We show how to efficiently solve a clustering problem that arises in a method to evaluate functions of matrices. The problem requires finding the connected components of a graph whose vertices are eigenvalues of a real or complex matrix and whose edges are pairs of eigenvalues that are at most \delta away from each other. Davies and Higham proposed solving this problem by enumerating the edges of the graph, which requires at least $\Omega(n^{2})$ work. We show that the problem can be solved by computing the Delaunay triangulation of the eigenvalues, removing from it long edges, and computing the connected components of the remaining edges in the triangulation. This leads to an $O(n\log n)$ algorithm. We have implemented both algorithms using CGAL, a mature and sophisticated computational-geometry software library, and we demonstrate that the new algorithm is much faster in practice than the naive algorithm. We also present a tight analysis of the naive algorithm, showing that it performs $\Theta(n^{2})$ work, and correct a misrepresentation in the original statement of the problem. To the best of our knowledge, this is the first application of computational geometry to solve a real-world problem in numerical linear algebra.

中文翻译：

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紧密关系的几何稀疏化：计算矩阵函数的特征值聚类

我们展示了如何有效地解决在评估矩阵函数的方法中出现的聚类问题。该问题需要找到一个图的连通分量，该图的顶点是实数或复数矩阵的特征值，边是成对的特征值，它们之间的距离至多为 δ。Davies 和 Higham 提出通过枚举图的边来解决这个问题，这至少需要 $\Omega(n^{2})$ 的工作。我们表明该问题可以通过计算特征值的 Delaunay 三角剖分、从中移除长边并计算三角剖分中剩余边的连通分量来解决。这导致了 $O(n\log n)$ 算法。我们已经使用 CGAL（一个成熟而复杂的计算几何软件库）实现了这两种算法，我们证明了新算法在实践中比朴素算法快得多。我们还对朴素算法进行了严格的分析，表明它执行了 $\Theta(n^{2})$ 工作，并纠正了问题原始陈述中的错误表述。据我们所知，这是计算几何首次应用于解决数值线性代数中的实际问题。