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 |
Numerical Derivative, Numerical Integration, Extrema of a Function
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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
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Download@article{10.11648/j.ajam.20241203.13,
author = {Rui Wang and Binh-Le Ly and Wei-Chau Xie and Mahesh Pandey},
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.},
year = {2024}
}
TY - JOUR T1 - Lagrange Interpolation in Matrix Form for Numerical Differentiation and Integration AU - Rui Wang AU - Binh-Le Ly AU - Wei-Chau Xie AU - Mahesh Pandey Y1 - 2024/06/19 PY - 2024 N1 - https://doi.org/10.11648/j.ajam.20241203.13 DO - 10.11648/j.ajam.20241203.13 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 66 EP - 78 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20241203.13 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. VL - 12 IS - 3 ER -
Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Canada
Atomic Energy of Canada, Ltd, Mississauga, Canada
Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Canada
Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Canada
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