拟可分矩阵的精确计算
Exact computations with quasiseparable matrices
摘要：
拟可分矩阵是一类秩结构矩阵，广泛应用于数值线性代数，在计算机代数中也越来越受到人们的关注，例如在多项式矩阵的线性化中的应用。各种表示格式存在的thesematrices很少被compared.We显示如何最核心的格式SSS和HSS可以适应符号计算，其中确切的秩thresholds基于数值秩。我们澄清了它们的链接并将其与Bruhat格式进行了比较。为此，我们陈述了它们基于快速矩阵乘法的空间和时间成本估计，并将它们与它们的前导常数进行比较。在Bruhat格式的基础上，我们提出了一种基于Crout消去法的Bruhat格式的生成算法，该算法专门研究了由稀疏矩阵或Bruhat表示之和构造Bruhat格式的快速算法。
英文摘要：
Quasi-separable matrices are a class of rank-structured matriceswidely used in numerical linear algebra and of growing interestin computer algebra, with applications in e.g. the linearization ofpolynomial matrices. Various representation formats exist for thesematrices that have rarely been compared.We show how the most central formats SSS and HSS can beadapted to symbolic computation, where the exact rank replacesthreshold based numerical ranks. We clarify their links and comparethem with the Bruhat format. To this end, we state their space andtime cost estimates based on fast matrix multiplication, and comparethem, with their leading constants. The comparison is supportedby software experiments.We make further progresses for the Bruhat format, for which wegive a generation algorithm, following a Crout elimination scheme,which specializes into fast algorithms for the construction from asparse matrix or from the sum of Bruhat representations.
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