|James Wanless on Sat, 12 Mar 2011 17:07:28 +0100|
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|Re: Polrootsmod query|
Hello Bill,Thanks very much once again for your valued help. pls see below for response(s)...
J On 12 Mar 2011, at 15:06, Bill Allombert wrote:
On Sat, Mar 12, 2011 at 02:03:40PM +0000, James Wanless wrote:Just realized I (probably?) sent this to the wrong list - I'll try and get it right in future!Why is PARI's implementation of polrootsmod so fast??? :))) I've implemented my own version using GMP and Algorithm 1.6.1 of Cohen ACCANT, but PARI's seems order(s) of magnitude faster. What algorithm does polrootsmod use, pls? Maybe Cantor-Zassenhaus split etc., ie Cohen Algorithms 3.4.x, using matrices?? I've looked at PARI's code and haven't (yet?) understood it, despite noticing a few references to 'Berlekamp'...Hello James, THe relevant C function is FpX_roots_i.
I am not totally clear (?) - would it be feasible to work from 'factormod' or 'factorcantor' and easily generate the roots from the results these give? [math question as opposed to coding question]. If so, would this be of a similar (or even better?) efficiency algorithm- wise, even though going via this route, ie via _factorization_, rather than just root-finding directly, a more complex algorithm involving matrices is needed?
PARI use Cantor-Zassenhauss and follow essentially Algorithm 1.6.1.
Interesting... I wonder why your version is so much quicker than mine, then.
Most of the time is spend in the computation of (X mod (p,P))^p (see remark (2) page 38). For that you need one of the algorithms of section 1.2 (binary powering).
Yes I believe I am already doing this correctly (I _was_ aware of remark 2, and have carefully checked I am not falling foul of that)
You also need to implement fast polynomial arithmetic over Fp[X].
Maybe that's it... would you say, in general, that there's a lot of optimization going on in PARI that could be having a large positive effect in this respect?
Note that PARI 2.4.3 is faster than PARI 2.3.5.
Which might seem to suggest 'yes' to my question immediately above.