       Re: Re: LatticeReduce

• To: mathgroup at smc.vnet.net
• Subject: [mg93602] Re: [mg93580] Re: LatticeReduce
• From: danl at wolfram.com
• Date: Mon, 17 Nov 2008 06:18:24 -0500 (EST)
• References: <200811151103.GAA16591@smc.vnet.net> <491F2366.3040401@csl.pl>

```> Dear Daniel,
> Many thanks for explanations! That helps!  I will be learning about
> ExtendedGCD and  HermiteDecomposition.
> What do you mean LinearSolve ? Could you give me one equation exercise
> (related to my) and solvng them by LinearSolve?
> Best wishes
> Artur
> [...]

The equations are homogeneous, so I should have used NullSpace instead of
LinearSolve. The idea is you can obtain a generating set of solutions to
the linear homogeneous equation simply by finding the null space
generators. If any are fractional (they are not, but I want to cover all
possibilities) then you would need to clear denominators to get integral
solutions.

{a0, a1, a2} = {1, 2^15, -3^8};
In:= NullSpace[{{a0, a1, a2}}]
Out= {{6561, 0, 1}, {-32768, 1, 0}}

In:= {a1, a2, a3} = {1, 3^4, 5^4};
NullSpace[{{a1, a2, a3}}]
Out= {{-625, 0, 1}, {-81, 1, 0}}

Again, whether these are useful (thay are, admittedly, fairly "obvious"
solutions) depends on what you want to do. If you want small solutions you
might be beter off with the methods indicated in my prior response.

Daniel Lichtblau
Wolfram Research

```

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