Lattice Basis from VAL

Hi Gene,

I'm working with Monz and I have a question about turning bicommas
into a lattice basis.

I don't have a good method for determining a basis from the VAL.
What I can do is to generate a set of bicommas from the VAL. From
the bicommas, I can generate a number of combinations I call 1st
generation vectors. Based on the bicommas and 1st-generation
vectors, I test various combinations to find a combination that
forms a proper basis.

When the dimensionality is roughly 5 or greater, however, the
combinatorial search takes way too long. Do you know of a better
way of forming the initial basis? (I then use a modified KZ
algorithm to reduce it to a TM reduced.)

Cheers

Chris Wittmann

--- In tuning-math@yahoogroups.com, "outthewazoo2002" <chris@t...> wrote:

> I'm working with Monz and I have a question about turning bicommas
> into a lattice basis.

Do you mean biprime commas?

> I don't have a good method for determining a basis from the VAL.

I use the Hermite reduction function of Maple applied to the biprime
commas, but simply Gaussian elimination using integral row operations
should work.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "outthewazoo2002" <chris@t...>
wrote:
>
> > I'm working with Monz and I have a question about turning
bicommas
> > into a lattice basis.
>
> Do you mean biprime commas?

YES, I mean the biprime commas..

>
> > I don't have a good method for determining a basis from the VAL.
>
> I use the Hermite reduction function of Maple applied to the
biprime
> commas, but simply Gaussian elimination using integral row
operations
> should work.

Ahaaa. :-) Thanks for the tip. I'll take a closer look at Hermite
reudction to see if it's better suited for coding than is the
Gaussian approach.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "outthewazoo2002" <chris@t...>
wrote:
>
> > I'm working with Monz and I have a question about turning
bicommas
> > into a lattice basis.
>
> Do you mean biprime commas?
>
> > I don't have a good method for determining a basis from the VAL.
>
> I use the Hermite reduction function of Maple applied to the
biprime
> commas, but simply Gaussian elimination using integral row
operations
> should work.

************************
Hi Gene,

I just need to see an example of the basis computation. Taking the
12-ET/2-3-5 case, the initial bi-prime commas are as follows:

[-19 12 0]
[ 7 0 -3]
[ 0 28 -19]

Using this example, can you show me what matrix you feed to the
Maple Hermite function and what it spits out??

--- In tuning-math@yahoogroups.com, "outthewazoo2002" <chris@t...> wrote:

> I just need to see an example of the basis computation. Taking the
> 12-ET/2-3-5 case, the initial bi-prime commas are as follows:
>
> [-19 12 0]
> [ 7 0 -3]
> [ 0 28 -19]
>
> Using this example, can you show me what matrix you feed to the
> Maple Hermite function and what it spits out??

Here's a text version of a Maple session, where I first use the
Hermite function and then LLL. As you see, the LLL reduction is what
does the real work, but I understand you have your own lattice-basis
reduction algorithm going. The Hermite reduction gives me an integral
Gaussian reduction in a standard form.

> tt:=[[-19,12,0],[7,0,-3],[0,28,-19]];

[[-19, 12, 0], [7, 0, -3], [0, 28, -19]]

> ihermite(tt);

[1 20 -14]
[ ]
[0 28 -19]
[ ]
[0 0 0]

> convert(%,listlist);

[[1, 20, -14], [0, 28, -19], [0, 0, 0]]

> [%[1],%[2]];

[[1, 20, -14], [0, 28, -19]]

> lattice(%);

[[4, -4, 1], [3, 4, -4]]

>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "outthewazoo2002" <chris@t...>
wrote:
>
> > I just need to see an example of the basis computation. Taking
the
> > 12-ET/2-3-5 case, the initial bi-prime commas are as follows:
> >
> > [-19 12 0]
> > [ 7 0 -3]
> > [ 0 28 -19]
> >
> > Using this example, can you show me what matrix you feed to the
> > Maple Hermite function and what it spits out??
>
> Here's a text version of a Maple session, where I first use the
> Hermite function and then LLL. As you see, the LLL reduction is
what
> does the real work, but I understand you have your own lattice-
basis
> reduction algorithm going. The Hermite reduction gives me an
integral
> Gaussian reduction in a standard form.
>
>
> > tt:=[[-19,12,0],[7,0,-3],[0,28,-19]];
>
> [[-19, 12, 0], [7, 0, -3], [0, 28, -19]]
>
>
> > ihermite(tt);
>
> [1 20 -14]
> [ ]
> [0 28 -19]
> [ ]
> [0 0 0]
>
> > convert(%,listlist);
>
> [[1, 20, -14], [0, 28, -19], [0, 0, 0]]
>
> > [%[1],%[2]];
>
> [[1, 20, -14], [0, 28, -19]]
>
> > lattice(%);
>
> [[4, -4, 1], [3, 4, -4]]
>
> >

Gene,

The LLL-reduced basis is not what I would expect for 12 ET. I
expected [7 0 -3] instead of [3 4 -4]? Do you normally use LLL
reduction? I thought you used a method to reduce the Tenney-height?

--- In tuning-math@yahoogroups.com, "outthewazoo2002" <chris@t...> wrote:

> The LLL-reduced basis is not what I would expect for 12 ET. I
> expected [7 0 -3] instead of [3 4 -4]? Do you normally use LLL
> reduction? I thought you used a method to reduce the Tenney-height?

Calculating the Minkowski (or KZ, for that matter) reduction is not
computationally easy compared to LLL, so it makes sense to compute
that first.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "outthewazoo2002" <chris@t...>
wrote:
>
> > The LLL-reduced basis is not what I would expect for 12 ET. I
> > expected [7 0 -3] instead of [3 4 -4]? Do you normally use LLL
> > reduction? I thought you used a method to reduce the Tenney-
height?
>
> Calculating the Minkowski (or KZ, for that matter) reduction is not
> computationally easy compared to LLL, so it makes sense to compute
> that first.

Makes sense. I presume your next step is to compute the TM-reduced
basis from the LLL-reduced basis.