Recursive determinantal framework for testing D-stability. I
For control theory and economics, this provides a practical test for D-stability in higher dimensions, though the conditions are only sufficient and the problem remains open.
The paper proposes a recursive algorithm for testing D-stability of matrices, a problem unsolved for dimensions n>4 since 1958. The algorithm generates recurrence relations for determinants, yielding sufficient conditions in terms of principal minors, with numerical experiments confirming feasibility.
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.