Given a square pencil $A+ \lambda B$, where $A$ and $B$ are $n\times n$ complex (resp. real) matrices, we consider the problem of finding the singular complex (resp. real) pencil nearest to it in the Frobenius distance. This problem is known to be very difficult, and the few algorithms available in the literature can only deal efficiently with pencils of very small size. We show that the problem is equivalent to minimizing a certain objective function $f$ over the Riemannian manifold $SU(n) \times SU(n)$ (resp. $SO(n) \times SO(n)$ if the nearest real singular pencil is sought), where $SU(n)$ denotes the special unitary group (resp. $SO(n)$ denotes the special orthogonal group). This novel perspective is based on the generalized Schur form of pencils, and yields competitive numerical methods, by pairing it with { algorithms} capable of doing optimization on { Riemannian manifolds. We propose one algorithm that directly minimizes the (almost everywhere, but not everywhere, differentiable) function $f$, as well as a smoothed alternative and a third algorithm that is smooth and can also solve the problem} of finding a nearest singular pencil with a specified minimal index. We provide numerical experiments that show that the resulting methods allow us to deal with pencils of much larger size than alternative techniques, yielding candidate minimizers of comparable or better quality. In the course of our analysis, we also obtain a number of new theoretical results related to the generalized Schur form of a (regular or singular) square pencil and to the minimal index of a singular square pencil whose nullity is $1$.

翻译：给定一个方束 $A+ \lambda B$，其中 $A$ 和 $B$ 是 $n\times n$ 的复（或实）矩阵，我们考虑在 Frobenius 距离下寻找与其最近的奇异复（或实）束的问题。该问题已知非常困难，现有文献中的少数算法仅能有效处理非常小尺寸的束。我们证明该问题等价于在黎曼流形 $SU(n) \times SU(n)$（若寻找最近的实奇异束，则为 $SO(n) \times SO(n)$）上最小化某个目标函数 $f$，其中 $SU(n)$ 表示特殊酉群（$SO(n)$ 表示特殊正交群）。这一新颖视角基于束的广义 Schur 形式，并利用能够进行黎曼流形优化的算法，从而产生具有竞争力的数值方法。我们提出一种算法直接最小化（几乎处处可微，但并非处处可微的）函数 $f$，以及一种平滑替代方案和第三种光滑算法，该算法还能解决寻找具有指定最小指标的最近奇异束的问题。数值实验表明，我们所提出的方法能够处理比现有技术大得多的束，并产生质量相当或更优的候选极小值。在分析过程中，我们还获得了一系列与（正则或奇异）方束的广义 Schur 形式以及零度为 1 的奇异方束的最小指标相关的新理论结果。