Systems of Diophantine equations

I found a reference for this:

<http://citeseer.nj.nec.com/aardal98solving.html>

It says they do lattice base reduction as a step to solving the equations.
Whereas we're trying to do a twist on base reduction. Here's a paper
about that:

<http://citeseer.nj.nec.com/248763.html>

It says "The Gram-Schmidt vectors, however, do not necessarily belong to
the lattice, but they do span the same real vector space as b1, ..., bi,
so they are used as a "reference" for the basis reduction algorithm."

Also defines a lattice: "The lattice L spanned by b1, ..., bi is the set
of vectors that can be obtained by taking integer linear combinations of
the vectors b1, ..., bi."

It looks like what we want is between the Gram-Schmidt space (integer
combinations) and lattice (real combinations) because it has rational
combinations. Or is that the same as Gram-Schmidt?

I don't know, it's starting to get complicated. I think I'll ignore it
for a while.

Graham

--- In tuning-math@y..., graham@m... wrote:
> I found a reference for this:
>
> <http://citeseer.nj.nec.com/aardal98solving.html>
>
> It says they do lattice base reduction as a step to solving the
equations.
> Whereas we're trying to do a twist on base reduction. Here's a
paper
> about that:

That's a fancy new method for an old classic problem, and presumably
becomes interesting mostly when the number of simultaneous
Diophantine equations are high. Can you solve a system of linear
equations over the rationals in Python?

In-Reply-To: <9v35h4+amle@eGroups.com>
In article <9v35h4+amle@eGroups.com>, genewardsmith@juno.com
(genewardsmith) wrote:

> That's a fancy new method for an old classic problem, and presumably
> becomes interesting mostly when the number of simultaneous
> Diophantine equations are high. Can you solve a system of linear
> equations over the rationals in Python?

With Numeric, you can solve for floating point numbers (using a wrapper
around the Fortran LAPACK) but there's no support for integers or
rationals.

Graham