Calculus and Numerics on Levi-Civita Fields
AbstractThe 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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