Higher-order Reverse Automatic Differentiation with emphasis on the third-order

It is commonly assumed that calculating third order information is too expensive for most applications. But we show that the directional derivative of the Hessian (D3f(x)· d) can be calculated at a cost proportional to that of a state-of-the-art method for calculating the Hessian matrix. We do this by first presenting a simple procedure for designing high order reverse methods and applying it to deduce several methods including a reverse method that calculates D3f(x)· d. We have implemented this method taking into account symmetry and sparsity, and successfully calculated this derivative for functions with a million variables. These results indicate that the use of third order information in a general nonlinear solver, such as Halley-Chebyshev methods, could be a practical alternative to Newton's method.