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May  2021, 14(5): 1673-1692. doi: 10.3934/dcdss.2020449

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)

Received  February 2020 Revised  June 2020 Published  May 2021 Early access  November 2020

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.

Keywords: Infinitesimal numbers, nonstandard analysis, euclidean numbers, non-archimedean scientific computing, fixed-length formats, hardware-friendly representations.
Mathematics Subject Classification: Primary: 68U01; Secondary: 26E35, 12J25, 14B10, 46N10, 34M03.
Citation: Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1673-1692. doi: 10.3934/dcdss.2020449
References:
[1]

A. Albeverio, J. E. Fenstad, R. Høegh-Krohn and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Dover Publications, Princeton, 2009.

[2]

V. Benci, I Numeri e gli Insiemi Etichettati, Laterza, Bari, Italia, 1995. Conferenze del seminario di matematica dell' Università di Bari, vol. 261, pp. 29.

[3]

V. Benci, An algebraic approach to nonstandard analysis, In G. Buttazzo, A. Marino, and M.K.V. Murthy, editors, Calculus of Variations and Partial Differential Equations: Topics on Geometrical Evolution Problems and Degree Theory, pages 285–326. Springer Berlin Heidelberg, Berlin, Heidelberg, 1999. doi: 10.1007/978-3-642-57186-2_12.

[4]

V. Benci, An algebraic approach to nonstandard analysis, In G. Buttazzo, editor, Calculus of Variations and Partial differential equations, volume 4 of 5, chapter 8, pages 285–326. Springer, Berlin, 2000.

[5]

V. Benci, Ultrafunctions and generalized solutions, Adv. Nonlinear Studies, 13 (2013), 461-486.  doi: 10.1515/ans-2013-0212.

[6]

V. Benci, Alla Scoperta dei Numeri Infinitesimi, Lezioni di Analisi Matematica Esposte in un Campo Non-Archimedeo, Aracne Editrice, Rome, 2018.

[7]

V. Benci and M. Di Nasso, Numerosities of labelled sets: A new way of counting, Adv. Math., 173 (2003), 50-67.  doi: 10.1016/S0001-8708(02)00012-9.

[8]

V. Benci and M. Di Nasso, How to Measure the Infinite: Mathematics with Infinite and Infinitesimal Numbers, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.

[9]

V. Benci, M. Di Nasso and M. Forti, An aristotelian notion of size, Ann. Pure Appl. Logic, 143 (2006), 43-53.  doi: 10.1016/j.apal.2006.01.008.

[10]

V. Benci and M. Forti, The Euclidean numbers, arXiv: 1702.04163v2, 2018.

[11]

V. Benci, M. Forti and M. Di Nasso, The eightfold path to nonstandard analysis, In D. A. Ross N. J. Cutland, M. Di Nasso, editor, Nonstandard Methods and Applications in Mathematics, volume 25 of Lecture Notes in Logic, pages 3–44. Association for Symbolic Logic, AK Peters, Wellesley, MA, 2006.

[12]

V. Benci and P. Freguglia, Alcune osservazioni sulla matematica non archimedea, Matem. Cultura e Soc., RUMI, 1 (2016), 105-122. 

[13]

V. Benci and L. Luperi Baglini, Ultrafunctions and applications, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 593-616.  doi: 10.3934/dcdss.2014.7.593.

[14]

M. Cococcioni, A. Cudazzo, M. Pappalardo and Y. D. Sergeyev, Solving the Lexicographic Multi-Objective Mixed-Integer Linear Programming Problem using Branch-and-Bound and Grossone Methodology, Commun. Nonlinear Sci. Numer. Simul., 84 (2020), 105177, 20 pp. doi: 10.1016/j.cnsns.2020.105177.

[15]

M. Cococcioni and L. Fiaschi, The Big-M method with the Numerical Infinite $M$, Optimization Letters, 2020 doi: 10.1007/s11590-020-01644-6.

[16]

M. Cococcioni, M. Pappalardo and Y. D. Sergeyev, Lexicographic Multi-Objective Linear Programming using Grossone Methodology: Theory and Algorithm, Applied Mathematics and Computation, 318 (2018), 298-311.  doi: 10.1016/j.amc.2017.05.058.

[17]

G. B. Dantzig and M. N. Thapa, Linear Programming 2: Theory and Extensions, Springer-Verlag, New York, 2003.

[18]

L. Fiaschi and M. Cococcioni, Numerical Asymptotic Results in Game Theory using Sergeyev's Infinity Computing, International Journal of Unconventional Computing, 14 (2018), 1-25. 

[19]

L. Fiaschi and M. Cococcioni, Non-Archimedean Game Theory: A Numerical Approach, Applied Mathematics and Computation, 2020, 125356. doi: 10.1016/j.amc.2020.125356.

[20]

P. Fletcher, K. Hrbacek, V. Kanovei, M. G. Katz, C. Lobry and S. Sanders, Approaches to analysis following Robinson, Nelson and others, Real Analysis Exchange, 42 (2017), 193-251.  doi: 10.14321/realanalexch.42.2.0193.

[21]

L. Lai, L. Fiaschi and M. Cococcioni, Solving Mixed Pareto-Lexicographic Multi-Objective Optimization Problems: The Case of Priority Chains, Swarm and Evolutionary Computation, 55 (2020), 100687. doi: 10.1016/j.swevo.2020.100687.

[22]

T. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti del R. Istituto Veneto di Scienze Lettere ed Arti, Venezia, Series 7, 1892.

