2026/07/17 by Vanni Noferini, Xuzhou Zhan
#math.NA #cs.NA
Assessing the asymptotic stability of linear self-adjoint homogeneous systems of differential-algebraic equations requires testing the Hurwitz stability of the associated Hermitian matrix polynomial P(λ). Tests for known necessary and sufficient conditions rely on linearizations and eigensolvers, solving matrix equations and testing matrix inequalities, or generalized Bézoutians, and scale with either O(d2 n3) or O(d3n3) complexity, where d and n are the degree and size of P(λ), respectively. We establish several novel sufficient conditions for stability, based on the numerical range of P(λ). Based on the new results, we propose algorithms with O(d n3) asymptotic complexity. Our methods rely on very efficient core numerical linear algebra routines, such as the Cholesky decomposition of n × n matrices or the computation of the largest eigenvalue of n × n definite pencils. Therefore, a significant computational advantage can be expected in favor of the proposed approach even for moderate values of d or n, and we verify this with numerical experiments.