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Paper details
Number 1 - March 2022
Volume 32 - 2022
Non-standard analysis revisited: An easy axiomatic presentation oriented towards numerical applications
Vieri Benci, Marco Cococcioni, Lorenzo Fiaschi
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
Alpha-Theory, non-standard analysis, non-Archimedean analysis, algorithmic numbers, non-Archimedean scientific computing