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Calculus and Numerics on Levi-Civita Fields


Abstract

The formal process of the evaluation of derivatives using some of the various modern methods of computational differentiation can be recognized as an example for the application of conventional "approximate" numerical techniques on a nonarchimedean extension of the real numbers. In many cases, the application of "infinitely small numbers" instead of "small but finite" numbers allows the use of the old numerical algorithm, but now with an error that in a rigorous way can be shown to become ifinitely small (and hence irrelevant). While intuitive ideas in this direction have accompanied analysis from the early days of Newton and Leibniz, the first rigorous work goes back to Levi Civita, who introduced a number field that in the past few years turned out to be particularly suitable for numerical problems. While Levi Civita's field appears to be of fundamental importance and simplicity, efforts to introduce advanced concepts of calculus on it are only very new. In this paper, we address several of the basic questions providing a foundation for such a calculus. After addressing questions of algebra and convergence, we study questions of differentiabilit, in particular with an eye to usefulness for practical ork.


M. Berz, in: "Computational Differentiation: Techniques, Applications, and Tools", M. Berz, C. Bischof, G. Corliss, A. Griewank (Eds.) (1996) 19-37, SIAM


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