Year: 2026
Type: preprint
Abstract: Finding the distance to singularity for a matrix is a ubiquitous problem in numerical linear algebra, and is elegantly solved by the Eckart-Young-Mirsky theorem. Its structured variant naturally emerges when one considers structured matrices, and wants to preserve their structure. Recent work has shown that this problem is particularly important for a class of matrix nearness problems that either entirely or partly reduce to a structured distance to singularity problem. In this work, we propose a new framework for addressing this problem, based on the concept of singular vector spaces, that is, linear subsets of the set of singular matrices. We analyze singular vector spaces in the context of this problem, prove new results, and detail how a specific subfamily of singular vector spaces can be incorporated into a practical algorithm. The resulting algorithm is based on globally minimizing a certain objective function alternatingly in its arguments. Numerical experiments demonstrate that this new algorithm, in many cases, offers a significant speedup over the state of the art, while the quality of the output remains comparable. This makes it possible to solve problems of much larger size than what was previously possible.
Source: arXiv (Cornell University)
Author: Nyman, Lauri
Institution: University of Manchester
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7133355331
Cites:
Cited by:
Related to: