Re: [Caml-list] Integral solutions of rational linear equations

From: Sebastian Egner <sebastian.egner@philips.com>
To: Thomas Gazagnaire <thomas.gazagnaire@irisa.fr>
Cc: caml-list@yquem.inria.fr, caml-list-bounces@yquem.inria.fr
Subject: Re: [Caml-list] Integral solutions of rational linear equations
Date: Wed, 30 Nov 2005 09:32:12 +0100	[thread overview]
Message-ID: <OF3052ADF4.D88702E4-ONC12570C9.002B4D99-C12570C9.002F0C5A@philips.com> (raw)
In-Reply-To: <438CA20C.2080200@irisa.fr>

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Thomas,

Ocaml comes with a library for arbitrary-precision rational arithmetics;
the library is called 'Num' and its documentation can be found from here:

        http://caml.inria.fr/pub/docs/manual-ocaml/manual036.html

Now for computing integer solutions of rational equations you might
want to keep in mind that Gauss reduction suffers from exponential
growth of the entries because every row operation on the matrix
can---and usually does---double the number of bits in the numbers.
For this reason, Gaussian elimination and simple Hermite NF algorithms
are only useful if the number of equations is very low (<< 10, say).

Otherwise, you need to use more advanced algorithms (e.g. solving
several "mod m" images of the equation and reconstructing using the
Chinese Remainder Theorem), but I am not aware of any 'off the shelf'
implementation of this functionality in Ocaml.

As an entry point for the algorithms consider this page from the manual
of the computer algebra system Magma:

        http://magma.maths.usyd.edu.au/magma/htmlhelp/text602.htm#5489

Sebastian.

----
Dr. Sebastian Egner
Senior Scientist
Philips Research Laboratories
Prof. Holstlaan 4 (WDC 1-051, 1st floor, room 51)
5656 AA Eindhoven
The Netherlands
tel:       +31 40 27-43166
fax:      +31 40 27-44004
email: sebastian.egner@philips.com

Thomas Gazagnaire <thomas.gazagnaire@irisa.fr> 
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Subject
[Caml-list] Integral solutions of rational linear equations
Classification

Hello,

in my one of my Ocaml programs, I need to find all integers solutions of 
a rational equation systems. This algo uses Gauss reduction and Hermite 
normal form, and need to know if a rational is an integer or not (ie. I 
don't want to use numerical approximation : (1/3) * 3 is an integer but 
0,333333*3 we don't know). I didn't find any integer algebra package for 
ocaml, so I tried to implement Gauss elimination (easy) and Hermite 
normal form (more difficult...). But I didn't implement optimized 
version of these algorithms...

So my question is : do you know if exists a native ocaml module or an 
interface with a C library which is able to do integer/rational matrix 
manipulation (essentialy the Hermite normal form) in an efficient way ?

Thomas

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