GSoC with Flint – Week 9
So this week I’ve been working on implementing the methods to find the Smith normal form of an integer matrix, rather than the Hermite normal form which I’ve worked on up until now. I’m mostly working off of the excellent paper of Saunders and Wan which describes an “engineered” algorithm to compute the Smith form which is highly practical. By engineered they mean that the algorithm switches between different methods based on practical observations regarding which method is likely to be the most efficient for different types of matrix, however the algorithm is structured in such a way that the asymptotic complexity remains as small as possible.
The algorithm is actually randomised of Monte Carlo type, meaning there is a controllable possibility of error present in the results. It becomes harder to be sure that the exponents of primes occurring in the Smith form are correct as the primes get smaller in the algorithm of Eberly, Giesbrecht and Villard (on which Saunders and Wan algorithm is based). So the computation is split into two parts, the smooth part concerning only primes less than a given limit and a rough part concerning all others. Currently I have an Eberly, Giesbrecht and Villard based routine working and finding rough parts fairly well and so I am currently working on a modular SNF algorithm to fill in the gap.
Saunders and Wan don’t stop with just this however, they also incorporate ideas described by Dumas, Saunders and Villard which use a so called valence based method which I would also like to add at some point. It seems this does require efficient computation of the minimal polynomial however, so I’m not sure how far I will get with this.