The theoretical convergence guarantees of the single shooting method with the approximate Fréchet derivative can be analyzed using standard convergence analysis techniques for numerical algorithms. The convergence properties of the method depend on factors such as the smoothness of the objective function, the choice of step size in the iterative process, and the accuracy of the approximation of the Fréchet derivative.
Compared to the exact Fréchet derivative approach, the approximate Fréchet derivative method may have slightly slower convergence rates due to the approximation error introduced. However, in practice, the approximate Fréchet derivative approach can still achieve convergence to a high degree of accuracy and is computationally more efficient, especially for large-scale problems where computing the exact Fréchet derivative may be prohibitively expensive.