Clenshaw algorithm in the interpolation problem by the Chebyshev collocation method

Cover Page

Cite item

Full Text

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.

Full Text

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
×

About the authors

P. Lovetskiy Konstantin

RUDN University

Author for correspondence.
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

A. Tiutiunnik Anastasiia

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

Email: 1032199094@rudn.ru
student of Department of Computational Mathematics and Artificial Intelligence 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

References

  1. Boyd, J. P. Chebyshev and Fourier Spectral Methods: Second Revised Edition. Dover Books on Mathematics (Courier Corporation, 2013).
  2. Fornberg, B. A practical guide to pseudospectral methods doi: 10.1017/cbo9780511626357 (Cambridge University Press, 1996).
  3. Mason, J. C. & Handscomb, D. C. Chebyshev Polynomials in Chebyshev Polynomials (Chapman and Hall/CRC Press, 2002).
  4. Orszag, S. A. Comparison of Pseudospectral and Spectral Approximation. Studies in Applied Mathematics 51, 253-259. doi: 10.1002/sapm1972513253 (1972).
  5. Clenshaw, C. W. A note on the summation of Chebyshev series. Mathematics of Computation 9, 118-120. doi: 10.1090/S0025-5718-1955-0071856-0 (1955).
  6. Fox, L. & Parker, I. B. Chebyshev polynomials in numerical analysis (Oxford, 1968).
  7. Shen, Z. & Serkh, K. Is polynomial interpolation in the monomial basis unstable? 2023. doi:10. 48550/arXiv.2212.10519.
  8. Zhang, X. & Boyd, J. P. Asymptotic Coefficients and Errors for Chebyshev Polynomial Approximations with Weak Endpoint Singularities: Effects of Different Bases 2021. doi: 10.48550/arXiv.2103.11841.
  9. Lovetskiy, K. P., Sevastianov, L. A. & Nikolaev, N. E. Regularized Computation of Oscillatory Integrals with Stationary Points. Procedia Computer Science 108, 998-1007. doi: 10.1016/j.procs. 2017.05.028 (2017).
  10. Lovetskiy, K. P., Sevastianov, L. A., Kulyabov, D. S. & Nikolaev, N. E. Regularized computation of oscillatory integrals with stationary points. Journal of Computational Science 26, 22-27. doi:10. 1016/j.jocs.2018.03.001 (2018).
  11. Lovetskiy, K. P., Kulyabov, D. S. & Hissein, A. W. Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order. Discrete and Continuous Models and Applied Computational Science 30, 127-138. doi: 10.22363/26584670-2022-30-2-127-138 (2022).
  12. Lovetskiy, K. P., Sevastianov, L. A., Hnatič, M. & Kulyabov, D. S. Numerical Integration of Highly Oscillatory Functions with and without Stationary Points. Mathematics 12, 307. doi: 10.3390/math12020307 (2024).
  13. Sevastianov, L. A., Lovetskiy, K. P. & Kulyabov, D. S. An Effective Stable Numerical Method for Integrating Highly Oscillating Functions with a Linear Phase in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 12138 LNCS (2020), 29-43. doi: 10.1007/978-3-030-50417-5_3.
  14. Sevastianov, L. A., Lovetskiy, K. P. & Kulyabov, D. S. Numerical integrating of highly oscillating functions: effective stable algorithms in case of linear phase 2021. doi: 10.48550/arXiv.2104.03653.
  15. Sevastianov, L. A., Lovetskiy, K. P. & Kulyabov, D. S. A new approach to the formation of systems of linear algebraic equations for solving ordinary differential equations by the collocation method. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics 23, 36-47. doi:10. 18500/1816-9791-2023-23-1-36-47 (2023).
  16. Berrut, J. & Trefethen, L. N. Barycentric Lagrange Interpolation. SIAM Review 46, 501-517. doi: 10.1137/S0036144502417715 (2004).
  17. Epperson, J. F. On the Runge Example. The American Mathematical Monthly 94, 329. doi:10. 2307/2323093 (1987).
  18. Amiraslani, A., Corless, R. M. & Gunasingam, M. Differentiation matrices for univariate polynomials. Numerical Algorithms 83, 1-31. doi: 10.1007/s11075-019-00668-z (2020).
  19. Wang, Z. Interpolation using type i discrete cosine transform. Electronics Letters 26, 1170. doi: 10.1049/el:19900757 (1990).
  20. Wang, Z. Interpolation using the discrete cosine transform: reconsideration. Electronics Letters 29, 198. doi: 10.1049/el:19930133 (1993).

Copyright (c) 2024 Konstantin P.L., Anastasiia A.T., do Nascimento Vicente F.J., Boa Morte C.T.

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies