- Tytuł:
- Non-standard analysis revisited: An easy axiomatic presentation oriented towards numerical applications
- Autorzy:
-
Benci, Vieri
Cococcioni, Marco
Fiaschi, Lorenzo - Powiązania:
- https://bibliotekanauki.pl/articles/2055153.pdf
- Data publikacji:
- 2022
- Wydawca:
- Uniwersytet Zielonogórski. Oficyna Wydawnicza
- Tematy:
-
Alpha Theory
non-standard analysis
non Archimedean analysis
algorithmic numbers
non Archimedean scientific computing - Opis:
- 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.
- Źródło:
-
International Journal of Applied Mathematics and Computer Science; 2022, 32, 1; 65--80
1641-876X
2083-8492 - Pojawia się w:
- International Journal of Applied Mathematics and Computer Science
- Dostawca treści:
- Biblioteka Nauki
Artykuł