In this paper we present two new classes of SQP secant methods for the equality constrained optimization problem. One class of methods uses the SQP augmented Lagrangian formulation, while the other class uses the SQP Lagrangian formulation. We demonstrate, under the standard assumptions, that in both cases the BFGS and DFP versions of the algorithm are locally q-superlinearly convergent. To our knowledge this is the first time that either local or q-superlinear convergence has been established for an SQP Lagrangian secant method which uses either the BFGS or DFP updating philosophy and assumes no more than the standard assumptions. Since the standard assumptions do not require positive definiteness of the Hessian of the Lagrangian at the solution, it is no surprise that our BFGS and DFP updates possess the hereditary positive definiteness property only on a proper subspace.