On the numerical approximation of the distance to singularity for matrix-valued functions

TL;DR

This paper presents a numerical method to approximate the distance to singularity for matrix-valued functions, extending techniques beyond polynomials to more general functions with structured matrices.

Contribution

It introduces an iterative approach to approximate the closest singular matrix-valued function, handling infinite roots and structural constraints.

Findings

Method effectively approximates the distance to singularity.

Handles functions with infinite roots in the determinant.

Supports structured matrices with sparsity patterns.

Abstract

Given a matrix-valued function $\mathcal{F}(\lambda)=\sum_{i=1}^d f_i(\lambda) A_i$, with complex matrices $A_i$ and $f_i(\lambda)$ entire functions for $i=1,\ldots,d$, we discuss a method for the numerical approximation of the distance to singularity of $\mathcal{F}(\lambda)$. The closest singular matrix-valued function $\widetilde{\mathcal{F}}(\lambda)$ with respect to the Frobenius norm is approximated using an iterative method. The property of singularity on the matrix-valued function is translated into a numerical constraint for a suitable minimization problem. Unlike the case of matrix polynomials, in the general setting of matrix-valued functions the main issue is that the function $\det ( \widetilde{\mathcal{F}}(\lambda) )$ may have an infinite number of roots. An important feature of the numerical method consists in the possibility of addressing different structures, such as sparsity patterns induced by the matrix coefficients, in which case the search of the closest singular function is restricted to the class of functions preserving the structure of the matrices.