In many problems involving the solution of a system of nonlinear equations, it is necessary to keep an approximation to the Jacobian matrix which is updated at each iteration. Computational experience indicates that the best updates are those that minimize some reasonable measure of the change to the current Jacobian approximation subject to the new approximation obeying a secant condition and perhaps some other approximation properties such as symmetry. All of the updates obtained thus far deal with updating an approximation to an nxn Jacobian matrix. In this thesis we consider extending most of the popular updates to the non-square case. Two applications are immediate: between-step updating of the approximate Jacobian of f(X,t) in a non-autonomous ODE system, and solving nonlinear systems of equations which depend on a parameter, such as occur in continuation methods. Both of these cases require extending the present updates to include the nx(n+l) Jacobian matrix, which is the issue we address here. Our approach is to stay with the least change secant formulation. Computational results for these new updates are also presented to illustrate their convergence behavior.