Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)4138810.22363/2658-4670-2024-32-2-202-212CRLKAJArticlesСтатьиResearch ArticleClenshaw algorithm in the interpolation problem by the Chebyshev collocation methodАлгоритм Кленшоу в задаче интерполяции методом Чебышевской коллокацииhttps://orcid.org/0000-0002-3645-1060LovetskiyKonstantin P.ЛовецкийК. П.

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

lovetskiy-kp@rudn.ruhttps://orcid.org/0000-0002-4643-327XTiutiunnikAnastasiia A.ТютюнникА. А.

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

tyutyunnik-aa@rudn.rudo Nascimento VicenteFelix JoseДу Нашсименту ВисентеФеликс Жозеstudent of Department of Computational Mathematics and Artificial Intelligence1032199092@rudn.ruBoa MorteCelmilton TeixeiraБоа МортеСелмилтон Тейшейра

student of Department of Computational Mathematics and Artificial Intelligence

1032199094@rudn.ruRUDN UniversityРоссийский университет дружбы народов15102024322VOL 32, NO2 (2024)ТОМ 32, №2 (2024)20221201112024Copyright ©; 2024, Lovetskiy K.P., Tiutiunnik A.A., do Nascimento Vicente F.J., Teixeira Boa M.C.Copyright ©; 2024, Ловецкий К.П., Тютюнник А.А., Ду Нашсименту Висенте Ф.Ж., Тейшейра Боа М.С.2024Lovetskiy K.P., Tiutiunnik A.A., do Nascimento Vicente F.J., Teixeira Boa M.C.Ловецкий К.П., Тютюнник А.А., Ду Нашсименту Висенте Ф.Ж., Тейшейра Боа М.С.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/41388

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.

В статье описан метод вычисления интерполяционных коэффициентов разложения по полиномам Чебышева. Метод справедлив, когда искомое функция ограничена и имеет конечное число максимумов и минимумов в конечной области интерполирования. Суть метода состоит в том, что интерполируемая искомая функция может быть представлена в виде разложения по полиномам Чебышева; затем коэффициенты разложения определяются по методу коллокаций сведением задачи к решению хорошо обусловленной системы линейных алгебраических уравнений относительно искомых коэффициентов. Использование известных полезных свойств полиномов Чебышева позволяет значительно упростить решение задачи интерполяции функций. Изложена методика использования алгоритма Кленшоу для суммирования рядов и определения коэффициентов разложения интерполируемой функции, основанная на дискретной ортогональности полиномов Чебышева 1-го рода.

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