Hi all,
I'm wondering if anyone is aware of an efficient mechanism by which to derive the null space of a "large" symbolic matrix. Here, large means on the order of 10^2 rows, not necessarily square, where element (i,j) is a polynomial in symbolic parameters p_1,...,p_n, n also on the order of 10^2.
Conventional desktop symbol toolboxes like Matlab, Maxima, and Mathematica tend to break down at this problem size. Any pointers most welcome! Thanks.
Try to specialize some of the variables. Given enough specializations, one can then reconstruct kernel elements by interpolation.
Another method: try to use series expansions in terms of the variables and lift the order of such series expansions.
Another useful trick (when working with rational coefficients or coefficients in a number field) is working over finite fields since large determinant computations use up huge amounts of memory. Use then either the Hensel lemma (working over $p-$adics) or combine the information coming from different primes in order to lift the solution to $\mathbb Z$ or $\mathbb Q$.
Last ressort: If everything fails and if you are really interested in just one very specific example, write a C-program for just your example using the best strategy you know and hope that it works.
There is probably no universally optimal way to do this, I guess you have to make advantage of any special features of your example.
