relative saturation?

[Daniel Allcock <allcock@math.utexas.edu>]

Dear all,

I have a basis for a saturated subgroup L of Z^n, and one extra vector V in Z^n but not in L.  I'd like to find an element of Z^n which together with L generates the saturation of the span of L and V.  I think of this as the "saturation of <L,V> relative to L".

I can't find any ready-made function that lends itself to an easy implementation of this.  The best I can think of right now is to find a basis for the saturation of <L,V>, express it in terms of V and my basis for L, look at the V-components, and follow Euclid's algorithm to find a Z-linear comb whose V-component is smallest possible.  This looks icky and I am probably missing something more obvious.  

I'd appreciate any tips.  (I'm using library mode, so hacks using it are welcome too).

Thanks,
Daniel