We provide numerical bounds on the Crouzeix ratiofor KLS matrices $A$ which have a line segment on the boundary of the numerical range. The Crouzeix ratio is the supremum over all polynomials $p$ of the spectral norm of $p(A)$ dividedby the maximum absolute value of $p$ on the numerical range of $A$.Our bounds confirm the conjecture that this ratiois less than or equal to $2$. We also give a precise description of these numerical ranges.

翻译：我们针对数值范围边界包含线段的KLS矩阵$A$，给出了Crouzeix比率的数值界。该比率定义为所有多项式$p$作用下$p(A)$谱范数与$p$在$A$数值范围上最大绝对值的比值上确界。本数值界证实了该比率不超过$2$的猜想。同时我们给出了这些数值范围的精确描述。