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A Simulink-based software solution using the infinity computer methodology for higher order differentiation. (English) Zbl 1510.65004

Summary: This paper is dedicated to numerical computation of higher order derivatives in Simulink. In this paper, a new module has been implemented to achieve this purpose within the Simulink-based Infinity Computer solution, recently introduced by the authors. This module offers several blocks to calculate higher order derivatives of a function given by the arithmetic operations and elementary functions. Traditionally, this can be done in Simulink using finite differences only, for which it is well-known that they can be characterized by instability and low accuracy. Moreover, the proposed module allows to calculate higher order Lie derivatives embedded in the numerical solution to Ordinary Differential Equations (ODEs). Traditionally, Simulink does not offer any practical solution for this case without using difficult external libraries and methodologies, which are domain-specific, not general-purpose and have their own limitations. The proposed differentiation module bridges this gap, is simple and does not require any additional knowledge or skills except basic knowledge of the Simulink programming language. Finally, the block for constructing the Taylor expansion of the differentiated function is also proposed, adding so another efficient numerical method for solving ODEs and for polynomial approximation of the functions. Numerical experiments on several classes of test problems confirm advantages of the proposed solution.

MSC:

65-04 Software, source code, etc. for problems pertaining to numerical analysis
65D25 Numerical differentiation
90C30 Nonlinear programming
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[1] EMSOFT ’11, Association for Computing Machinery,
[2] Alur, R., Principles of Cyber-physical Systems (2015), MIT Press
[3] Amodio, P.; Iavernaro, F.; Mazzia, F.; Mukhametzhanov, M. S.; Sergeyev, Y. D., A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic, Math Comput Simul, 141, 24-39 (2017) · Zbl 07313861
[4] ISBN 978-3-540-48060-0
[5] ISBN 978-3-540-32834-6
[6] Bar-Yam, Y., Dynamics of Complex Systems (2019), CRC Press
[7] Baydin, A. G.; Pearlmutter, B. A.; Radul, A. A.; Siskind, M., Automatic differentiation in machine learning: a survey, The Journal of Machine Learning Research, 18, 1-43 (2018) · Zbl 06982909
[8] Bocciarelli, P.; D’Ambrogio, A.; Falcone, A.; Garro, A.; Giglio, A., A model-driven approach to enable the simulation of complex systems on distributed architectures, SIMULATION: Transactions of the Society for Modeling and Simulation International, 95, 12 (2019)
[9] Calude, C. S.; Dumitrescu, M., Infinitesimal probabilities based on grossone, SN Computer Science, 1 (2020)
[10] Cococcioni, M.; Pappalardo, M.; Sergeyev, Y. D., Lexicographic multiobjective linear programming using grossone methodology: theory and algorithm, Appl Math Comput, 318, 298-311 (2018) · Zbl 1426.90226
[11] Cococcioni, M.; Cudazzo, A.; Pappalardo, M.; Sergeyev, Y. D., Solving the lexicographic multi-objective mixed-integer linear programming problem using branch-and-bound and grossone methodology, Commun. Nonlinear Sci. Numer. Simul., 105177 (2020) · Zbl 1451.90140
[12] Committee, I. C.S. S.; Institute, A. N.S., IEEE Standard for Binary Floating-point Arithmetic, vol. 754 (1985), IEEE
[13] D’Alotto, L., Cellular automata using infinite computations, Appl Math Comput, 218, 16, 8077-8082 (2012) · Zbl 1252.37017
[14] Daponte, P.; Grimaldi, D.; Molinaro, A.; Sergeyev, Y. D., An algorithm for finding the zero-crossing of time signals with Lipschitzean derivatives, Measurement, 16, 1, 37-49 (1995)
[15] De Cosmis, S.; De Leone, R., The use of grossone in mathematical programming and operations research, Appl Math Comput, 218, 16, 8029-8038 (2012) · Zbl 1273.90117
[16] De Leone, R., Nonlinear programming and grossone: Quadratic programming and the role of constraint qualifications, Appl Math Comput, 318, 290-297 (2018) · Zbl 1426.90235
[17] De Leone, R.; Fasano, G.; Sergeyev, Y. D., Planar methods and grossone for the conjugate gradient breakdown in nonlinear programming, Comput Optim Appl, 71, 1, 73-93 (2018) · Zbl 1465.90123
[18] Doyen, L.; Frehse, G.; Pappas, G. J.; Platzer, A., Verification of Hybrid Systems, 1047-1110 (2018), Springer International Publishing: Springer International Publishing Cham · Zbl 1392.68246
[19] ISBN 9780534382162
[20] Dime University of Genoa, 2016. ISBN 978-889799970-6
[21] (August): ISSN 1569-190X
[22] Institute of Electrical and Electronics Engineers Inc., 2015. ISBN 978-1-4673-7822-2
[23] Institute of Electrical and Electronics Engineers Inc., 2017. ISBN 978-1-5386-3428-8
[24] Institute of Electrical and Electronics Engineers Inc., 2017
[25] A. Falcone, A. Garro, S.J.E. Taylor, A. Anagnostou, Simplifying the development of HLA-based distributed simulations with the HLA development kit software framework (DKF), 2017c, 21st IEEE/ACM International Symposium on Distributed Simulation and Real Time Applications, DS-RT 2017, Rome, Italy, October 18-20, 216-217, 10.1109/DISTRA.2017.8167691. Institute of Electrical and Electronics Engineers Inc., 2017d.
