Research Article | | Peer-Reviewed

### Lagrange Interpolation in Matrix Form for Numerical Differentiation and Integration

Received: 15 April 2024     Accepted: 6 May 2024     Published: 19 June 2024
Abstract

Numerical differentiation has been widely applied in engineering practice due to its remarkable simplicity in the approximation of derivatives. Existing formulas rely on only three-point interpolation to compute derivatives when dealing with irregular sampling intervals. However, it is widely recognized that employing five-point interpolation yields a more accurate estimation compared to the three-point method. Thus, the objective of this study is to develop formulas for numerical differentiation using more than three sample points, particularly when the intervals are irregular. Based on Lagrange interpolation in matrix form, formulas for numerical differentiation are developed, which are applicable to both regular and irregular intervals and can use any desired number of points. The method can also be extended for numerical integration and for finding the extremum of a function from its samples. Moreover, in the proposed formulas, the target point does not need to be at a sampling point, as long as it is within the sampling domain. Numerical examples are presented to illustrate the accuracy of the proposed method and its engineering applications. It is demonstrated that the proposed method is versatile, easy to implements, efficient, and accurate in performing numerical differentiation and integration, as well as the determination of extremum of a function.

 Published in American Journal of Applied Mathematics (Volume 12, Issue 3) DOI 10.11648/j.ajam.20241203.13 Page(s) 66-78 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2024. Published by Science Publishing Group
Keywords

Numerical Derivative, Numerical Integration, Extrema of a Function

References
 [1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol. 55. US Government printing office, 1968. [2] R. Hamming, Numerical methods for scientists and engineers, 2nd Edition. New York: McGraw-Hill, 2012. [3] T. Sauer, Numerical Analysis, 2nd Edition. Addison- Wesley Publishing Company: Pearson, 2011. [4] J. Grabmeier, E. Kaltofen, and V. Weispfenning, Computer algebra handbook: foundations, applications, systems. Berlin: Springer, 2003. https://doi.org/10.1007/978-3-642-55826-9 [5] S. Wolfram, The Mathematica Book, 4th Edition. Cambridge: Cambridge university press, 1999. [6] A. Heck and W. Koepf, Introduction to MAPLE, vol. 3. New York: Springer, 1993. [7] N. Ketkar, J. Moolayil, N. Ketkar, and J. Moolayil, “Automatic differentiation in deep learning,” Deep Learning with Python: Learn Best Practices of Deep Learning Models with PyTorch, pp. 133–145, 2021. https://doi.org/10.1007/978-1-4842-5364-9_4 [8] A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, “Automatic differentiation in machine learning: a survey,” Journal of Marchine Learning Research, vol. 18, pp. 1–43, 2018. https://doi.org/10.48550/arXiv.1502.05767 [9] C. C. Margossian, “A review of automatic differentiation and its efficient implementation,” Wiley interdisciplinary reviews: data mining and knowledge discovery, vol. 9, no. 4, p. e1305, 2019. https://doi.org/10.1002/widm.1305 [10] N. Yoshikawa and M. Sumita, “Automatic differentiation for the direct minimization approach to the hartree– fock method,” The Journal of Physical Chemistry A, vol. 126, no. 45, pp. 8487–8493, 2022. https://doi.org/10.1021/acs.jpca.2c05922 [11] B. Van Merriënboer, O. Breuleux, A. Bergeron, and P. Lamblin, “Automatic differentiation in ML: Where we are and where we should be going,” Advances in neural information processing systems, vol. 31, 2018. https://doi.org/10.48550/arXiv.1810.11530 [12] J. R. Martins and A. Ning, Engineering design optimization. Cambridge University Press, 2021. https://doi.org/10.1017/9781108980647 [13] J. D. Hoffman and S. Frankel, Numerical methods for engineers and scientists. New York: CRC press, 2018. https://doi.org/10.1201/9781315274508 [14] R. J. LeVeque, Finite difference methods for ordinary and partial differential equations: steady-state and time- dependent problems. Philadelphia, USA: SIAM, 2007. https://doi.org/10.1137/1.9780898717839 [15] P. J. Davis and P. Rabinowitz, Methods of numerical integration, second ed. Courier Corporation, 1984. https://doi.org/10.1016/C2013-0-10566-1 [16] D. Chen, X. Zhang, and H. Ding, “Generalized numerical differentiation method for stability calculation of periodic delayed differential equation: Application for variable pitch cutter in milling,” International Journal of Precision Engineering and Manufacturing, vol. 21, pp. 2027–2039, 2020. https://doi.org/10.1007/s12541-020-00409-6
• APA Style

