English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We show how to efficiently solve a clustering problem that arises in a methodto evaluate functions of matrices. The problem requires finding the connectedcomponents of a graph whose vertices are eigenvalues of a real or complexmatrix and whose edges are pairs of eigenvalues that are at most \delta awayfrom each other. Davies and Higham proposed solving this problem by enumeratingthe edges of the graph, which requires at least $\Omega(n^{2})$ work. We showthat the problem can be solved by computing the Delaunay triangulation of theeigenvalues, removing from it long edges, and computing the connectedcomponents of the remaining edges in the triangulation. This leads to an$O(n\log n)$ algorithm. We have implemented both algorithms using CGAL, amature and sophisticated computational-geometry software library, and wedemonstrate that the new algorithm is much faster in practice than the naivealgorithm. We also present a tight analysis of the naive algorithm, showingthat it performs $\Theta(n^{2})$ work, and correct a misrepresentation in theoriginal statement of the problem. To the best of our knowledge, this is thefirst application of computational geometry to solve a real-world problem innumerical linear algebra.

Geometric Sparsification of Closeness Relations: Eigenvalue Clustering for Computing Matrix Functions