[26] ISBN 978-3-030-40616-5 · Zbl 07250769
[27] Falcone, A.; Garro, A.; Mukhametzhanov, M. S.; Sergeyev, Y. D., Representation of grossone-based arithmetic in simulink and applications to scientific computing, Soft comput (2020)
[28] Fiaschi, L.; Cococcioni, M., Numerical asymptotic results in game theory using sergeyev’s infinity computing, International Journal of Unconventional Computing, 14, 1, 1-25 (2018)
[29] Fornberg, B., Numerical differentiation of analytic functions, ACM Trans. Math. Software, 7, 4, 512-526 (1981) · Zbl 0465.65012
[30] A. Fuller, Z. Fan, C. Day, Digital twin: Enabling technology, challenges and open research, 2019, ArXiv preprint arXiv:1911.01276.
[31] Association for Computing Machinery Inc.
[32] Gaudioso, M.; Giallombardo, G.; Mukhametzhanov, M. S., Numerical infinitesimals in a variable metric method for convex nonsmooth optimization, Appl Math Comput, 318, 312-320 (2018) · Zbl 1426.90197
[33] Gergel, V. P.; Sergeyev, Y. D., Sequential and parallel algorithms for global minimizing functions with Lipschitzian derivatives, Computers & Mathematics with Applications, 37, 4-5, 163-179 (1999) · Zbl 0931.90049
[34] Grossman, R. L.; Nerode, A.; Ravn, A. P.; Rischel, H., Hybrid systems, volume 736 (1993), Springer · Zbl 0825.00044
[35] Iavernaro, F.; Mazzia, F.; Mukhametzhanov, M. S.; Sergeyev, Y. D., Conjugate-symplecticity properties of Euler-Maclaurin methods and their implementation on the infinity computer, Appl. Numer. Math., 155, 58-72 (2020) · Zbl 1440.65273
[36] Iavernaro, F.; Mazzia, F.; Mukhametzhanov, M. S.; Sergeyev, Y. D., Computation of higher order Lie Derivatives on the Infinity Computer, Journal of Computational and Applied Mathematics (2020), in Press · Zbl 1440.65273
[37] Isidori, A., Nonlinear Control Systems, Third Edition, Communications and Control Engineering (1995), Springer-Verlag: Springer-Verlag London · Zbl 0878.93001
[38] Kvasov, D. E.; Sergeyev, Y. D., A univariate global search working with a set of Lipschitz constants for the first derivative, Optimization Letters, 3, 2, 303-318 (2009) · Zbl 1173.90544
[39] Lantoine, G.; Russell, R.; Dargent, T., Using multicomplex variables for automatic computation of high-order derivatives, ACM Trans. Math. Software, 38, 3 (2012) · Zbl 1365.65055
[40] Lee, J. M., Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics (2006), Springer: Springer New York
[41] Lunze, J.; Lamnabhi-Lagarrigue, F., Handbook of Hybrid Systems Control: Theory, Tools, Applications (2009), Cambridge University Press · Zbl 1180.93001
[42] MathWorks, Simulink home page, 2020, https://www.mathworks.com/products/simulink.html. Accessed 23 Apr 2020.
[43] Institute of Electrical and Electronics Engineers Inc.
