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Adapting a 1915 method of Macaulay, one can give a calculation of the local ring of an isolated zero of a polynomial system {f1, f2, . . . , ft} ⊆ C[x1, x2, . . . , xs] using oating point arithmetic. Using an approximate reverse reduced row echelon form algorithm (ARRREF) one gets a Gröbner basis with respect to a global, rather than local, ordering which leads to the usual representation as a matrix algebra. This can be exploited by relaxing the tolerance in the ARRREF to get information on zeros in a small Euclidean neighborhood of the given zero. The technique, which may be useful in the endgame stage of the homotopy continuation method, is applied to analytic as well as polynomial systems.