Higher accuracy order in differentiation-by-integration
In this text explicit forms of several higher precision order kernel functions (to be used in the differentiation-by-integration procedure) are given for several derivative orders. Also, a system of linear equations is formulated which allows to construct kernels with an arbitrary precision for an arbitrary derivative order. A computer study is realized and it is shown that numerical differentiation based on higher precision order kernels performs much better (w.r.t. errors) than the same procedure based on the usual Legendre-polynomial kernels. Presented results may have implications for numerical implementations of the differentiation-by-integration method.
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N. Cioranescu. La généralisation de la première formule de la moyenne. Enseign. Math., 37:292–302, 1938.
E. Diekema. The fractional orthogonal derivative. Mathematics, 3:273–298, 2015. https://doi.org/10.3390/math3020273
E. Diekema. The fractional orthogonal difference with applications. Mathematics, 3(2):487–509, 2015. https://doi.org/10.3390/math3020487
E. Diekema. The fractional orthogonal derivative for functions of one and two variables. PhD thesis, 2018. Available from Internet: https://pure.uva.nl/ws/files/29401750/Thesis.pdf
E. Diekema and T.H. Koornwinder. Differentiation by integration using orthogonal polynomials, a survey. Journal of Approximation Theory, 164(5):637–667, 2012. https://doi.org/10.1016/j.jat.2012.01.003
J.W. Eaton, D.Bateman, S. Hauberg and R. Wehbring. GNU Octave version 5.1.0 manual: a high-level interactive language for numerical computations, 2019. Available from Internet: https://www.gnu.org/software/octave/doc/v5.1.0/
A.H. Galeana, R.L. Vázquez, J.L.-Bonilla and L.-I. Pişcoran. On the Cioranescu - (Haslam - Jones) - Lanczos generalized derivative. Global Journal of Advanced Researchon Classical and Modern Geometries, 3(1):44–99, 2014.
C.W. Groetsch. Lanczos’ generalized derivative. The American Mathematical Monthly, 105(4):320–326, 1998. https://doi.org/10.1080/00029890.1998.12004888
D.L. Hicks and L.M. Liebrock. Lanczos’ generalized derivative: Insights and applications. Applied Mathematics and Computation, 112(1):63–73, 2000. https://doi.org/10.1016/S0096-3003(99)00048-X
X. Huang, Ch. Wu and J. Zhou. Numerical differentiation by integration. Mathematics of Computation, 83(286):789–807, 2014. https://doi.org/10.1090/S0025-5718-2013-02722-6
C. Lanczos. Applied analysis. Prentice-Hall Mathematics, Englewood Cliffs, N.J., 1956.
A. Liptaj. Maximal generalization of Lanczos’ derivative using one-dimensional integrals. arXiv e-prints, p. arXiv:1906.04921, June 2019.
D.-Y. Liu, O. Gibaru and W. Perruquetti. Differentiation by integration with Jacobi polynomials. Journal of Computational and Applied Mathematics, 235(9):3015–3032, 2011. https://doi.org/10.1016/j.cam.2010.12.023
D.-Y. Liu, O. Gibaru, W. Perruquetti and T.-M. Laleg-Kirati. Fractional order differentiation by integration and error analysis in noisy environment. IEEE Transactions on Automatic Control, 60(11):2945–2960, 2015. https://doi.org/10.1109/TAC.2015.2417852
Maxima. Maxima, a computer algebra system. version 5.34.1, 2014. Available from Internet: http://maxima.sourceforge.net/
T.J. McDevitt. Discrete Lanczos derivatives of noisy data. International Journal of Computer Mathematics, 89(7):916–931, 2012. https://doi.org/10.1080/00207160.2012.666348
S.K. Rangarajan and S.P. Purushothaman. Lanczos generalized derivative for higher orders. Journal of Computational and Applied Mathematics, 177(2):461– 465, 2005. https://doi.org/10.1016/j.cam.2004.10.016
A. Savitzky and M.J.E. Golay. Smoothing and differentiation of data by simplified least squares procedures. Analytical Chemistry, 36(8):1627–1639, 1964. https://doi.org/10.1021/ac60214a047
J. Shen. On the generalized “Lanczos’ generalized derivative”. The American Mathematical Monthly, 106(8):766–768, 1999. https://doi.org/10.1080/00029890.1999.12005116
G.R.P. Teruel. A new class of generalized Lanczos derivatives. Palestine Journal of Mathematics, 7(1):211–221, 2018.
Z. Wang and R. Wen. Numerical differentiation for high orders by an integration method. Journal of Computational and Applied Mathematics, 234(3):941–948, 2010. https://doi.org/10.1016/j.cam.2010.01.056