April 30, 2010, all from Alex Ghitza...

---
On Mon, 26 Apr 2010 07:36:27 -0700, William Stein  wrote:
>
> Wow, that is awesome.     I'll add a note about this to the website.
>
> Could you go further?  E.g., to 4096?
>

Definitely to 4096, it should only take me a couple of days (I just
tried 4096 itself and it took a bit less than 3.5 hours).

Here are a couple of unexpected (by me!) things that I discovered in the
process:

1. Linbox is disabled by default on 64-bit Linux for charpoly
computations because it sometimes hangs; I tried 4096 with "generic" and
with "linbox" and the difference in speed was minimal, with "linbox"
being only about 2% faster.  Naively, I would have thought it to be much
faster than that.

2. This is really impressive: I tried the same computation for 2796 in
Magma, and it's still running after 260 minutes, right now it's doing the
computation of the characteristic polynomial, so after this it will have
to test irreducibility.  So Sage is significantly faster at this; I'll
keep track of the timings for the individual steps, but it seems that
computing characteristic polynomials for large integer matrices with
pretty large entries is much quicker in Sage than in Magma.

I'll get the computation to 4096 started and let you know how it goes.
I'll save the characteristic polynomials to file as I go.  Is there
anything else that would be good to keep track of?

---


OK, so I've made it to 4096.  No counterexamples yet :)
I have saved the characteristic polynomials onto disk, it's about 13 Gb
of stuff.

The whole thing went fairly smoothly; out of about 550 weights, there
were about 25 that I had to run again (some a couple more times) because
they appeared to had hung.  I don't know which part of the computation
they choked on, but I can investigate this at some point.

I think this was a fairly serious test of Sage linear algebra over
(really big) integers, and it went pretty well.  I'm impressed!