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
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