[Numpy-discussion] Reduced row echelon form
jason-sage@creativetra...
jason-sage@creativetra...
Tue Nov 18 17:17:00 CST 2008
Anne Archibald wrote:
> 2008/11/18 Robert Young <rob@roryoung.co.uk>:
>
>
>> Is there a method in NumPy that reduces a matrix to it's reduced row echelon
>> form? I'm brand new to both NumPy and linear algebra, and I'm not quite sure
>> where to look.
>>
>
> Unfortunately, reduced row-echelon form doesn't really work when using
> approximate values: any construction of the reduced row-echelon form
> forces you many times to ask "is this number exactly zero or just very
> small?"; if it's zero you do one thing, but if it's very small you do
> something totally different - usually divide by it. With
> floating-point numbers, every calculation is approximate, and such a
> method will blow up completely. If you really need reduced row echelon
> form, you have to start with exact numbers and use a package that does
> exact computations (I think SymPy might be a place to start).
>
I also recommend Sage here (http://www.sagemath.org). For example, here
is a session in which we calculate the reduced echelon form of a matrix
over the rationals (QQ):
sage: a=matrix(QQ,[[1,2,3],[4,5,6],[7,8,9]])
sage: a.echelon_form()
[ 1 0 -1]
[ 0 1 2]
[ 0 0 0]
Sage has quite a bit of advanced functionality for exact linear
algebra. (It also uses numpy as a backend to provide some nice
functionality for approximate linear algebra). Here is a short tutorial
on constructions in linear algebra:
http://sagemath.org/doc/const/node28.html
Jason
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