Built-in Mathematica Symbol

LatticeReduce

LatticeReduce[{v₁, v₂, ...}]
gives a reduced basis for the set of vectors v_i.

The elements of the v_i can be integers, Gaussian integers, or Gaussian rational numbers.

Find the reduced norm basis for a lattice:

Starting with trivial integer linear relationships, LatticeReduce can produce more interesting ones:

Find integer linear relationships for

and

of the form

:

LatticeReduce preserves linear relationships, and the third row provides

,

and

:

Find polynomial relationships

for

:

The trivial initial relationships:

The reduced relationships:

The first relationship:

Find linear relationships x₀+x₁ ArcTan[1]+x₂ ArcTan[1/5]+x₃ ArcTan[1/239]

0:

Initial trivial relationships:

Reduced relationships:

The first relationship:

LatticeReduce produces a new reduced basis for the same lattice:

The product of the norms will decrease:

The determinant, or volume of the generator cell, is preserved:

The lattice is generated by {v₁, v₂}, but also by {w₁, w₂} produced by LatticeReduce:

The original cell is pink, and the one produced by LatticeReduce is cyan:

The set of vectors must have rational or Gaussian rational coefficients: