Solving Sparse Integer Linear Systems

Wayne Eberly (1) , Mark Giesbrecht (2) , Pascal Giorgi (2, 3) , Arne Storjohann (2) , Gilles Villard (4)

1 CPSC - Department of Computer Science [Calgary]
2 SCG - Symbolic Computation Group
3 LP2A - Laboratoire de Physique Appliquée et d'Automatique
4 LIP - Laboratoire de l'Informatique du Parallélisme

We propose a new algorithm to solve sparse linear systems of equations over the integers. This algorithm is based on a $p$-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox-based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art.

Mots clés: Sparse integer matrix; linear system solving; structured matrix

Domaines: Calcul formel [cs.SC]

Wayne Eberly, Mark Giesbrecht, Pascal Giorgi, Arne Storjohann, Gilles Villard. Solving Sparse Integer Linear Systems. 2006. ⟨hal-00021456⟩