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Computing the Kreiss Constant of a Matrix
SIAM Journal on Matrix Analysis and Applications ( IF 1.5 ) Pub Date : 2020-01-01 , DOI: 10.1137/19m1275127
Tim Mitchell

We establish the first globally convergent algorithms for computing the Kreiss constant of a matrix to arbitrary accuracy. We propose three different iterations for continuous-time Kreiss constants and analogues for discrete-time Kreiss constants. With standard eigensolvers, the methods do $\mathcal{O}(n^6)$ work, but we show how this theoretical work complexity can be lowered to $\mathcal{O}(n^4)$ on average and $\mathcal{O}(n^5)$ in the worst case via divide-and-conquer variants. Finally, locally optimal Kreiss constant approximations can be efficiently obtained for large-scale matrices via optimization.

中文翻译:

计算矩阵的克赖斯常数

我们建立了第一个全局收敛算法,用于以任意精度计算矩阵的 Kreiss 常数。我们为连续时间 Kreiss 常数和离散时间 Kreiss 常数的类似物提出了三种不同的迭代。使用标准特征求解器,这些方法可以执行 $\mathcal{O}(n^6)$ 工作,但我们展示了如何将这种理论工作复杂度平均降低到 $\mathcal{O}(n^4)$ 和 $\ mathcal{O}(n^5)$ 在最坏的情况下通过分而治之的变体。最后,通过优化可以有效地为大规模矩阵获得局部最优的 Kreiss 常数近似值。
更新日期:2020-01-01
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