The aim of this work is to develop a fast algorithm for approximating the matrix function $f(A)$ of a square matrix $A$ that is symmetric and has hierarchically semiseparable (HSS) structure. Appearing in a wide variety of applications, often in the context of discretized (fractional) differential and integral operators, HSS matrices have a number of attractive properties facilitating the development of fast algorithms. In this work, we use an unconventional telescopic decomposition of $A$, inspired by recent work of Levitt and Martinsson on approximating an HSS matrix from matrix-vector products with a few random vectors. This telescopic decomposition allows us to approximate $f(A)$ by recursively performing low-rank updates with rational Krylov subspaces while keeping the size of the matrices involved in the rational Krylov subspaces small. In particular, no large-scale linear system needs to be solved, which yields favorable complexity estimates and reduced execution times compared to existing methods, including an existing divide-and-conquer strategy. The advantages of our newly proposed algorithms are demonstrated for a number of examples from the literature, featuring the exponential, the inverse square root, and the sign function of a matrix. Even for matrix inversion, our algorithm exhibits superior performance, even if not specifically designed for this task.

翻译：本研究旨在开发一种快速算法，用于近似计算对称且具有层次半可分离（HSS）结构的方阵$A$的矩阵函数$f(A)$。HSS矩阵广泛出现在各类应用场景中，常与离散化（分数阶）微分和积分算子相关，其诸多优良特性为快速算法的开发提供了便利。受Levitt与Martinsson近期关于通过少量随机向量进行矩阵-向量乘积来近似HSS矩阵的研究启发，本文采用一种非传统的 telescopic 分解方法。该分解允许我们通过有理Krylov子空间递归执行低秩更新来近似$f(A)$，同时确保有理Krylov子空间涉及的矩阵规模保持较小。特别地，该方法无需求解大规模线性系统，因而相较于现有方法（包括现有分治策略）具有更优的复杂度估计与更短的计算时间。通过文献中多个算例验证，本算法在矩阵指数、逆平方根及符号函数计算中均展现出优势。即便针对非专门设计的矩阵求逆任务，本算法仍表现出卓越性能。