  As an attempt to bridge between numerical analysis and algebraic geometry,
this paper formulates the multiplicity for the general nonlinear system at an
isolated zero, presents an algorithm for computing the multiplicity structure,
proposes a depth-deflation method for accurate computation of multiple zeros,
and introduces the basic algebraic theory of the multiplicity. Furthermore,
this paper elaborates and proves some fundamental properties of the
multiplicity, including local finiteness, consistency, perturbation invariance,
and depth-deflatability. As a justification of this formulation, the
multiplicity is proved to be consistent with the multiplicity defined in
algebraic geometry for the special case of polynomial systems. The proposed
algorithms can accurately compute the multiplicity and the multiple zeros using
floating point arithmetic even if the nonlinear system is perturbed.
