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Re: Smith Form


From: Laurent Decreusefond
Subject: Re: Smith Form
Date: Thu, 18 Oct 2007 08:43:20 +0200

My bigger goal is to compute the first Betti number of the R-Cech complex of a random configurations in [0,1]x[0,1]. It means, I have 1)-to generate the random positions of a random number (Poisson) of points,
2)-compute the matrix of distance between pair of points,
3)-for a fixed R, find all the pair of points lying within this distance R (this involves concatenation of list of couples) 4)-find all the triple of points (x_1,x_2,x_3) such that d(x_i,x_j) <R, i=1,2,3,j=1,2,3 (again concatenation of list of triples) 5)-compute the representative matrices of the boundary operators (again list manipulation). The result are matrices with coefficients -1,1 or 0. The first matrix has one -1 and one 1 per column. The other has two 1 and one -1 per column. So they are very sparse. 6)-at last compute the rank of these two matrices, which is easily done once they are in Smith Form.

I know pretty well how to do 1-5 in Octave but not in Sage. I am looking for a program doing Smithification.

Actually, since my first post, P. Sondergaard indicated me this smith.m program

http://www.univie.ac.at/nuhag-php/mmodule/download.php?id=12

I didn't have time yet to test it but certainly will. I also scanned the GAP documentation and saw that there exists also in this soft some list-manipulation tools. GAP also has a package optimized to smithify sparse integer matrices, which is exactly what happened to be done in my kind of problem.

But if I could just combine what I know in Octave and what GAP knows about smithification, that would be the easiest and most efficient way to proceed.


Laurent,



Le 18 oct. 07 à 02:07, Jordi Gutiérrez Hermoso a écrit :

On 17/10/2007, Laurent Decreusefond <address@hidden> wrote:

Le 16 oct. 07 à 20:47, Jordi Gutiérrez Hermoso a écrit :

On 16/10/2007, Laurent Decreusefond <address@hidden>
wrote:
I looked through the documentation and find no occurrence of the term
Smith Form so my question is : does there exist a package which
computes the Smith normal form of a matrix with integer
coefficients ?


The Smith normal form seems in general to be of little use in
numerical analysis. Perhaps it is better suited for Maxima,


I know very well how to generate a random matrix in
octave but not in Maxima/GAP or any other discrete math oriented soft
and the reverse is true for the computation of the Smith form.

What are you trying to do? How do you need the Smith normal form in
Octave? What's the bigger goal?

It's not difficult to write a Smithification algorithm. We could write
one and submit it Octave-forge if it's absolutely indispensable. Do
you think we should?

- Jordi G. H.

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