[GAP Forum] Use of GAP for Homological Algebra

[Shaun Van Ault]

Dear GAP forum,
   In my PhD research, I have been using GAP to calculate the homology 
of various types of chain complex, essentially by setting up the 
differential maps as matrices (with rational or integral entries), then 
applying NormalFormIntMat iteratively.  For low-dimensional chain groups 
(i.e., differential matrices on the order of 100 x 100), this process 
works quite well and quickly.  However, I have recently found it 
necessary to construct differential matrices on the order of 10000 x 
10000 and significantly larger.  The computations for homology get 
bogged down considerably.  Even computing rationally (i.e., just making 
computations of ranks of matrices instead of doing a full SNF), I run 
out of resources.
   My question is this:  Is there a method for computing rank of a 
sparse matrix using much less resources than for a dense matrix?  The 
large differential matrices I am constructing are incredibly sparse. 
Additionally, are there any existing methods for dealing with spectral 
sequence calculations?
   Here is a link to the preprint outlining the thesis and the work done 
by myself and my advisor, Zbigniew Fiedorowicz.  The homology 
calculations in question would be those of the chain complexes 
Sym_*^{(p)} defined on p.6.

http://www.math.ohio-state.edu/~ault/Papers/symmetric_arxiv-1.pdf

  Thank you,
   -Shaun Ault