Non-standard analysis revisited: An easy axiomatic presentation oriented towards numerical applications

• Autorzy: Benci Vieri , Cococcioni Marco , Fiaschi Lorenzo

Identyfikatory

DOI

10.34768/amcs-2022-0006

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Alpha Theory; non-standard analysis; non Archimedean analysis; algorithmic numbers; non Archimedean scientific computing

PL

analiza niestandardowa; analiza niearchimedesowa; liczby algorytmiczne

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2022
• Tom: Vol. 32, no. 1
• Strony: 65--80
• Opis fizyczny: Bibliogr. 39 poz., rys., tab.

Twórcy

autor

Benci Vieri

• Department of Mathematics University of Pisa Via Filippo Buonarroti, 1/c, Pisa, 56122, Italy

autor

Cococcioni Marco

• Department of Information Engineering University of Pisa L.go Lucio Lazzarino, 1, Pisa, 56122, Italy

autor

Fiaschi Lorenzo

• Department of Information Engineering University of Pisa L.go Lucio Lazzarino, 1, Pisa, 56122, Italy

Bibliografia

• [1] Amodio, P., Iavernaro, F., Mazzia, F., Mukhametzhanov, M. and Sergeyev, Y. (2017). A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic, Mathematics and Computers in Simulation 141: 24–39, DOI: 10.1016/j.matcom.2016.03.007.
• [2] Arora, J.S. (2004). Introduction to Optimum Design, Elsevier, San Diego.
• [3] Astorino, A. and Fuduli, A. (2020). Spherical separation with infinitely far center, Soft Computing 24(23): 17751–17759.
• [4] Benci, V. and Cococcioni, M. (2020). The algorithmic numbers in non-Archimedean numerical computing environments, Discrete and Continuous Dynamical Systems S 14(5): 1673–1692, DOI: 10.3934/dcdss.2020449.
• [5] Benci, V. and Di Nasso, M. (2003). Numerosities of labelled sets: A new way of counting, Advances in Mathematics 173(1): 50–67.
• [6] Benci, V. and Di Nasso, M. (2018). How to Measure the Infinite: Mathematics with Infinite and Infinitesimal Numbers, World Scientific, Singapore.
• [7] Benci, V., Di Nasso, M. and Forti, M. (2006). The eightfold path to nonstandard analysis, in N. J. Cutland et al. (Eds), Nonstandard Methods and Applications in Mathematics, AK Peters, Wellesley, pp. 3–44.
• [8] Benci, V., Horsten, L. and Wenmackers, S. (2018). Infinitesimal probabilities, British Journal for the Philosophy of Science 69(2): 509–552.
• [9] Bierman, G.J. (2006). Factorization Methods for Discrete Sequential Estimation, Courier Corporation, North Chelmsford.
• [10] Cococcioni, M., Cudazzo, A., Pappalardo, M. and Sergeyev, Y. (2020). Solving the lexicographic multi-objective mixed-integer linear programming problem using branch-and-bound and grossone methodology, Communications in Nonlinear Science and Numerical Simulation 84: 105177, DOI: 10.1016/j.cnsns.2020.105177.
• [11] Cococcioni, M. and Fiaschi, L. (2020). The Big-M method with the numerical infinite M, Optimization Letters 15: 2455–2468, DOI: 10.1007/s11590-020-01644-6.
• [12] Cococcioni, M., Fiaschi, L. and Lambertini, L. (2021). Non-Achimedean zero-sum games, Journal of Applied and Computational Mathematics 113483: 1–17, DOI: 10.1016/j.cam.2021.113483.
• [13] Conway, J.H. (2000). On Numbers and Games, CRC Press, New York.
• [14] De Leone, R. (2018). Nonlinear programming and grossone: Quadratic programming and the role of constraint qualifications, Applied Mathematics and Computation 318: 290–297, DOI: 10.1016/j.amc.2017.03.029.
• [15] De Leone, R., Egidi, N. and Fatone, L. (2020a). The use of grossone in elastic net regularization and sparse support vector machines, Soft Computing 24(23): 17669–17677.
• [16] De Leone, R., Fasano, G., Roma, M. and Sergeyev, Y.D. (2020b). Iterative grossone-based computation of negative curvature directions in large-scale optimization, Journal of Optimization Theory and Applications 186(2): 554–589.
• [17] Dehn, M. (1900). Die Legendre’schen Sätze über die Winkelsumme im Dreieck, Mathematische Annalen 53(1900): 404–439, DOI: 10.1007/BF01448980.