[23] E. Nelson, Radically Elementary Probability Theory, Princeton University Press, Princeton, New Jersey, 1987.  doi: 10.1515/9781400882144.
[24]

K. Ogata, Modern Control Engineering, Prentice Hall, New Jersey, 5 edition, 2010.

[25]

Y. D. Sergeyev, Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Mathematical Sciences, 4 (2017), 219-320.  doi: 10.4171/EMSS/4-2-3.

show all references

References:
[1]

A. Albeverio, J. E. Fenstad, R. Høegh-Krohn and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Dover Publications, Princeton, 2009.

[2]

V. Benci, I Numeri e gli Insiemi Etichettati, Laterza, Bari, Italia, 1995. Conferenze del seminario di matematica dell' Università di Bari, vol. 261, pp. 29.

[3]

V. Benci, An algebraic approach to nonstandard analysis, In G. Buttazzo, A. Marino, and M.K.V. Murthy, editors, Calculus of Variations and Partial Differential Equations: Topics on Geometrical Evolution Problems and Degree Theory, pages 285–326. Springer Berlin Heidelberg, Berlin, Heidelberg, 1999. doi: 10.1007/978-3-642-57186-2_12.

[4]

V. Benci, An algebraic approach to nonstandard analysis, In G. Buttazzo, editor, Calculus of Variations and Partial differential equations, volume 4 of 5, chapter 8, pages 285–326. Springer, Berlin, 2000.

[5]

V. Benci, Ultrafunctions and generalized solutions, Adv. Nonlinear Studies, 13 (2013), 461-486.  doi: 10.1515/ans-2013-0212.

[6]

V. Benci, Alla Scoperta dei Numeri Infinitesimi, Lezioni di Analisi Matematica Esposte in un Campo Non-Archimedeo, Aracne Editrice, Rome, 2018.

[7]

V. Benci and M. Di Nasso, Numerosities of labelled sets: A new way of counting, Adv. Math., 173 (2003), 50-67.  doi: 10.1016/S0001-8708(02)00012-9.

[8]

V. Benci and M. Di Nasso, How to Measure the Infinite: Mathematics with Infinite and Infinitesimal Numbers, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.

[9]

V. Benci, M. Di Nasso and M. Forti, An aristotelian notion of size, Ann. Pure Appl. Logic, 143 (2006), 43-53.  doi: 10.1016/j.apal.2006.01.008.

[10]

V. Benci and M. Forti, The Euclidean numbers, arXiv: 1702.04163v2, 2018.

[11]

V. Benci, M. Forti and M. Di Nasso, The eightfold path to nonstandard analysis, In D. A. Ross N. J. Cutland, M. Di Nasso, editor, Nonstandard Methods and Applications in Mathematics, volume 25 of Lecture Notes in Logic, pages 3–44. Association for Symbolic Logic, AK Peters, Wellesley, MA, 2006.

[12]

V. Benci and P. Freguglia, Alcune osservazioni sulla matematica non archimedea, Matem. Cultura e Soc., RUMI, 1 (2016), 105-122. 

[13]

V. Benci and L. Luperi Baglini, Ultrafunctions and applications, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 593-616.  doi: 10.3934/dcdss.2014.7.593.

[14]

M. Cococcioni, A. Cudazzo, M. Pappalardo and Y. D. Sergeyev, Solving the Lexicographic Multi-Objective Mixed-Integer Linear Programming Problem using Branch-and-Bound and Grossone Methodology, Commun. Nonlinear Sci. Numer. Simul., 84 (2020), 105177, 20 pp. doi: 10.1016/j.cnsns.2020.105177.

[15]

M. Cococcioni and L. Fiaschi, The Big-M method with the Numerical Infinite $M$, Optimization Letters, 2020 doi: 10.1007/s11590-020-01644-6.

[16]

M. Cococcioni, M. Pappalardo and Y. D. Sergeyev, Lexicographic Multi-Objective Linear Programming using Grossone Methodology: Theory and Algorithm, Applied Mathematics and Computation, 318 (2018), 298-311.  doi: 10.1016/j.amc.2017.05.058.

[17]

G. B. Dantzig and M. N. Thapa, Linear Programming 2: Theory and Extensions, Springer-Verlag, New York, 2003.

[18]

L. Fiaschi and M. Cococcioni, Numerical Asymptotic Results in Game Theory using Sergeyev's Infinity Computing, International Journal of Unconventional Computing, 14 (2018), 1-25. 

[19]

L. Fiaschi and M. Cococcioni, Non-Archimedean Game Theory: A Numerical Approach, Applied Mathematics and Computation, 2020, 125356. doi: 10.1016/j.amc.2020.125356.

[20]

P. Fletcher, K. Hrbacek, V. Kanovei, M. G. Katz, C. Lobry and S. Sanders, Approaches to analysis following Robinson, Nelson and others, Real Analysis Exchange, 42 (2017), 193-251.  doi: 10.14321/realanalexch.42.2.0193.

[21]

L. Lai, L. Fiaschi and M. Cococcioni, Solving Mixed Pareto-Lexicographic Multi-Objective Optimization Problems: The Case of Priority Chains, Swarm and Evolutionary Computation, 55 (2020), 100687. doi: 10.1016/j.swevo.2020.100687.

[22]

T. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti del R. Istituto Veneto di Scienze Lettere ed Arti, Venezia, Series 7, 1892.

[23] E. Nelson, Radically Elementary Probability Theory, Princeton University Press, Princeton, New Jersey, 1987.  doi: 10.1515/9781400882144.
[24]

K. Ogata, Modern Control Engineering, Prentice Hall, New Jersey, 5 edition, 2010.

[25]

Y. D. Sergeyev, Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Mathematical Sciences, 4 (2017), 219-320.  doi: 10.4171/EMSS/4-2-3.

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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