We show how, in parallel to the accumulation of derivatives, error bounds of all functional dependencies can be carried along the computation. The additional effort is minor, and the resulting bounds are usually rather sharp, in particular at higher orders. This Remainder Differential Algebraic Method is more straight forward and can yield tighter bounds than the mere interval bounding of the Taylor remainders ( n + 1 )st order derivative obtained via forward differentiation. The method can be applied to various numerical problems: Here we focus on global optimization, where blow-up can often be substantially reduced compared with interval methods, in particular for the cases of complicated functions or many variables. This problem is at the core of many questions of nonlinear dynamics and can help facilitate a detailed, quantitativ understanding.
K. Makino, M. Berz, in: "Computational Differentiation: Techniques, Applications, and Tools", M. Berz, C. Bischof, G. Corliss, A. Griewank (Eds.) (1996) 63-75, SIAM
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