6 Solving Sparse Integer Linear Systems
Wayne Eberly, Mark W. Giesbrecht, Pascal Giorgi, Arne Storjohann, Gilles Villard · 2006
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.