Method of interpolation using root-fractional-rational functions of different orders
DOI:
https://doi.org/10.17721/1812-5409.2025/1.17Keywords:
interpolation, root-fractional-rational function, electrodynamic tasks, functions of probability theory, fuzzy logic, membership functionsAbstract
The possibility of interpolating different kinds of mathematical functions using different order root-fractional-rational functions, namely, second, third, and fourth, is considered in the article. Generally, it is shown that root-fractional-rational functions are given a precision interpolation with a small value of error even for stiff analytical dependences. Root-fractional-rational functions from second to fourth order are considered, and corresponding analytical relations for defining polynomial coefficients in the nominator and the denominator are given. It is also proven that the number of necessary points for interpolation corresponds to the value 2n+1, where n is the order of the root-fractional-rational function. Examples of interpolation of different functions for electrodynamics problems, simulation of magnetic lenses, probability tasks, and fuzzy-logic tasks are given; the error of interpolation for all considered examples is also defined. All presumptions of theoretical analysis are tested and verified using the original elaborated computer software, created using the Python programming language means. In the most considered examples, the resulting error of interpolation is smaller than a few percent. The graphic results of testing the proposed method of interpolation are also given.
Pages of the article in the issue: 130 - 138
Language of the article: English
References
Akhonin, S. V., Berezos, B. A., Pikulin, A. N., Severin, A. Yu., Shvab, S. L., & Erokhin, A. G. (2018). Production of high-strength titanium alloy VT22 by electron beam melting method. Advances in Electrometallurgy, 3(131), 8–14 [in Russian]. https://patonpublishinghouse.com/sem/pdf/2018/sem201803all.pdf
Bronshtein, I. N., Semendyayev, K. A., Musiol, G., & Mühlig, H. (2007). Handbook of Mathematics (5th ed.). Springer. https://doi.org/10.1007/978-3-662-46221-8
Chapra, S. C., & Canale, R. P. (2014). Numerical Methods for Engineers (7th ed.). McGraw Hill.
Chollet, F. (2022). Deep Learning with Python (2nd ed.). Manning. https://doi.org/10.31211/interacoes.n42.2022.r1
Druzhinin, A. A., Ostrovskii, I. P., Khoverko, Y. N., Liakh-Kaguy, N. S., & Vuytsyk, A. M. (2014). Low temperature characteristics of germanium whiskers. Functional materials, 21(2), 130–136. http://functmaterials.org.ua/contents/21-2/2; http://dx.doi.org/10.15407/fm21.02.130
Druzhinin, A. A., Bolshakova, I. A., Ostrovskii, I. P., Khoverko, Y. N., & Liakh-Kaguy, N. S. (2015). Low temperature magnetoresistance of InSb whiskers. Materials Science in Semiconductor Processing, 40, 550–555. https://doi.org/10.1016/j.mssp.2015.07.030
Epperson, J. F. (2013). An Introduction to Numerical Methods and Analysis (2nd ed.). Wiley-Interscience. https://doi.org/10.1002/9781119604570
Grechanyuk, I. M., Grechanyuk, M. I., Bagliuk, G. A., Grechanyuk, V. G., Khomenko, O. V., Dudnik, O. V., & Gots, V. I. (2022). Electron-Beam and Plasma Oxidation-Resistant and Thermal-Barrier Coatings Deposited on Turbine Blades Using Cast and Powder Ni(Co)CrAlY(Si) Alloys Produced by Electron Beam Melting IV. Chemical and Phase Composition and Structure of Cocralysi Powder Alloys and Their Use. Powder Metallurgy and Metal Ceramics, 61(7–8), 459–464. https://doi.org/10.1007/s11106-023-00333-0
Grivet, P. (1972). Electron Optics. Elsevier. https://doi.org/10.1016/C2013-0-02400-0
Gubner, J. A. (2006). Probability and random processes for electrical and computer engineers. Cambridge University Press. http://www.cambridge.org/gb/academic/subjects/engineering/communications-and-signal-processing/probability-and-random-processes-electrical-and-computer-engineers
Jain, M. K., Iengar, S. R. K., & Jain, R. K. (2010). Numerical Methods for Scientific and Engineering Computation. New Age International Pvt. Ltd. https://www.academia.edu/8565134/Numerical_Methods_For_Scientific_And_Engineering_Computation_M_K_Jain_S_R_K_Iyengar_And_R_K_Jain
Kasper, E., & Hawkes, P. (2018). Principles of Electron Optics. Applied Geometrical Optics. Elsevier Science. https://doi.org/10.1016/C2015-0-06653-9
Kemmotsu, T., Nagai, T., & Maeda, M. (2011). Removal Rate of Phosphorous form Melting Silicon. High Temperature Materials and Processes, 30(1–2), 17–22. https://doi.org/10.1515/htmp.2011.002
Lutz, M. (2013). Learning Python (5th ed.). O'Reilly. https://www.oreilly.com/library/view/learning-python-5th/9781449355722/
Mathews, J. H., & Fink, K. D. (1998). Numerical Methods. Using MATLAB (3rd ed.). Prentice Hall. https://convexoptimization.com/TOOLS/Fink.pdf
McKinney, W. (2023). Python for Data Analysis: Data Wrangling with Pandas, NumPy, and Jupyter (3rd ed.). O'Reilly Media. https://www.oreilly.com/library/view/python-for-data/9781098104023/