Wang, R., Ly, B., Xie, W., Pandey, M. (2024). Lagrange Interpolation in Matrix Form for Numerical Differentiation and Integration. American Journal of Applied Mathematics, 12(3), 66-78. https://doi.org/10.11648/j.ajam.20241203.13

ACS Style

Wang, R.; Ly, B.; Xie, W.; Pandey, M. Lagrange Interpolation in Matrix Form for Numerical Differentiation and Integration. Am. J. Appl. Math. 2024, 12(3), 66-78. doi: 10.11648/j.ajam.20241203.13

AMA Style

Wang R, Ly B, Xie W, Pandey M. Lagrange Interpolation in Matrix Form for Numerical Differentiation and Integration. Am J Appl Math. 2024;12(3):66-78. doi: 10.11648/j.ajam.20241203.13

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title = {Lagrange Interpolation in Matrix Form for Numerical Differentiation and Integration},
journal = {American Journal of Applied Mathematics},
volume = {12},
number = {3},
pages = {66-78},
doi = {10.11648/j.ajam.20241203.13},
url = {https://doi.org/10.11648/j.ajam.20241203.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241203.13},
abstract = {Numerical differentiation has been widely applied in engineering practice due to its remarkable simplicity in the approximation of derivatives. Existing formulas rely on only three-point interpolation to compute derivatives when dealing with irregular sampling intervals. However, it is widely recognized that employing five-point interpolation yields a more accurate estimation compared to the three-point method. Thus, the objective of this study is to develop formulas for numerical differentiation using more than three sample points, particularly when the intervals are irregular. Based on Lagrange interpolation in matrix form, formulas for numerical differentiation are developed, which are applicable to both regular and irregular intervals and can use any desired number of points. The method can also be extended for numerical integration and for finding the extremum of a function from its samples. Moreover, in the proposed formulas, the target point does not need to be at a sampling point, as long as it is within the sampling domain. Numerical examples are presented to illustrate the accuracy of the proposed method and its engineering applications. It is demonstrated that the proposed method is versatile, easy to implements, efficient, and accurate in performing numerical differentiation and integration, as well as the determination of extremum of a function.},
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AB  - Numerical differentiation has been widely applied in engineering practice due to its remarkable simplicity in the approximation of derivatives. Existing formulas rely on only three-point interpolation to compute derivatives when dealing with irregular sampling intervals. However, it is widely recognized that employing five-point interpolation yields a more accurate estimation compared to the three-point method. Thus, the objective of this study is to develop formulas for numerical differentiation using more than three sample points, particularly when the intervals are irregular. Based on Lagrange interpolation in matrix form, formulas for numerical differentiation are developed, which are applicable to both regular and irregular intervals and can use any desired number of points. The method can also be extended for numerical integration and for finding the extremum of a function from its samples. Moreover, in the proposed formulas, the target point does not need to be at a sampling point, as long as it is within the sampling domain. Numerical examples are presented to illustrate the accuracy of the proposed method and its engineering applications. It is demonstrated that the proposed method is versatile, easy to implements, efficient, and accurate in performing numerical differentiation and integration, as well as the determination of extremum of a function.
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Author Information
• Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Canada