Given an analytic function f and a Jordan curve g that does not pass through any zero of f, we consider the problem of computing all the zeros of f that lie inside g together with their respective multiplicities. If f has one or several clusters of zeros, then the mapping from the ordinary moments associated with this form to the zeros and their respective multiplicities is very ill-conditioned. We present numerical methods to calculate the centre of a cluster and its size, i.e., the arithmetic mean of the zeros that form a certain cluster and the total number of zeros in this cluster, respectively. Our approach relies on formal orthogonal polynomials and rational interpolation at roots of unity.
Numerical examples illustrate the effectiveness of our techniques.