The algorithmic numbers in non-archimedean numerical computing environments

Vieri Benci; Marco Cococcioni

doi:10.3934/dcdss.2020449

Article Contents

2021, Volume 14, Issue 5: 1673-1692. Doi: 10.3934/dcdss.2020449

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The algorithmic numbers in non-archimedean numerical computing environments

Vieri Benci^1, and
Marco Cococcioni^2, ,

1.
Department of Mathematics, University of Pisa, 56127 Tuscany, IT
2.
Department of Information Engineering, University of Pisa, 56126 Tuscany, IT

^* Corresponding author: Marco Cococcioni (marco.cococcioni@unipi.it)
^* Corresponding author: Marco Cococcioni (marco.cococcioni@unipi.it)

Received: January 31, 2020

Revised: May 31, 2020

Early access: November 2020

Published: May 2021

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Abstract

There are many natural phenomena that can best be described by the use of infinitesimal and infinite numbers (see e.g. [1,5,13,23]. However, until now, the Non-standard techniques have been applied to theoretical models. In this paper we investigate the possibility to implement such models in numerical simulations. First we define the field of Euclidean numbers which is a particular field of hyperreal numbers. Then, we introduce a set of families of Euclidean numbers, that we have called altogether algorithmic numbers, some of which are inspired by the IEEE 754 standard for floating point numbers. In particular, we suggest three formats which are relevant from the hardware implementation point of view: the Polynomial Algorithmic Numbers, the Bounded Algorithmic Numbers and the Truncated Algorithmic Numbers. In the second part of the paper, we show a few applications of such numbers.

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Mathematics Subject Classification: Primary: 68U01; Secondary: 26E35, 12J25, 14B10, 46N10, 34M03.

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Figure 1. Feasible region (the vertical line on the left is positioned at $ x_1 = -\alpha $, while the vertical constraint on the right is at $ x_1 = \alpha $. The upper horizontal constraint at $ x_2 = 1 $ and the lower constraint at $ x_2 = -1. $

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