Computing with quasiseparable matrices

TL;DR

This paper explores the properties and algorithms for quasiseparable matrices, introducing new structured representations and efficient algorithms for computing their orders, decompositions, and matrix operations, with applications in linear algebra.

Contribution

It presents novel algorithms for determining quasiseparable orders and introduces two new structured matrix representations with improved computational efficiency.

Findings

Algorithms compute quasiseparable orders in quadratic time.

New structured representations reduce storage from quadratic to linearithmic or linear.

Matrix-vector and matrix-matrix multiplications are performed efficiently using these representations.

Abstract

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^($\omega$--2)) where s = max(rL, rU) and $\omega$ 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^($\omega$--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^($\omega$--2)).