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Higher derivative tensors from univariate Taylor series with comparison to Maple on an engineering problem

Abstract:

We consider the problem of evaluating partial derivatives of a vector function defined by a computer program. The natural way seems to propagate derivative tensors according to the multivariate chain rule. Instead much simpler data access patterns and similar or lower computational cost can be achieved through propagating families of univariate Taylor series of a suitable degree. The partial derivatives are then obtained by a conversion process at a negligible cost. Both techniques are applied to calculate the sensitivities of bevel gears, where directional derivatives up to order three are needed. To evaluate the derivatives desired we use Maple for the multivariate approach and the automatic differentiation (AD) tool ADOL-C for the univariate approach. The run times achieved are reported.