Recursive determinantal framework for testing D-stability. I

Olga Y. Kushel

Pith reviewed 2026-05-10 08:46 UTC · model grok-4.3

classification 🧮 math.SP cs.NAmath.NA

keywords stabilitymatrixalgorithmmathbfrecursiverelationstestingapplications

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The pith

A recursive algorithm generates a hierarchy of sufficient conditions for matrix D-stability via recurrence relations on principal minors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

D-stability means that multiplying a matrix by any positive diagonal matrix keeps all eigenvalues with negative real parts. For small matrices this can be checked directly, but for larger ones it is difficult. The authors build an algorithm that repeatedly deletes or zeros entries in a structured way, creating a tree of simpler matrices whose determinants satisfy recurrence relations. These relations are then turned into inequalities on the principal minors that, if satisfied, guarantee D-stability. Numerical tests are said to show the method runs in practice.

Core claim

we propose a recursive delete/zero algorithm for testing matrix D-stability. The algorithm generates a binary tree of parameter-dependent matrices A_s and yields recurrence relations for the real and imaginary parts of det(A_s). These relations lead to a hierarchy of sufficient for D-stability conditions, expressed in terms of principal minors.

Load-bearing premise

That the recurrence relations derived from the delete/zero operations on the binary tree correctly produce sufficient (and practically useful) conditions for D-stability without missing cases or requiring post-hoc adjustments.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a recursive delete/zero algorithm for testing matrix D-stability. The algorithm constructs a binary tree of parameter-dependent matrices A_s and derives recurrence relations for the real and imaginary parts of det(A_s). These relations are used to generate a hierarchy of sufficient conditions for D-stability expressed in terms of principal minors. The abstract states that numerical experiments confirm the practical feasibility of the approach.

Significance. If the recurrence relations are shown to be correct and the resulting conditions are both sufficient and computationally useful for n>4, the framework could provide a new algorithmic route to sufficient tests for D-stability, a problem whose full characterization remains open. The recursive structure based on determinant recurrences is potentially extensible, but its value depends on verification that the derived conditions are non-vacuous and correctly capture the D-stability requirement.

major comments (2)

[Derivation of recurrence relations (around the binary-tree construction and the Re/Im recurrences)] The sufficiency claim rests on the recurrence relations for Re(det(A_s)) and Im(det(A_s)) obtained from the delete/zero operations on the binary tree. No explicit verification of these relations against direct expansion of det(DA) is provided even for n=3 or n=4, where D-stability is completely characterized by known criteria on principal minors. Any sign error or omitted branch in the recurrences would invalidate the hierarchy of sufficient conditions.
[Numerical experiments section] The abstract asserts that numerical experiments confirm feasibility, yet the manuscript supplies no description of the test matrices, the dimensions examined, the specific sufficient conditions evaluated, or any error analysis. Without these details it is impossible to assess whether the hierarchy yields non-trivial tests or merely reproduces known low-dimensional criteria.

minor comments (2)

Notation for the binary tree indices s and the parameter-dependent matrices A_s should be introduced with a small illustrative example (e.g., n=2) before the general recurrence is stated.
The paper is labeled “Part I”; a brief forward reference to what will be treated in subsequent parts would help readers understand the scope of the present claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The approach rests on standard properties of determinants and matrix stability; the binary tree and recurrence relations are algorithmic inventions without independent external validation in the abstract.

axioms (2)

standard math Determinant recurrence relations hold under the delete/zero operations on parameter-dependent matrices
Invoked to generate the relations for real and imaginary parts of det(A_s)
domain assumption Satisfaction of the derived minor inequalities implies D-stability
Central to turning the recurrences into sufficient conditions

invented entities (1)

Binary tree of parameter-dependent matrices A_s no independent evidence
purpose: To recursively reduce the D-stability test
Core construct of the delete/zero algorithm

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discussion (0)

Reference graph

Works this paper leans on

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