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

2023/09/03 by Miryam Gnazzo, Gnazzo, Miryam, Nicola Guglielmi +1
Computer Science · Mathematics · #15A18 #47A56 #65F99 #65K05 #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2309.01220

openalex publication_date 2023/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

Given a matrix-valued function F(λ)=∑_i=1^d f_i(λ) A_i, with complex matrices A_i and f_i(λ) entire functions for i=1,…,d, we discuss a method for the numerical approximation of the distance to singularity of F(λ). The closest singular matrix-valued function \widetildeF(λ) 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 ( \widetildeF(λ) ) 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.

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