1. bookVolume 32 (2022): Edition 1 (March 2022)
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eISSN
2083-8492
Première parution
05 Apr 2007
Périodicité
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Anglais
access type Accès libre

Non–Standard Analysis Revisited: An Easy Axiomatic Presentation Oriented Towards Numerical Applications

Publié en ligne: 31 Mar 2022
Volume & Edition: Volume 32 (2022) - Edition 1 (March 2022)
Pages: 65 - 80
Reçu: 12 Feb 2021
Accepté: 21 Sep 2021
Détails du magazine
License
Format
Magazine
eISSN
2083-8492
Première parution
05 Apr 2007
Périodicité
4 fois par an
Langues
Anglais
Abstract

Alpha-Theory was introduced in 1995 to provide a simplified version of Robinson’s non-standard analysis which overcomes the technicalities of symbolic logic. The theory has been improved over the years, and recently it has been used also to solve practical problems in a pure numerical way, thanks to the introduction of algorithmic numbers. In this paper, we introduce Alpha-Theory using a novel axiomatic approach oriented towards real-world applications, to avoid the need to master mathematical logic and model theory. To corroborate the strong link of this Alpha-Theory axiomatization and scientific computations, we report numerical illustrative applications never carried out by means of non-standard numbers within a computer, i.e., the computation of the eigenvalues of a non-Archimedean matrix, some computations related to non-Archimedean Markov chains, and the Cholesky factorization of a non-Archimedean matrix. We also highlight the differences between our numerical routines and pure symbolic approaches: as expected, the former scales better when the dimension of the problem increases.

Keywords

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