We develop a theory of quasi-Newton and least-change update methods for solving systems of nonlinear equations F(x) = 0. In this theory, no differentiability conditions are necessary. Instead, we assume that F can be approximated, in a weak sense, by an affine function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-differentiable functions and to partially differentiable functions.164192111931209