GSoC with Flint - Week 7

This week I've been working on improving the algorithm of Pernet-Stein as much as possible. After introducing as many of the ideas given in the original paper as I could I found the main bottleneck to actually be the computation of the nullspace of a certain submatrix of the matrix given, this is needed in order to efficiently solve a linear system which (likely) has a nasty final row. If we know a basis for a nullspace of the first n-1 rows of the system we can replace the final row with a random nice (having small entries) row and then find a solution to the original system by adding on a suitable multiple of the nullspace basis vector (the nullspace should be 1 dimensional for random input).
Flint uses the reduced row echelon form of a matrix to compute nullspaces (the nullspace of a matrix in this form can be easily read off and the transformation does not alter it) and so a natural place to improve nullspace computation is to improve the row echelon form computation. We can use a multimodular approach for this problem (this is described in Stein's book on computing with modular forms) and I've achieved some nice speed-ups with this method in the past couple of days. For example the multimodular method is around 200x faster for 125x125 matrices with 32 bit entries. While this has made Hermite form computations a lot faster (nullspace computation is over 90% of the work for large matrices) I still want to try and see if this can be improved upon further, after all, in this case we don't need the whole row echelon form just a vector in the kernel and the knowledge that the nullspace is 1-dimensional. So I plan to work on this further in the next couple of days, and depending on how I feel about this approach I will either spend the rest of this week making improvements to Pernet-Stein or possibly work on implementing the algorithm of Pauderis and Storjohann.