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LLLBases > gcdLLL

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

Synopsis

Usage:
(g,z) = gcdLLL m
Inputs:
- m, a list, of integers
Outputs:
- g, an integer, the gcd of the integers in the list s
- z, a matrix, of integers
Optional inputs:
- Strategy => ...,
- Threshold => ...,

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 = {347049960, 344357052, 291802380, 171072384, 198187092}

o1 : List

i2 : (g,z) = gcdLLL s

o2 = (372, | -13 -14 3   -31 19  |)
           | 6   10  -31 2   -1  |
           | 14  5   24  -22 5   |
           | 2   -13 -2  57  -30 |
           | -10 11  15  34  -13 |

o2 : Sequence

i3 : matrix{s} * z

o3 = | 0 0 0 0 372 |

              1        5
o3 : Matrix ZZ  <--- ZZ

See also

LLLBases -- lattice reduction (Lenstra-Lenstra-Lovasz bases)
LLL -- compute an LLL basis
kernelLLL -- compute the kernel of an integer matrix using LLL bases
hermite -- compute the Hermite normal form and small multiplier matrix using LLL bases

Ways to use `gcdLLL` :

gcdLLL(List)