GSoC with Flint – Week 8
This week I’ve been working on improving the computation of the nullspace of an integer by implementing a p-adic lifting based method. The algorithm is also described in Stein’s book on computing with modular forms and is closely related to Dixon’s p-adic method for linear system solving. This is pretty much finished now (modulo some weird memory leaks and determining the best values for parameters such as p-adic precision and range of primes used) and does appear to be asymptotically faster than the multimodular algorithm I worked on before, though experiments suggest that currently the matrix needs to be fairly large before the p-adic algorithm becomes better.
I was originally thinking of implementing Pauderis and Storjohann’s algorithm if I had time this week, but have spent much of the past week looking at efficient nullspace computation (and being plagued a little by unexpected power cuts!) so haven’t got much further than a skeleton implementation. Maybe I can flesh this out at some point but for the next week I’m going to move on to algorithms for the Smith normal form, another important normal form for matrices over the integers, where both column and row operations are allowed.
Hopefully building on what I’ve done so far I should be able to get some fast Smith normal form algorithms implemented shortly, indeed one simple approach is to repeatedly transpose and compute Hermite normal forms until the resulting matrix is diagonal, this won’t necessarily be the Smith form but it can then be computed with relative ease. Something more sophisticated than this will undoubtedly be best, but many SNF algorithms involve computing the HNF at some point and working from there so the current HNF algorithms provide a good basis for those computations.