拟可分矩阵的计算
Computing with quasiseparable matrices
摘要：
拟可分矩阵类是由完全位于其严格下三角部分和严格上三角部分的极大子矩阵的秩上的一对界（称为拟可分阶）定义的。这些自然出现在应用程序中，例如带矩阵的逆，并被广泛使用，因为它们允许结构化表示，允许用它们在时间上线性地计算维度和二次的准可分阶。本文给出了[Dumas \& al. ISSAC'15]中提出的拟可分性概念与秩轮廓矩阵不变量之间的联系。这使我们能够提出一个算法，计算准可分阶（rL，rU）在时间O（n^2 s^（$ω$ -2）），其中s = max（rL，rU）和$ω$矩阵乘法的指数。然后，我们提出了两个新的结构化表示，PLUQ分解的二叉树，和Bruhat生成器，分别使用O（ns log n/s）和O（ns）字段元素，而不是O（ns^2）以前已知的生成器。我们提出了在时间O（n^2 s^（$ω$ --2））内计算这些表示的算法。这些表示允许矩阵-向量乘积在时间上与其表示的大小成线性。最后，我们展示了如何在时间O（n^2 s^（$ω$ -2））内将两个这样的结构矩阵相乘。
英文摘要：
The class of quasiseparable matrices is defined by a pair of bounds, called the quasiseparable orders, on the ranks of the maximal sub-matrices entirely located in their strictly lower and upper triangular parts. These arise naturally in applications, as e.g. the inverse of band matrices, and are widely used for they admit structured representations allowing to compute with them in time linear in the dimension and quadratic with the quasiseparable order. We show, in this paper, the connection between the notion of quasisepa-rability and the rank profile matrix invariant, presented in [Dumas \& al. ISSAC'15]. This allows us to propose an algorithm computing the quasiseparable orders (rL, rU) in time O(n^2 s^($ω$--2)) where s = max(rL, rU) and $ω$ the exponent of matrix multiplication. We then present two new structured representations, a binary tree of PLUQ decompositions, and the Bruhat generator, using respectively O(ns log n/s) and O(ns) field elements instead of O(ns^2) for the previously known generators. We present algorithms computing these representations in time O(n^2 s^($ω$--2)). These representations allow a matrix-vector product in time linear in the size of their representation. Lastly we show how to multiply two such structured matrices in time O(n^2 s^($ω$--2)).
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