Clenshaw algorithm in the interpolation problem by the Chebyshev collocation method

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Abstract

The article describes a method for calculating interpolation coefficients of expansion using Chebyshev polynomials. The method is valid when the desired function is bounded and has a finite number of maxima and minima in a finite domain of interpolation. The essence of the method is that the interpolated desired function can be represented as an expansion in Chebyshev polynomials; then the expansion coefficients are determined using the collocation method by reducing the problem to solving a well-conditioned system of linear algebraic equations for the required coefficients. Using the well-known useful properties of Chebyshev polynomials can significantly simplify the solution of the problem of function interpolation. A technique using the Clenshaw algorithm for summing the series and determining the expansion coefficients of the interpolated function, based on the discrete orthogonality of Chebyshev polynomials of the 1st kind, is outlined.

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1. Introduction The construction of efficient numerical methods for solving differential and integral equations is an important element in solving applied problems in various fields, such as aerospace engineering, meteorology, physical oceanography, mechanical engineering, and nuclear energy. Taking this into account, we will consider and analyze the efficiency of some spectral algorithms for function interpolation, which are often used when solving equations of mathematical physics. Spectral methods are a class of methods used in applied mathematics for the numerical solution of certain differential and integral equations, sometimes using the fast Fourier transform [1-4]. The idea is to present the desired solution
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About the authors

Konstantin P. Lovetskiy

RUDN University

Email: lovetskiy-kp@rudn.ru
ORCID iD: 0000-0002-3645-1060

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Computational Mathematics and Artificial Intelligence

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Anastasiia A. Tiutiunnik

RUDN University

Email: tyutyunnik-aa@rudn.ru
ORCID iD: 0000-0002-4643-327X

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Computational Mathematics and Artificial Intelligence

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Felix Jose do Nascimento Vicente

RUDN University

Email: 1032199092@rudn.ru
student of Department of Computational Mathematics and Artificial Intelligence 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Celmilton Teixeira Boa Morte

RUDN University

Author for correspondence.
Email: 1032199094@rudn.ru

student of Department of Computational Mathematics and Artificial Intelligence

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

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Copyright (c) 2024 Lovetskiy K.P., Tiutiunnik A.A., do Nascimento Vicente F.J., Teixeira Boa M.C.

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