Ordering algorithms for irreducible sparse linear systems

Abstract

Ordinary algorithms aim to pre-order a matrix in order to achieve a favourable structure for factorization. Two prototype structures are described which are commonly aimed at by current algorithms. A different structure, based on a lower Hessenberg form, is introduced. We show that the common structures may be obtained from the Hessenberg form in a simple way. A fundamental and very simple algorithm is presented for deriving the lower Hessenberg form. This algorithm is inherently a part of other common algorithms, but is usually obscured by other detailed heuristics. Some of these heuristics are seen to correspond to different tie-break rules in the fundamental algorithm. We describe a particularly simple tie-break rule used in SPK1 [11], which is effective and not at all well known. Ordinary algorithms need to be modified to enable pivoting for numerical stability to be carried out. We describe how the idea of threshold pivoting can be used in the context of these algorithms. Most ordering algorithms in common use employ some sort of look-ahead to promote a favourable structure. It is argued that look-ahead is generally ineffective in practice, and that numerical evidence supports the use of very simple strategies. A new ordering algorithm is presented which aims to obtain a narrower bandwidth in the lower Hessenberg form. Some limited numerical experience is presented which enables us to draw tentative conclusions about various ordering strategies, and how they compare with those in common use.