In this paper, we investigate the role of the secant or quasi-Newton condition in the sparse Broyden or Schubert update method for solving systems of nonlinear equations whose Jacobians are either sparse, or can be approximated acceptably by conveniently sparse matrices. We develop a general theory on perturbations to the secant equation that will still allow a proof of local q-linear convergence. To illustrate the theory, we show how to generalize the standard secant condition to the case when the function difference is contaminated by noise.