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 = {208711344, 194936928, 3312660, 190020948, 46494420}
o1 : List
|
i2 : (g,z) = gcdLLL s
o2 = (372, | -7 -11 11 -28 4 |)
| 20 -1 3 21 5 |
| -3 19 0 -41 17 |
| -14 14 -11 18 -13 |
| 5 -5 -17 -33 13 |
o2 : Sequence
|
i3 : matrix{s} * z
o3 = | 0 0 0 0 372 |
1 5
o3 : Matrix ZZ <--- ZZ
|