优化Rayleigh商商数,使其具有对称限制,并适用于干扰结构化多金属元元值问题 (Optimizing Rayleigh quotient with symmetric constraints and their applications to perturbations of the structured polynomial eigenvalue problem)

For a Hermitian matrix $H \in \mathbb C^{n,n}$ and symmetric matrices $S_0, S_1,\ldots,S_k \in \mathbb C^{n,n}$, we consider the problem of computing the supremum of $\left\{ \frac{v^*Hv}{v^*v}:~v\in \mathbb C^{n}\setminus \{0\},\,v^TS_iv=0~\text{for}~i=0,\ldots,k\right\}$. For this, we derive an estimation in the form of minimizing the second largest eigenvalue of a parameter depending Hermitian matrix, which is exact when the eigenvalue at the optimal is simple. The results are then applied to compute the eigenvalue backward errors of higher degree matrix polynomials with T-palindromic, T-antipalindromic, T-even, T-odd, and skew-symmetric structures. The results are illustrated by numerical experiments.

翻译：赫米提亚矩阵 $H $H\\ mathbb C ⁇ n,n}$ $和对称矩阵 $S_0, S_1,\ldots, S_k\ in\mathbb C ⁇ n, n}$, 我们考虑计算 left\\\\\ frac{v}Hv\v ⁇ vv} 的问题: ~v\in\mathbb C ⁇ ⁇ setminus ⁇ 0 ⁇,\\, v ⁇ TS_iv=0}text{for ⁇ i=0,\ldots, k\right}$。对于这一点, 我们得出一个估计, 以最大限度地减少取决于赫米提亚矩阵的参数的第二大电子值为形式。当最优值为简单时, 。其结果应用于计算高度矩阵多级矩阵、 T- contipalicromic, T- even, T-ddd, 和 skew-symymatical 结构的精度矩阵错误错误错误。。以数值实验为图示。