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IntegerRelations

 LLL
 find a reduced basis of a lattice

 Calling Sequence LLL(lvect, integer)

Parameters

 lvect - list, set, or rtable of lists or Vectors, or a Matrix integer - (optional) literal name

Description

 • The LLL(lvect) command finds a reduced basis (in the sense of Lovasz) of the lattice specified by the (row) vectors of lvect using the LLL algorithm.
 • If the lattice is generated by vectors with integer coefficients and the option integer is specified, then the reduction is performed using only integer arithmetic. This version is sometimes faster than the default version, which uses rational arithmetic.
 • This function requires that the dimension of the subspace generated by the vectors equals the number of vectors.

Examples

 > $\mathrm{with}\left(\mathrm{IntegerRelations}\right):$
 > $\mathrm{LLL}\left(\left[\left[1,2,3\right],\left[2,1,6\right]\right]\right)$
 $\left[\left[{0}{,}{-3}{,}{0}\right]{,}\left[{1}{,}{-1}{,}{3}\right]\right]$ (1)
 > $\mathrm{LLL}\left(\left[\left[1,2,3\right],\left[2,1,6\right]\right],'\mathrm{integer}'\right)$
 $\left[\left[{0}{,}{-3}{,}{0}\right]{,}\left[{1}{,}{-1}{,}{3}\right]\right]$ (2)
 > $\mathrm{LLL}\left(\mathrm{Matrix}\left(\left[\left[1,2,3\right],\left[-1,0,1\right],\left[0,1,1\right]\right]\right)\right)$
 $\left[\begin{array}{ccc}{-1}& {0}& {1}\\ {0}& {1}& {1}\\ {0}& {-1}& {1}\end{array}\right]$ (3)

References

 Lenstra, A.K.; Lenstra, H.W.; and Lovasz, L. "Factoring Polynomials with Rational Coefficients." Math. Ann., Vol. 261. (1982): 515-534.