  For a pair of polynomials with real or complex coefficients, given in any
particular basis, the problem of finding their GCD is known to be ill-posed. An
answer is still desired for many applications, however. Hence, looking for a
GCD of so-called approximate polynomials where this term explicitly denotes
small uncertainties in the coefficients has received significant attention in
the field of hybrid symbolic-numeric computation. In this paper we give an
algorithm, based on one of Victor Ya. Pan, to find an approximate GCD for a
pair of approximate polynomials given in a Lagrange basis. More precisely, we
suppose that these polynomials are given by their approximate values at
distinct known points. We first find each of their roots by using a Lagrange
basis companion matrix for each polynomial, cluster the roots of each
polynomial to identify multiple roots, and then "marry" the two polynomials to
find their GCD. At no point do we change to the monomial basis, thus preserving
the good conditioning properties of the original Lagrange basis. We discuss
advantages and drawbacks of this method. The computational cost is dominated by
the rootfinding step; unless special-purpose eigenvalue algorithms are used,
the cost is cubic in the degrees of the polynomials. In principle, this cost
could be reduced but we do not do so here.
