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# LatticeReduce

 LatticeReduce[{v1, v2, ...}]gives a reduced basis for the set of vectors vi.
• The elements of the vi can be integers, Gaussian integers, or Gaussian rational numbers.
Find the reduced norm basis for a lattice:
Find the reduced norm basis for a lattice:
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 Applications   (3)
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:
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