In a current research project, I have a number of matrices with coefficients in ℚ[𝑥] for which I want to understand how their rank depends on the value of the parameter 𝑥.
These matrices are:
- sparse (a typical example is a matrix of size 160x200 with less than 10 non-zero entries per row),
- rather 'regular' (each entry is a polynomial of degree at most one), and relatively
- close to full rank (in the above example of 160x200 rank is at least 150 no matter what the parameter is).
I would like to find software that can try to speed up calculations knowing at least some of these special features. So far I mostly tried Magma, and the calculations get endlessly long very fast for the sizes of matrices that I need. (I would also appreciate advice on Magma since perhaps I am forgetting some options that one should specify, I just tried the default versions of HermiteForm and ElementaryDivisors).