Efficient derivation of null space of large symbolic matrices? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:44:15Z http://mathoverflow.net/feeds/question/23490 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23490/efficient-derivation-of-null-space-of-large-symbolic-matrices Efficient derivation of null space of large symbolic matrices? Paul 2010-05-04T21:42:09Z 2010-05-05T08:22:22Z <p>Hi all,</p> <p>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.</p> <p>Conventional desktop symbol toolboxes like Matlab, Maxima, and Mathematica tend to break down at this problem size. Any pointers most welcome! Thanks.</p> http://mathoverflow.net/questions/23490/efficient-derivation-of-null-space-of-large-symbolic-matrices/23560#23560 Answer by Roland Bacher for Efficient derivation of null space of large symbolic matrices? Roland Bacher 2010-05-05T08:22:22Z 2010-05-05T08:22:22Z <p>Try to specialize some of the variables. Given enough specializations, one can then reconstruct kernel elements by interpolation.</p> <p>Another method: try to use series expansions in terms of the variables and lift the order of such series expansions.</p> <p>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\$.</p> <p>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.</p> <p>There is probably no universally optimal way to do this, I guess you have to make advantage of any special features of your example.</p>