METHOD OF OBTAINING APPROXIMATE FORMULAS

Konstantin Ludanov

Abstract


The two-parameter method of approximating the sum of a power series in terms of its first three terms of the expansion, which allows one to obtain analytic approximations of various functions, decomposes into a Maclaurin series. As an approximation function of this approximation, it is proposed to use elementary functions constructed in the Nth degree, but with a "compressed" or "stretched" variable x due to the introduction of the numerical factor M (x ≡ ε ∙ m, M ≠ 0) into it. The use of this method makes it possible to significantly increase the range of very accurate approximation of the obtained approximate function with respect to a similar range of the output fragment of a series of three terms. Expressions for both the approximation parameters (M and N) are obtained in a general form and are determined by the coefficients of the second and third terms of the Maclaurin series. Also expressions of both approximation parameters are found for the case if the basis function and the approximant function decompose into the Maclaurin series in even powers of the argument. A number of examples of approximation of functions on the basis of the analysis of power series into which they decompose are given.


Keywords


approximation of functions; power series expansion; Maclaurin series; sum of a power series; two-parameter approximation; Newton's binomial

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References


Koul, Dzh. (1972). Metody vozmushheniy v prikladnoy matematike. Moscow: MIR, 275.

Van-Dayk, M. (1967). Metody vozmushheniy v dinamike zhidkosti. Moscow: MIR, 311.

Popov, V. A., Tesler, G. S. (1980). Priblizhenie funktsiy dlya tekhnicheskikh prilozheniy. Kyiv: Naukova dumka, 353.

Krylov, V. I. (1988). Matematicheskiy analiz. Uskorenie skhodimosti. Minsk: N i T, 175.

Baker, G. A. Jr. (1965). The theory and application of the Pade approximant method. Advances in Theoretical Physics, 1, 1–58.

Baker, G. A. Jr. (1975). Essentials of Pade approximants. New York: Academic Press, 318.

Baker, G. A. Jr., Graves-Morris, P. (1981). Pade Approximants. Addison-Wesley Publishing Co.

Vinogradov, V. N., Gay, E. V., Rabotnov, N. S. (1987). Analiticheskaya approksimatsiya dannykh v yadernoy i neytronnoy fizike. Moscow: Energoatomizdat, 128.

Ludanov, K. I. (1984). Metod approksimatsii summy stepennogo ryada po ego pervym trem chlenam. Uk-84, Dep. № 763 v UkrNIINTI. Kyiv, 7.

Fisher, M. E. (1965). The nature of critical points. Boulder: University of Colorado Press, 159.




DOI: http://dx.doi.org/10.21303/2461-4262.2018.00589

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ISSN 2461-4262 (Online), ISSN 2461-4254 (Print)