Given two square matrices $A$ and $B$, we propose a new approach for computing the smallest value $\varepsilon \geq 0$ such that $A+E$ and $A+F$ share an eigenvalue, where $\|E\|=\|F\|=\varepsilon$. In 2006, Gu and Overton proposed the first algorithm for computing this quantity, called $\mathrm{sep}_\lambda(A,B)$ ("sep-lambda"), using ideas inspired from an earlier algorithm of Gu for computing the distance to uncontrollability. However, the algorithm of Gu and Overton is extremely expensive, which limits it to the tiniest of problems, and until now, no other algorithms have been known. Our new algorithm can be orders of magnitude faster and can solve problems where $A$ and $B$ are of moderate size. Moreover, our method consists of many "embarrassingly parallel" computations, and so it can be further accelerated on multi-core hardware. Finally, we also propose the first algorithm to compute an earlier version of sep-lambda where $\|E\| + \|F\|=\varepsilon$.
翻译：以两平方基体$A$和$B$,我们提出一种新的方法来计算最小值$varepsilon\ geq 0美元,这样一来,美元+E$和美元+F$共享一个egenvalue,美元+E ⁇ F ⁇ F ⁇ varepsilon$。2006年,Gu和Overton提出了计算这一数量的第一个算法,称为$mathrm{sep ⁇ lambda(A,B),使用古古早期算法的灵感来计算离失控的距离。然而,Gu和Overton的算法非常昂贵,它仅限于最小的问题,而且直到现在还不知道其他算法。我们的新算法可以更快的排序,解决美元和B$之间的问题。此外,我们的方法包括许多“惊人的平行”计算,因此可以进一步加速多核心硬件的计算。最后,我们还提议第一种算法来计算早期版本的Sep-lambda $___\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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