PS. Apologies for my apparent inability to compute the determinant of
a 2x2 integer matrix correctly!
David
On 6 September 2012 14:03, David Loeffler <d.a.loeff...@warwick.ac.uk> wrote:
> On 6 September 2012 13:28, Cindy <cindy425192...@gmail.com> wrote:
>> Hi, David,
>>
>> Thanks for your explanation about the minimize function in sage. I didn't
>> realize it's only for differentiable functions.
>>
>> For the stuff regarding lattice, I think there may be some misunderstanding
>> here.
>>
>> What I want is to find the minimum of a lattice.
>>
>> A lattice L can be defined as
>>
>> L={x=\lambda M| \lambda\in Z^m},
>>
>> where M is the generator matrix of L and the gram matrix of L is equal to
>> MM^T.
>
> OK, I've never heard of this definition but if you want to take that
> to be the definition that's up to you -- apparently for you all
> lattices come with a fixed embedding into Euclidean space. But that
> then changes the interpretation of your previous question, because in
> your previous thread I assumed you wanted a Gram matrix, and that is
> what the code I suggested calculates; the lattices coming from trace
> pairings on number fields won't have any preferred embedding into
> Euclidean space. To get *a* generator matrix (in your sense) from the
> Gram matrix, you could use Cholesky decomposition, for example. But to
> do this you will have to introduce square roots all over the place and
> hence the computation becomes inexact; it is far simpler to just work
> with the Gram matrix, which will be integer-valued in the examples
> you've mentioned so far.
>
> To find the shortest vector, you might want to use some of the
> routines in Sage's quadratic forms module.
>
> David
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