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The local convergence properties of a class of primal-dual interior point methods are analyzed. These methods are designed to minimize a nonlinear, nonconvex, objective function subject to linear equality constraints and general inequalities. They involve an inner iteration in which the log-barrier merit function is approximately minimized subject to satisfying the linear equality constraints, and an outer iteration that specifies both the decrease in the barrier parameter and the level of accuracy for the inner minimization. Under nondegeneracy assumptions, it is shown that, asymptotically, for each value of the barrier parameter, solving a single primal-dual linear system is enough to produce an iterate that already matches the barrier subproblem accuracy requirements. The asymptotic rate of convergence of the resulting algorithm is Q-superlinear and may be chosen arbitrarily close to quadratic. Furthermore, this rate applies componentwise. These results hold in particular for the method described in [A. R. Conn, N. I. M. Gould, D. Orban, and P. L. Toint, Math. Program. Ser. B, 87 (2000), pp. 215-249] and indicate that the details of its inner minimization are irrelevant in the asymptotics, except for its accuracy requirements.