Exact computations with quasiseparable matrices

TL;DR

This paper compares different exact representation formats for quasiseparable matrices, analyzing their efficiency and relationships, and introduces algorithms for their construction and manipulation in symbolic computation.

Contribution

It adapts SSS and HSS formats to symbolic computation, clarifies their links with Bruhat format, and provides algorithms for efficient generation and manipulation.

Findings

SSS and HSS formats can be adapted for symbolic computation.

The paper provides space and time cost estimates for these formats.

Algorithms for generating Bruhat format from sparse matrices are introduced.

Abstract

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.