On the errors incurred calculating derivatives using Chebyshev polynomials. (English) Zbl 0747.65009

The round-off errors arising from the computation of derivatives of functions approximated by Chebyshev polynomials are studied on an example function. It is found that the error grows as \(N^ 2\) for standard Chebyshev transform methods, where \(N+1\) is the number of Chebyshev polynomials used to approximate the function.
It is also shown that matrix multiplication techniques exhibit errors of the same order of magnitude but the computation of the matrix elements is ill-conditioned (errors grow as \(N^ 2\)). The errors are most severe near the boundaries of the domain. A method for reducing the error is discussed.


65D25 Numerical differentiation
65F30 Other matrix algorithms (MSC2010)
65G50 Roundoff error
Full Text: DOI


[1] Gottlieb, D.; Orszag, S. A., Numerical Analysis of Spectral Methods: Theory and Applications (1977), SIAM: SIAM Philadelphia · Zbl 0412.65058
[2] Kim, J.; Moin, P.; Moser, R., J. Fluid Mech., 177, 133 (1987)
[3] Trefethen, L. N.; Trummer, M. R., SIAM J. Numer. Anal., 24, No. 5, 1008 (1987)
[4] Greengard, L., Spectral Integration and Two-Point Boundary Value Problems, Yale University, Department of Computer Science Research Report (1988), submitted · Zbl 0731.65064
[5] Solomonoff, A.; Turkel, E., J. Comput. Phys., 18, 239 (1989)
[6] Heinrichs, W., Math. Comput., 53, 103 (1989)
[7] Gottlieb, D.; Hussaini, M. Y.; Orszag, S. A., (Voigt, D. G.; Gottlieb, D.; Hussaini, M. Y., Spectral Methods for Partial Differential Equations (1984), SIAM-CMBS: SIAM-CMBS Philadelphia), 1
[8] Peyret, R., Introduction to Spectral Methods (1986), von Karman Institute Lecture Series: von Karman Institute Lecture Series Rhode-St-Genese
[9] Golub, G. H.; van Loan, C. F., Matrix Computations (1983), Johns Hopkins University Press: Johns Hopkins University Press Baltimore · Zbl 0559.65011
[10] Orszag, S. A., J. Comput. Phys., 37, 70 (1980)
[11] Boyd, J. P., Chebyshev and Fourier Spectral Methods, (Lecture Notes in Engineering, Vol. 49 (1989), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0987.65122
[12] Fox, L.; Parker, I. B., Chebyshev Polynomials in Numerical Analysis (1968), Oxford Univ. Press: Oxford Univ. Press London · Zbl 0153.17502
[13] Zebib, A., J. Comput. Phys., 53, 443 (1984)
[14] Korczak, K. Z.; Patera, A. T., J. Comput. Phys., 62, 361 (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.