Monday, 23 June 2014

Week of 6/16 Update

This week I worked on improving parallel performance and on computing the Smith Normal Form (SNF) of matrix min-polys. I've finally got the entire pipeline of code up and running. Given a matrix supplied in Matrix Market Format I have a single program that will read it, compute the matrix min-poly in parallel, change the representation of the min-poly to a matrix entries over a polynomial ring, compute the SNF of that matrix and return the main diagonal. This particular program is in the Linbox repository under benchmarks/invariant-factors-benchmark. So far the results look good, with the last entry being the min-poly and the others looking like the correct invariant factors.

While this is a big step forward there's still a lot that needs to be done. In particular the algorithm I'm using for computing the SNF (The Kannan-Bachem algorithm) works great for matrices over the integers, but is horrendously slow for polynomial rings. I'm 90% sure that the problem is that it's generating intermediate values that are polynomials of enormous degree, and spending way too much time manipulating those. Fortunately, there's a way around this blow-up in polynomial degree, and that's to use the fact that all the computations of the SNF of $A$ can be done modulo $det(A)$. Unfortunately Linbox doesn't have a good way of finding the determinant of a polynomial matrix right now so that's my next step. Once I've got $det(A)$, I'll either extend the Kannan-Bachem class or just write a wrapper for the polynomial ring class that does everything modulo a given polynomial.

My basic strategy for computing $det(A)$ is to evaluate $A$ at $n$ points (over a field extension if there aren't enough distinct points in the base field), then compute $n$ determinant of these scalar matrices and finally interpolate the polynomial that goes through those points. This shouldn't be too hard, but the interpolation step might be a bit tricky since the usual way to do this (Inverse FFT) requires a root of unity with an order that's greater than $n$ and also a power of 2. This isn't always easy to find over an arbitrary field so I'm looking into alternative algorithms with less strict requirements for the interpolation points.

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