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 = {260925636, 262823952, 195935376, 41777832, 321344016}
o1 : List
|
i2 : (g,z) = gcdLLL s
o2 = (372, | -12 0 -6 -38 11 |)
| -4 3 -13 52 -16 |
| 18 13 -17 -19 21 |
| 8 -26 -1 7 -5 |
| 1 -7 26 -1 -8 |
o2 : Sequence
|
i3 : matrix{s} * z
o3 = | 0 0 0 0 372 |
1 5
o3 : Matrix ZZ <--- ZZ
|