Dear all, I am hoping for some advice about efficiency and good practice to implement the algebras I am building in GAP. The overall method is building a larger space, then finding a quotient which satisfies my axioms.
I have a vector space W over the rationals of (at least) several hundred dimensions. It has a small subspace V (of dimension < sqrt(dim(W))) which is endowed with an algebra product V x V -> W. To implement this product, I keep a list of lists of elements of W, i.e. structure constants a table---call it M---which has ~dim(W)^2 entries in the rationals. The product of two vectors u, v is the naive implementation: u * v = Sum( u[i] * v[j] * M[i][j] ). Typically M has not many zero entries, and few entries which are entirely zero vectors. I would estimate around 25% of the rationals are 0, but there are proportionally much fewer i,j for which M[i][j] is the zero vector. I (at one stage) keep track of unknowns in the algebra by unbound entries in M. There is more information than this, because the algebra is linked to a group, so I keep another list B whose elements I know are ordered to correspond with the basis elements. Often V, W (and M) will be modified. In particular, I take quotients by subspaces. Currently I am doing this manually, i.e. rewriting the vectors and tables myself. This is because I wanted to keep close control of the basis, and the documentation of NaturalHomomorphismBySubspace didn't seem to let me do that. Sometimes, I want to blow W up, maybe adding dim(V) or maybe increasing by an order of magnitude, and of course all of the old information should still be there, but the new dimensions correspond to new unknowns, so again I keep my basis under close control. I use quite a lot of products, so I am wondering if this is really a good way of doing things. Is this a good way to store and use the table M? I also have the action of my group G on W recorded; I take the generators of G, build corresponding dim(W) x dim(W) matrices (these are relatively sparse, but definitely in the rationals), and generate the matrix group from these, then find the GroupHomomorphismByImagesNC. I very often take orbits of vectors from W under the group G; the action is via this homomorphism. Is this sensible? I would be pleased to hear any suggestions. Also, of course I can provide any further information. Thanks, Felix _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
