Recursive determinantal framework for testing D-stability. I

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.