Describe sd2-padicArithmetic here.
---------- Forwarded message ---------- Date: Wed, 27 Sep 2006 12:26:36 -0400 From: David Harvey <[email protected]> To: Iftikhar Burhanuddin <[email protected]> Cc: [email protected] Subject: Re: [SAGEdev] p-adic nonsense? On Sep 27, 2006, at 10:42 AM, Iftikhar Burhanuddin wrote: > BTW can you make a todo list (more than just bugs ... we have trac for > that) for p-adic arithmetic (possibly on the SD2 wiki page)? It'll > come in > handy when we discuss overhauling the current p-adic nonsense :) at > Seattle. The reason I'm working on the bugs now is that I need p-adics to work a bit better for the p-adic heights application. I propose the following long-term plan for p-adics in SAGE: (1) Research how other people have done it, both *implementation* and *interface*. This includes studying MAGMA, PARI, and that local fields library that was mentioned on this list a few weeks back. Possibly others that I don't know about. It also includes a literature search, both in mathematics and computer science. It includes analyses of performance, both theoretical and maybe even actual performance. (2) Come up with our own ideal interface. The interface should support an implementation which takes into account (at least) the following issues: (a) Obviously, fast arithmetic, for small p, large p, p = 2, large precision, small precision... all bases must be covered. (b) Fast conversions between p-adic integers, p-adic rationals, plain integers and rationals, and the quotient rings Z/p^n Z (wherever these conversions make sense) (c) Finite extensions of p-adic fields... heh... ramified and unramified cases... :-). I want to be able to take a general number field and construct the local field corresponding to a selected prime ideal. (d) Fast polynomial arithmetic with p-adic coefficients (this is much trickier than it sounds at first... you can throw all the usual algorithms out the window if the precision of the coefficients varies a lot) (3) Throw out the current p-adic implementation/interface entirely, and quickly write a new, inefficient (but correct!) implementation in python which has the *right interface*. I'm not at all convinced that our current interface (prec, ordp, big_oh, etc) is the right one. In fact I often find it quite confusing. Is this what MAGMA does? (4) Later, we can write the efficient version, either in C/Pyrex ourselves, or using an external library. I think it's crucial that someone (who knows what they are doing) puts in the hard work in step (1). Otherwise someone is going to waste a lot of time on the other steps. William I think mentioned earlier that MAGMA has already been through three or four implementations (even interfaces!) for p-adics. You can stick any of this on the wiki if you like. David ---------- Forwarded message ---------- Date: Wed, 27 Sep 2006 10:59:10 -0700 (PDT) From: Iftikhar Burhanuddin <[email protected]> To: David Harvey <[email protected]> Cc: [email protected], Iftikhar Burhanuddin <[email protected]> Subject: Re: [SAGEdev] p-adic nonsense? > (1) Research how other people have done it, both *implementation* and > *interface*. This includes studying MAGMA, PARI, and that local > fields library that was mentioned on this list a few weeks back. > Possibly others that I don't know about. It also includes a > literature search, both in mathematics and computer science. It > includes analyses of performance, both theoretical and maybe even > actual performance. > (b) Fast conversions between p-adic integers, p-adic rationals, plain > integers and rationals, and the quotient rings Z/p^n Z (wherever > these conversions make sense) For the theory the first place to look at is von zur Gathen and Gerhard's excellent Modern Computer Algebra http://www-math.uni-paderborn.de/mca/. I was searching for references on fast p-adic arithmetic for my torsion comp work and Pierrick Gaudry suggested the aforementioned book. It has detailed time complexity analysis for all the algorithms they discuss and implementation timings for some (I think integer multiplication, etc.) For example: Integers to p-adic integers is done efficiently (using fast integer multiplication) by something called radix conversion. This is discussed in MCA, Thm 9.17: An integer of size N can be expressed p-adically using a recursive procedure in O(M(N log p) log N) time, where M(N) is the number of bit operations required to multiply two numbers of size N. (M(N)=O(N log N log log N) = soft-O(N).) Another place to look at would be the standard reference Cohen's computational books but he does do complexity analysis a la MCA. > You can stick any of this on the wiki if you like. Done. http://wiki.sagemath.org/sd2-padicArithmetic Ifti.