Geodesic
Fast automatic differentiation as applied to the computation of second derivatives of composite functions
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 11, pp. 1923-1949 
E. S. Zasukhina. Fast automatic differentiation as applied to the computation of second derivatives of composite functions. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 11, pp. 1923-1949. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_11_a1/
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     author = {E. S. Zasukhina},
     title = {Fast automatic differentiation as applied to the computation of second derivatives of composite functions},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     year = {2006},
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     number = {11},
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     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_11_a1/}
}
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A technique for deriving formulas for the second derivatives of a composite function with constrained variables is proposed. The original system of constraint equations is associated with a linear system of equations, whose solution is used to determine the Hessian of the function. The resulting formulas are applied to discrete problems obtained by approximating optimal control problems with the use of Runge–Kutta methods of various orders. For a particular optimal control problem, the numerical results obtained by the gradient method and Newton's method with the resulting formulas are described and analyzed in detail.

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