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 = {100694820, 123088848, 161628048, 359290248, 304519944}
o1 : List
|
i2 : (g,z) = gcdLLL s
o2 = (372, | 12 12 -44 -26 -9 |)
| 14 21 14 -8 2 |
| 14 -7 1 29 21 |
| -17 7 2 8 -11 |
| 3 -17 6 -13 4 |
o2 : Sequence
|
i3 : matrix{s} * z
o3 = | 0 0 0 0 372 |
1 5
o3 : Matrix ZZ <--- ZZ
|