Recursive determinantal framework for testing D-stability. I

TL;DR

This paper introduces a recursive algorithm to test matrix D-stability by generating a binary tree of matrices and deriving conditions based on principal minors, with numerical validation.

Contribution

It proposes a novel recursive delete/zero algorithm for testing D-stability, providing a hierarchy of sufficient conditions based on principal minors.

Findings

Algorithm generates a binary tree of parameter-dependent matrices.

Recurrence relations for determinants lead to stability conditions.

Numerical experiments confirm the approach's practical feasibility.

Abstract

The concept of matrix $D$-stability, introduced in 1958 by Arrow and McManus is of major importance due to the variety of its applications. However, characterization of matrix $D$-stability for dimensions $n > 4$ is considered as a hard open problem. In this paper, we propose a recursive delete/zero algorithm for testing matrix $D$-stability. The algorithm generates a binary tree of parameter-dependent matrices ${\mathbf A}_s$ and yields recurrence relations for the real and imaginary parts of $\det({\mathbf A}_s)$. These relations lead to a hierarchy of sufficient for $D$-stability conditions, expressed in terms of principal minors. Numerical experiments confirm the practical feasibility of the approach.