[44] Mukhametzhanov, M. S.; Sergeyev, Y. D., The infinity computer vs. symbolic computations: First steps in comparison (2020), AIP Publishing, in press: AIP Publishing, in press New York
[45] Pintér, J. D., Global optimization: software, test problems, and applications, (Pardalos, P. M.; Romeijn, H. E., Handbook of Global Optimization, volume 2 (2002), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 515-569 · Zbl 1029.90058
[46] ISSN 0168-7433 · Zbl 1181.03035
[47] Rizza, D., Numerical methods for infinite decision-making processes, International Journal of Unconventional Computing, 14, 2, 139-158 (2019)
[48] Y.D. Sergeyev, Arithmetic of infinity, 2013, Edizioni Orizzonti Meridionali, CS, 2003, 2nd ed. · Zbl 1076.03048
[49] Sergeyev, Y. D., Higher order numerical differentiation on the infinity computer, Optimization Letters, 5, 4, 575-585 (2011) · Zbl 1230.65028
[50] Sergeyev, Y. D., Solving ordinary differential equations by working with infinitesimals numerically on the infinity computer, Appl Math Comput, 219, 22, 10668-10681 (2013) · Zbl 1303.65061
[51] Sergeyev, Y. D., Numerical infinities and infinitesimals: methodology, applications, and repercussions on two hilbert problems, EMS Surveys in Mathematical Sciences, 4, 219-320 (2017) · Zbl 1390.03048
[52] Sergeyev, Y. D., Independence of the grossone-based infinity methodology from non-standard analysis and comments upon logical fallacies in some texts asserting the opposite, Found Sci, 24, 1, 153-170 (2019) · Zbl 1428.03076
[53] Sergeyev, Y. D.; Garro, A., Single-tape and multi-tape turing machines through the lens of the Grossone methodology, Journal of Supercomputing, 65, 2, 645-663 (2013)
[54] Sergeyev, Y. D.; Mukhametzhanov, M. S.; Kvasov, D. E.; Lera, D., Derivative-free local tuning and local improvement techniques embedded in the univariate global optimization, J Optim Theory Appl, 171, 1, 186-208 (2016) · Zbl 1351.90134
[55] Sergeyev, Y. D.; Mukhametzhanov, M. S.; Mazzia, F.; Iavernaro, F.; Amodio, P., Numerical methods for solving initial value problems on the infinity computer, International Journal of Unconventional Computing, 12, 1, 3-23 (2016)
[56] Sergeyev, Y. D.; Kvasov, D. E.; Mukhametzhanov, M. S., On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales, Commun. Nonlinear Sci. Numer. Simul., 59, 319-330 (2018) · Zbl 1510.90292
[57] Srajer, F.; Kukelova, Z.; Fitzgibbon, A., A benchmark of selected algorithmic differentiation tools on some problems in computer vision and machine learning, Optimization Methods and Software, 33, 4-6, 889-906 (2018) · Zbl 1453.65050
[58] Suenaga, K.; Ishizawa, T., Generalized property-directed reachability for hybrid systems, International Conference on Verification, Model Checking, and Abstract Interpretation, 293-313 (2020), Springer · Zbl 07228512
[59] Wagg, D.; Worden, K.; Barthorpe, R.; Gardner, P., Digital twins: state-of-the-art future directions for modelling and simulation in engineering dynamics applications, ASCE-ASME J Risk and Uncert in Engrg Sys Part B Mech Engrg (2020)
[60] Weinstein, M.; Rao, A., Algorithm 984: adigator, a toolbox for the algorithmic differentiation of mathematical functions in MATLAB using source transformation via operator overloading, ACM Trans. Math. Software, 44, 2 (2017) · Zbl 1484.65363
[61] Yu, W.; Blair, M., Dnad, a simple tool for automatic differentiation of fortran codes using dual numbers, Comput Phys Commun, 184, 5, 1446-1452 (2013) · Zbl 1315.65023
[62] Zhao, D.; Wang, Z.; Dai, Y., Importance of the first-order derivative formula in the Obrechkoff method, Comput Phys Commun, 167, 2, 65-75 (2005) · Zbl 1196.65129
[63] Zhigljavsky, A. A., Computing sums of conditionally convergent and divergent series using the concept of Ggrossone, Appl Math Comput, 218, 16, 8064-8076 (2012) · Zbl 1254.03123
[64] Žilinskas, A., On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions, Appl Math Comput, 218, 16, 8131-8136 (2012) · Zbl 1245.90094
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