厄米矩阵多项式的快速稳定性测试
Fast stability tests for Hermitian matrix polynomials
AI总结：
研究评估线性自伴齐次微分代数方程组渐近稳定性时厄米矩阵多项式\(P(\lambda)\)的稳定性测试问题，基于\(P(\lambda)\)数值范围建立新充分条件，提出\(O(d n^3)\)复杂度算法，经实验验证有计算优势。
AI中文摘要：
评估线性自伴齐次微分代数方程组的渐近稳定性需要测试相关厄米矩阵多项式\(P(\lambda)\)的赫尔维茨稳定性。已知的充要条件测试依赖于线性化和特征求解器、求解矩阵方程和测试矩阵不等式或广义贝祖蒂安，其复杂度为\(O(d^2 n^3)\)或\(O(d^3n^3)\)。我们基于\(P(\lambda)\)的数值范围建立了几个新的稳定性充分条件，并提出了渐近复杂度为\(O(d n^3)\)的算法。我们的方法依赖于高效的核心数值线性代数例程，通过数值实验验证了该方法的计算优势。
英文摘要：
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 $O(d^2 n^3)$ or $O(d^3n^3)$ 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 $O(d n^3)$ asymptotic complexity. Our methods rely on very efficient core numerical linear algebra routines, such as the Cholesky decomposition of $n \times n$ matrices or the computation of the largest eigenvalue of $n \times n$ 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.
