# GSoC with Flint – Week 3

This week I’ve been able to get started working on my GSoC project full time after finishing my university exams on Tuesday (hooray!).

I finished off what I was working on at the time of the last update, a more efficient algorithm that still computes the Hermite normal form in a straightforward way but takes less time by using the extended Euclidean algorithm to reduce rows rather than simply repeatedly reducing one row using another. The coefficients can still grow very fast in this algorithm which leaves it highly non-optimal in practice however small changes can be made to reduce this problem such as changing the order in which rows are reduced (this is described by Bradley here).

One way to mitigate the coefficient explosion problem is to perform the computations modulo some number and then use the result of this computation to find the actual HNF. If we use a multiple of the determinant of the HNF matrix we are trying to find (which we can in fact determine from the original matrix) the real HNF can be found fairly easily and this is what I have been working on for the past couple of days. This method of finding the HNF is due to Domich, Kannan and Trotter and the paper describing it can be found here. The function I have implemented seems to be working fine on the test cases it is given but I am getting some memory leaks that I’m really having trouble getting to the bottom of which is my main problem currently.

Once this is done I’ll start working on a multimodular algorithm which will involve computing the HNF modulo several primes and combining the resulting matrices via the Chinese remainder theorem.