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Higher-order reverse automatic differentiation with emphasis on the third-order. (English) Zbl 1332.65034
Summary: 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 ($$D^3 f(x)\cdot d$$) 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 $$D^3f(x)\cdot d$$. 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. Furthermore, high-order sensitivity information is used in methods for robust aerodynamic design. An efficient high-order differentiation tool could facilitate the use of similar methods in the design of other mechanical structures.

##### MSC:
 65D25 Numerical differentiation 15A69 Multilinear algebra, tensor calculus 65F50 Computational methods for sparse matrices 49Q12 Sensitivity analysis for optimization problems on manifolds
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