树拟可分矩阵：顺序与层次半可分表示的同时推广
Tree quasi-separable matrices: a simultaneous generalization of sequentially and hierarchically semi-separable representations
AI总结：
本文提出树拟可分（TQS）矩阵，统一并推广了顺序半可分（SSS）与层次半可分（HSS）表示，证明其在加法、乘法和求逆下保持优良代数性质，并给出快速算法构造。
摘要：
我们提出对文献中称为顺序半可分（SSS）和层次半可分（HSS）矩阵表示的统一与推广。针对邻接图为树的稀疏矩阵（的逆），我们描述其秩结构化表示，并证明这些所谓的树拟可分（TQS）矩阵在加法、乘法和求逆运算下继承了SSS和HSS的所有优良代数性质。为得到这些性质，我们证明了一个关键结果，刻画了任意稠密矩阵转换为TQS表示的过程。在此，我们通过显式构造过程具体表明，生成元大小由矩阵的某些Hankel块的秩决定。与SSS和HSS类似，只要生成元大小较小，TQS矩阵就支持快速的矩阵-向量乘法和直接求解器。我们给出了相关算法的概要。
英文摘要：
We present a unification and generalization of what is known in the literature as sequentially and hierarchically semi-separable (SSS and HSS) representations for matrices. Describing rank-structured representations of (inverses of) sparse matrices whose adjacency graph is a tree, it is shown that these so-called tree quasi-separable (TQS) matrices inherit all the favorable algebraic properties of SSS and HSS under addition, products, and inversion. To arrive at these properties, we prove a key result that characterizes the conversion of any dense matrix into a TQS representation. Here, we specifically show through an explicit construction procedure that the generator sizes are dictated by the ranks of certain Hankel blocks of the matrix. Analogous to SSS and HSS, TQS matrices admit fast matrix-vector products and direct solvers provided the generator sizes are small. A sketch of the associated algorithms is provided.
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