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

IF 1.6 4区计算机科学 Q3 AUTOMATION & CONTROL SYSTEMS

International Journal of Applied Mathematics and Computer Science Pub Date : 2022-03-01 DOI:10.34768/amcs-2022-0006

V. Benci, M. Cococcioni, Lorenzo Fiaschi

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引用次数: 4

Abstract

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.

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重新审视非标准分析:面向数值应用的简单公理表示

1995年引入的阿尔法理论是罗宾逊非标准分析的简化版本，它克服了符号逻辑的技术性。多年来，这个理论得到了改进，最近，由于引入了算法数，它也被用于以纯粹的数值方式解决实际问题。在本文中，我们使用一种面向现实世界应用的新的公理方法来介绍Alpha-Theory，以避免需要掌握数学逻辑和模型理论。为了证实这一α理论公理化与科学计算的紧密联系，我们报告了从未在计算机中使用非标准数字进行的数值说明应用，即非阿基米德矩阵的特征值的计算，与非阿基米德马尔可夫链相关的一些计算，以及非阿基米德矩阵的Cholesky分解。我们还强调了我们的数值例程和纯符号方法之间的区别:正如预期的那样，当问题的维度增加时，前者的伸缩性更好。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal of Applied Mathematics and Computer Science 工程技术-计算机：人工智能

CiteScore

4.10

自引率

21.10%

发文量

审稿时长

4.2 months

期刊介绍： The International Journal of Applied Mathematics and Computer Science is a quarterly published in Poland since 1991 by the University of Zielona Góra in partnership with De Gruyter Poland (Sciendo) and Lubuskie Scientific Society, under the auspices of the Committee on Automatic Control and Robotics of the Polish Academy of Sciences. The journal strives to meet the demand for the presentation of interdisciplinary research in various fields related to control theory, applied mathematics, scientific computing and computer science. In particular, it publishes high quality original research results in the following areas: -modern control theory and practice- artificial intelligence methods and their applications- applied mathematics and mathematical optimisation techniques- mathematical methods in engineering, computer science, and biology.