# 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 125×125 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.