The use of the DFP or the BFGS secant updates requires the Hessian at the solution to be positive definite. The second order sufficiency conditions insure the positive definiteness only in a subspace of R^n. Conditions are given so we can safely update with either update. A new class of algorithms is proposed which generate a sequence {xk} converging 2-step q-superlinearly. We also propose two specific algorithms: One converges q-superlinearly if the Hessian is positive definite in R^n and converges 2-step q-superlinearly if the Hessian is positive definite only in a subspace; the second one generates a sequence converging 1-step q-superlinearly. While the former costs one extra gradient evaluation, the latter costs one extra gradient evaluation and one extra function evaluation on the constraints.