next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
LLLBases > gcdLLL

gcdLLL -- compute the gcd of integers, and small multipliers

Synopsis

Description

This function is provided by the package LLLBases.

The first n-1 columns of the matrix z form a basis of the kernel of the n integers of the list s, and the dot product of the last column of z and s is the gcd g.

The method used is described in the paper:

Havas, Majewski, Matthews, Extended GCD and Hermite Normal Form Algorithms via Lattice Basis Reduction, Experimental Mathematics 7:2 p. 125 (1998).

For an example,

i1 : s = apply(5,i->372*(random 1000000))

o1 = {292835796, 51167112, 365399232, 36053868, 106038972}

o1 : List
i2 : (g,z) = gcdLLL s

o2 = (372, | -9 10  20  13  0  |)
           | 27 18  15  16  -7 |
           | 1  -7  -22 -7  3  |
           | 7  -27 1   25  6  |
           | 6  -3  13  -28 -9 |

o2 : Sequence
i3 : matrix{s} * z

o3 = | 0 0 0 0 372 |

              1        5
o3 : Matrix ZZ  <--- ZZ

See also

Ways to use gcdLLL :