Melnyk, I. (2005). Numerical simulation of distribution of electric field and particle trajectories in electron sources based on high-voltage glow discharge. Radioelectronic and Communication Systems, 48(6), 61–71. http://radioelektronika.org/article/view/S0735272705060087
Melnik, I. (2006). Simulation of geometry of high voltage glow discharge electrodes' systems, formed profile electron beams. In A. M. Filachev (Ed.), Proceedings of SPIE: Vol. 6278. Seventh Seminar on Problems of Theoretical and applied Electron and Ion Optics. 627809-1–627809-13. https://doi.org/10.1117/12.693202
Melnyk, I., & Luntovskyy, A. (2022). Estimation of Energy Efficiency and Quality of Service in Cloud Realizations of Parallel Computing Algorithms for IBN. In M. Klymash, M. Beshley, & A. Luntovskyy (Eds.), Lecture Notes in Electrical Engineering: Vol. 831. Future Intent-Based Networking (pp. 339–379). Springer, Cham. https://doi.org/10.1007/978-3-030-92435-5_20
Melnyk, I., & Pochynok, A. (2020). Study of a Class of Algebraic Functions for Interpolation of Boundary Trajectories of Short-Focus Electron Beams. System Researches and Information Technologies, 3, 23–39 [in Ukrainian]. [Мельник, І. В., Починок, А. В. (2020). Дослідження класу алгебричних функцій для інтерполяції межових траєкторій короткофокусних електронних пучків, Системні дослідження та інформаційні технології, 3, 8–14] https://doi.org/10.20535/SRIT.2308-8893.2020.3.02
Melnyk, I., & Pochynok, A. (2023). Theoretical justification of application possibility of different order root-polynomial functions for interpolation and approximation of boundary trajectory of electron beam. Radioelectronics and Communications Systems, 66(2), 53–73. https://doi.org/10.3103/S0735272723020024
Melnyk, I., Tuhai, S., Surzhykov, M., Shved, I., Melnyk, V., & Kovalchuk, D. (2022). Analytical Estimation of the Deep of Seam Penetration for the Electron-Beam Welding Technologies with Application of Glow Discharge Electron Guns. In 2022 IEEE 41st International Conference on Electronics and Nanotechnology (pp. 1–5). IEEE. https://doi.org/10.1109/ELNANO54667.2022.9927071
Paton, B. E., Akhonin, S. V., & Berezos, V. A. (2018). Development of technologies for electron-beam melting of metals at the E.O. Paton Electric Welding Institute NAS of Ukraine. Advances Electrometallurgy, 4(132), 19–35 [in Russian]. http://dx.doi.org/10.15407/sem2018.04.01
Phillips, G. M. (2023). Interpolation and Approximation by Polynomials. Springer. https://doi.org/10.1007/b97417
Prikhna, T. O., Grechanyuk, I. M., Karpets, M. V., Grechanyuk, M. I., Bagliuk, G. A., Grechanyuk, V. G., & Chornovol, V. A. (2022а). Electron-Beam and Plasma Oxidation-Resistant and Thermal-Barrier Coatings Deposited on Turbine Blades Using Cast and Powder Ni(Co)CrALY(Si) Alloys I. Fundamentals of the Production Technology. Structure, and Phase Composition of Cast NiCrAlY Alloys. Powder Metallurgy and Metal Ceramics, 61(1–2), 70–76. https://doi.org/10.1007/s11106-022-00296-8
Prikhna, T. O., Grechanyuk, I. M., Karpets, M. V., Grechanyuk, M. I., Bagliuk, G. A., Grechanyuk, V. G., & Khomenko, O. V. (2022b). Electron-Beam and Plasma Oxidation-Resistant and Thermal-Barrier Coatings Deposited on Turbine Blades Using Cast and Powder Ni(Co)CrAlY(Si) Alloys Produced by Electron-Beam Melting II. Structure and Chemical and Phase Composition of Cast CoCrAlY Alloys. Powder Metallurgy and Metal Ceramicsthis, 61(3–4), 230–237. https://doi.org/10.1007/s11106-022-00310-z
Raizer, Yu. P. (1991). Gas Discharge Physics. Springer. https://link.springer.com/book/9783642647604
Rudychev, V. G., Lazurik, V. T., & Rudychev, Y. V. (2021). Influence of the electron beams incidence angles on the depth-dose distribution of the irradiated object. Radiation Physics and Chemistry, 186, 109527. https://doi.org/10.1016/j.radphyschem.2021.109527
Schiller, S., Heisig, U., & Panzer, S. (1982). Electron Beam Technology. John Wiley & Sons.
Smirnov, B. M. (2015). Theory of Gas Discharge Plasma. Springer Cham. https://doi.org/10.1007/978-3-319-11065-3
Szilagyi, M. (2012). Electron and Ion Optics. Springer. https://doi.org/10.1007/978-1-4613-0923-9
Tseluyko, A. F., Lazurik, V. T., Ryabchikov, D. L., Maslov, V. I., & Sereda, I. N. (2008). Experimental study of radiation in the wavelength range 12.2-15.8 nm from a pulsed high-current plasma diode. Plasma Physics Reports. 34(11), 963–968. https://doi.org/10.1134/S1063780X0811010X
Wentzel, E., & Ovcharov, L. (2002). Applied Problems of Probability Theory. Mir Publisher.
Zadeh, L. A., & Aliev, R. A. (2018). Fuzzy Logic Theory and Applications. World Scientific.
Zakharov, A., Rozenko, S., Litvintsev, S., & Ilchenko, M. (2020). Trisection Bandpass Filter with Mixed Cross-Coupling and Different Paths for Signal Propagation. IEEE Microwave Wireless Component Letters, 30(1), 12–15. https://doi.org/10.1109/LMWC.2019.2957207
Zakharov, A., Rozenko, S., & Ilchenko, M. (2019). Varactor-Tuned Microstrip Bandpass Filter with Loop Hairpin and Combline Resonators. IEEE Transactions on Circuits Systems. II. Experimental Briefs, 66(6), 953–957. https://doi.org/10.1109/LMWC.2019.2929650
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