• Previous Article
    Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations
  • DCDS-S Home
  • This Issue
  • Next Article
    Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms
doi: 10.3934/dcdss.2020449

The algorithmic numbers in non-archimedean numerical computing environments

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  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.

Citation: Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

V. BenciM. 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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. CococcioniM. 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.  Google Scholar

[17]

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

[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.   Google Scholar

[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.  Google Scholar

[20]

P. FletcherK. HrbacekV. KanoveiM. G. KatzC. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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

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

[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.  Google Scholar

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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

V. BenciM. 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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. CococcioniM. 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.  Google Scholar

[17]

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

[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.   Google Scholar

[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.  Google Scholar

[20]

P. FletcherK. HrbacekV. KanoveiM. G. KatzC. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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

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

[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.  Google Scholar

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. $
[1]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[2]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[3]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[4]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[5]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[6]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

[7]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[8]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[9]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[10]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[11]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[12]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[13]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026

[14]

Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331

[15]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[16]

Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111

[17]

Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097

[18]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[19]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[20]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (15)
  • HTML views (34)
  • Cited by (0)

Other articles
by authors

[Back to Top]