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Abstract
We analyse the application of automatic differentiation (AD) to the quadrature (numerical integration) of a function integrand to determine the sensitivities of the integral to variations in the limits of integration. We derive an expression for the truncation errors of such AD-derived sensitivities and relate them to the truncation error of the original, and a closely related, function quadrature. Our results hold provided the integrand is one degree higher continuously differentiable than that sufficient for convergence of its quadrature. Numerical results validate our analysis. However, utilization of algebraic expressions for such sensitivities, instead of directly applying AD, results in an approach that proves more efficient for the tetrachoric correlation estimation example we considered using our Matlab AD framework.
Acknowledgements
The authors acknowledge permission granted by Cranfield University to include copyright material from Citation13 and from BMT Publishing Group Ltd. for the data of . The authors thank the anonymous referees for their comments leading to improvements of this paper. We also thank Dr John D. Pryce for an early discussion regarding the error analysis described in Section 2 and Dr Trevor Ringrose for suggesting Cudeck's test case of Section 3.3. M. Menshikova thanks the Department of Engineering Systems, Cranfield University for a bursary supporting this work.