Higher Order Multivariate Automatic Differentiation and Validated Computation of Remainder Bounds

Abstract

High order automatic differentiation has been used in recent years for simultaneous computation of Taylor expansions of functional dependencies as well as validated enclosures for the remainder over a certain domain. The resulting Taylor model methods offer certain advantages compared to other validated methods, including an approximation by a functional form with an accuracy scaling with a high order, as well as the ability to suppress the dependency problem.

In this paper we describe the implementation of the method in a validated setting. Several steps are taken to increase computational effciency. Computations of coefficients are not performed in interval arithmetic, but rather in floating point arithmetic with simultaneous accurate accounting of the possible computational errors. Furthermore, the storage and elementary operations of the objects supports sparsity of the coefficients.

M. Berz, K. Makino, Transactions on Mathematics 3(1) (2004) 37-44

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