Monday, 14 July 2014

Week of 7/7 Update

This week I've been trying to speed up PolyDet, and to get it working with SmithFormIliopolus. Testing has revealed that, while asymptotically fast, the PolyDet code is way slower than any other code in the invariant factor pipeline. The bottleneck is at the evaluation stage, where we take $b^{2}$ polynomials of degree $d$ and evaluate them at about $bn$ points. By contrast the determinant and interpolation steps are relatively cheap. I suspect that the problem is in the GivaroPoly implementation of polynomial arithmetic. GivaroPoly and GivaroExtension extension both represent a polynomial by a std::vector. This means that a polynomial over an extension field is a vector of vectors and I suspect the doubly nested loops involved are killing performance. I'm looking into FLINT which also implements polynomial arithmetic in pure C and may be faster. Some other Linboxers have also been trying out FLINT and it sounds promising. I'm also looking into automatically switching to a Zech-log implementation which the extension field is sufficiently small, but this probably won't work for very large matrices with correspondingly large values of $d$ (roughly proportional to $\frac{n}{b}$).

I've also written a blackbox class which represents a matrix which we've been trying to get the invariant factors of. It has the structure of Pascal's triangle and the $(i,j)$ element is given by $c_{i+j}{i+j\choose{}j}$. It turns out that, with a bit of dynamic programming, ${n\choose{}k}$ can be computed in just 2 field operations using $O(n)$ space for precomputation. I took advantage of this and the regularity of the matrix to come up with a very space efficient representation that should help for the larger cases.

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