Publication: Calculus and Numerics on Levi-Civita Fields
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Calculus and Numerics on Levi-Civita Fields

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Author(s)
Martin Berz

Published in
Computational Differentiation: Techniques, Applications, and Tools

Editor(s)
Martin Berz, Christian Bischof, George Corliss, Andreas Griewank

Year
1996

Publisher
SIAM

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 non-archimedean 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 infinitely 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 differentiability, in particular with an eye to usefulness for practical work.

Cross-References
Berz1996CDT

BibTeX
@INPROCEEDINGS{
         Berz1996CaN,
       author = "Martin Berz",
       editor = "Martin Berz and Christian Bischof and George Corliss and Andreas Griewank",
       title = "Calculus and Numerics on {Levi-Civita} Fields",
       booktitle = "Computational Differentiation: Techniques, Applications, and Tools",
       pages = "19--35",
       publisher = "SIAM",
       address = "Philadelphia, PA",
       key = "Berz1996CaN",
       crossref = "Berz1996CDT",
       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 non-archimedean 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 infinitely 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 differentiability, in particular with an eye to
         usefulness for practical work.",
       keywords = "Levi-Civita, Non-standard Analysis, Non-Archimedean Analysis, Analysis with
         Infinitesimals, Differentials, Infinitesimals, Derivatives as Differential Quotients, Computer
         Functions, Differential Quotients, Computation of Derivatives.",
       referred = "[Berz2002TaU], [Pusch1996JSa], [Shamseddine1996EHi].",
       year = "1996"
}


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