Conclusions and further work

Key Result

Lemma 3.2

Consider a continuously differentiable path of matrices $D(t)\in\mathbb{C}^{p \times q}$ for $t$ in an open interval $I$. Let $\sigma(t)$, $t\in I$, be a path of simple singular values of $D(t)$, with $\sigma(t) \neq 0$, for all $t \in I$. Let $u(t)$ and $v(t)$ be left and right singular vectors of

Theorems & Definitions (35)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 25 more