• [18] Deveau, M. and Teismann, H. (2014). 72+ 42: Characterizations of the completeness and Archimedean properties of ordered fields, Real Analysis Exchange 39(2): 261–304.
• [19] Falcone, A., Garro, A., Mukhametzhanov, M.S. and Sergeyev, Y.D. (2020a). Representation of grossone-based arithmetic in simulink for scientific computing, Soft Computing 24(23): 17525–17539.
• [20] Falcone, A., Garro, A., Mukhametzhanov, M.S. and Sergeyev, Y.D. (2020b). A Simulink-based software solution using the infinity computer methodology for higher order differentiation, Applied Mathematics and Computation 409: 125606, DOI: 10.1016/j.amc.2020.125606.
• [21] Fiaschi, L. and Cococcioni, M. (2018). Numerical asymptotic results in game theory using Sergeyev’s infinity computing, International Journal of Unconventional Computing 14: 1–25.
• [22] Fiaschi, L. and Cococcioni, M. (2020). Non-Archimedean game theory: A numerical approach, Applied Mathematics and Computation 409: 125356, DOI: 10.1016/j.amc.2020.125356.
• [23] Fiaschi, L. and Cococcioni, M. (2021). A non-Archimedean interior point method for solving lexicographic multi-objective quadratic programming problems, EURO Journal on Computational Optimization, (submitted).
• [24] Gagniuc, P.A. (2017). Markov Chains: From Theory to Implementation and Experimentation, Wiley, Hoboken.
• [25] Golub, G.H. and Van Loan, C.F. (2013). Matrix Computations, JHU Press, Baltimore.
• [26] Iavernaro, F., Mazzia, F., Mukhametzhanov, M. and Sergeyev, Y. (2020). Conjugate-symplecticity properties of Euler–Maclaurin methods and their implementation on the infinity computer, Applied Numerical Mathematics 155: 58–72, DOI: 10.1016/j.apnum.2019.06.011.
• [27] Keisler, H.J. (1976). Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston.
• [28] Krishnamoorthy, A. and Menon, D. (2013). Matrix inversion using Cholesky decomposition, 2013 IEEE Conference on Signal Processing: Algorithms, Architectures, Arrangements, and Applications (SPA’13), Poznan, Poland, pp. 70–72.
• [29] Lai, L., Fiaschi, L., Cococcioni, M. and Deb, K. (2021a). Handling priority levels in mixed Pareto-lexicographic many-objective optimization problems, Evolutionary Multi-Criterion Optimization, Shenzhen, China, pp. 362–374, DOI: 10.1007/978-3-030-72062-9 29.
• [30] Lai, L., Fiaschi, L., Cococcioni, M. and Deb, K. (2021b). Solving mixed pareto-lexicographic multi-objective optimization problems: The case of priority levels, IEEE Transactions on Evolutionary Computation 25(5): 971–985, DOI: 10.1109/TEVC.2021.3068816.
• [31] Levi-Civita, T. (1892). Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti del R. Istituto Veneto di Scienze Lettere ed Arti Series 7: 1892–1893.
• [32] Mises, R. and Pollaczek-Geiringer, H. (1929). Praktische verfahren der gleichungsauflösung, Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 9(1): 58–77.
• [33] Pohlhausen, E. (1921). Berechnung der eigenschwingungen statisch-bestimmter fachwerke, Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 1(1): 28–42.
• [34] Robinson, A. (1996). Non-Standard Analysis, 2nd Edn, Princeton University Press, Princeton.
• [35] Sergeyev, Y. (2017). Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Mathematical Sciences 4(2): 219––320.
• [36] Sergeyev, Y.D., Mukhametzhanov, M., Mazzia, F., Iavernaro, F. and Amodio, P. (2016). Numerical methods for solving initial value problems on the infinity computer, Journal of Unconventional Computing 12(1): 3–23.
• [37] Thompson, G.L. and Weil, Jr, R.L. (1969). Further relations between game theory and eigensystems, SIAM Review 11(4): 597–602.
• [38] Thompson, G.L. and Weil, R.L. (1972). The roots of matrix pencils (Ay = λBy): Existence, calculations, and relations to game theory, Linear Algebra and Its Applications 5(3): 207–226.
• [39] Weil, Jr, R.L. (1968). Game theory and eigensystems, SIAM Review 10(3): 360–367.