We use results in [M. Crouzeix and A. Greenbaum,Spectral sets: numerical range and beyond, SIAM Jour. Matrix Anal. Appl., 40 (2019), pp. 1087-1101] to derive a variety of K-spectral sets and show how they can be used in some applications. We compare the K-values derived here to those that can be derived from a straightforward application of the Cauchy integral formula, by replacing the norm of the integral by the integral of the resolvent norm. While, in some cases, the new upper bounds on the optimal K-value are much tighter than those from the Cauchy integral formula, we show that in many cases of interest, the two values are of the same order of magnitude, with the bounds from the Cauchy integral formula actually being slightly smaller. We give a partial explanation of this in terms of the numerical range of the resolvent at points near an ill-conditioned eigenvalue.

翻译：我们利用[M. Crouzeix 和 A. Greenbaum 的成果，《谱集：数值范围及其拓展》，SIAM 矩阵分析与应用杂志，第40卷 (2019)，第1087-1101页]推导出一系列K-谱集，并展示它们在某些应用中的使用方式。我们将本文导出的K值与通过直接应用柯西积分公式（将积分范数替换为预解式范数的积分）所得K值进行比较。虽然在部分情形下，最优K值的新上界远优于柯西积分公式给出的上界，但我们证明在许多感兴趣案例中，两者具有相同量级，实际上柯西积分公式的界值略小。我们根据接近病态特征值点处的预解式数值范围，对此现象给出了部分解释。