Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form

Zhengquan Cheng; Wei-Chau Xie; Mahesh Pandey; Binh-Le Ly

doi:doi:10.11648/j.ajam.20261401.13

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Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form

Zhengquan Cheng^*

, Wei-Chau Xie

, Mahesh Pandey

, Binh-Le Ly

Published in American Journal of Applied Mathematics (Volume 14, Issue 1)

Received: 13 October 2025 Accepted: 8 November 2025 Published: 20 January 2026

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Abstract

In this paper, a generalized Lagrange interpolation formula expressed in matrix form is developed to systematically expand a sampled function with enhanced flexibility and computational rigor. The proposed formulation employs appropriate coordinate functions that not only satisfy prescribed boundary conditions but also exploit the symmetry or anti-symmetry inherent in the function under consideration. When such conditions are absent, the coordinate functions naturally degenerate into polynomial bases, thereby reproducing the classical Lagrange interpolation as a special case. The expansion coefficients are efficiently obtained through the collocation method, ensuring numerical simplicity and stability. The matrix-based generalized Lagrange interpolation exhibits substantial versatility beyond traditional interpolation tasks. It can be readily applied to numerical differentiation and integration under both uniform and non-uniform sampling schemes. Moreover, the approach proves useful in solving ordinary differential equations with specified boundary constraints, as well as in problems involving root-finding and extremum detection of functions. Numerical experiments demonstrate the accuracy and robustness of the proposed method, revealing a marked reduction in the Runge phenomenon even when the number of sampling points is limited. The results further indicate that computational efficiency and precision improve progressively as the number of samples increases. Overall, the generalized interpolation framework developed herein provides a unified and reliable computational tool for interpolation, differentiation, integration, and boundary-value problems, thereby offering broad potential for applications in numerical analysis and scientific computing.

Published in	American Journal of Applied Mathematics (Volume 14, Issue 1)
DOI	10.11648/j.ajam.20261401.13
Page(s)	14-26
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), 2026. Published by Science Publishing Group

Keywords

Lagrange Interpolation, Numerical Differentiation, Numerical Integration, Boundary Condition, Runge Phenomenon

References

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APA Style

Cheng, Z., Xie, W., Pandey, M., Ly, B. (2026). Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form. American Journal of Applied Mathematics, 14(1), 14-26. https://doi.org/10.11648/j.ajam.20261401.13

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ACS Style

Cheng, Z.; Xie, W.; Pandey, M.; Ly, B. Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form. Am. J. Appl. Math. 2026, 14(1), 14-26. doi: 10.11648/j.ajam.20261401.13

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AMA Style

Cheng Z, Xie W, Pandey M, Ly B. Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form. Am J Appl Math. 2026;14(1):14-26. doi: 10.11648/j.ajam.20261401.13

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@article{10.11648/j.ajam.20261401.13,
  author = {Zhengquan Cheng and Wei-Chau Xie and Mahesh Pandey and Binh-Le Ly},
  title = {Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {1},
  pages = {14-26},
  doi = {10.11648/j.ajam.20261401.13},
  url = {https://doi.org/10.11648/j.ajam.20261401.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261401.13},
  abstract = {In this paper, a generalized Lagrange interpolation formula expressed in matrix form is developed to systematically expand a sampled function with enhanced flexibility and computational rigor. The proposed formulation employs appropriate coordinate functions that not only satisfy prescribed boundary conditions but also exploit the symmetry or anti-symmetry inherent in the function under consideration. When such conditions are absent, the coordinate functions naturally degenerate into polynomial bases, thereby reproducing the classical Lagrange interpolation as a special case. The expansion coefficients are efficiently obtained through the collocation method, ensuring numerical simplicity and stability. The matrix-based generalized Lagrange interpolation exhibits substantial versatility beyond traditional interpolation tasks. It can be readily applied to numerical differentiation and integration under both uniform and non-uniform sampling schemes. Moreover, the approach proves useful in solving ordinary differential equations with specified boundary constraints, as well as in problems involving root-finding and extremum detection of functions. Numerical experiments demonstrate the accuracy and robustness of the proposed method, revealing a marked reduction in the Runge phenomenon even when the number of sampling points is limited. The results further indicate that computational efficiency and precision improve progressively as the number of samples increases. Overall, the generalized interpolation framework developed herein provides a unified and reliable computational tool for interpolation, differentiation, integration, and boundary-value problems, thereby offering broad potential for applications in numerical analysis and scientific computing.},
 year = {2026}
}

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TY  - JOUR
T1  - Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form
AU  - Zhengquan Cheng
AU  - Wei-Chau Xie
AU  - Mahesh Pandey
AU  - Binh-Le Ly
Y1  - 2026/01/20
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261401.13
DO  - 10.11648/j.ajam.20261401.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 14
EP  - 26
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261401.13
AB  - In this paper, a generalized Lagrange interpolation formula expressed in matrix form is developed to systematically expand a sampled function with enhanced flexibility and computational rigor. The proposed formulation employs appropriate coordinate functions that not only satisfy prescribed boundary conditions but also exploit the symmetry or anti-symmetry inherent in the function under consideration. When such conditions are absent, the coordinate functions naturally degenerate into polynomial bases, thereby reproducing the classical Lagrange interpolation as a special case. The expansion coefficients are efficiently obtained through the collocation method, ensuring numerical simplicity and stability. The matrix-based generalized Lagrange interpolation exhibits substantial versatility beyond traditional interpolation tasks. It can be readily applied to numerical differentiation and integration under both uniform and non-uniform sampling schemes. Moreover, the approach proves useful in solving ordinary differential equations with specified boundary constraints, as well as in problems involving root-finding and extremum detection of functions. Numerical experiments demonstrate the accuracy and robustness of the proposed method, revealing a marked reduction in the Runge phenomenon even when the number of sampling points is limited. The results further indicate that computational efficiency and precision improve progressively as the number of samples increases. Overall, the generalized interpolation framework developed herein provides a unified and reliable computational tool for interpolation, differentiation, integration, and boundary-value problems, thereby offering broad potential for applications in numerical analysis and scientific computing.
VL  - 14
IS  - 1
ER  -

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Author Information

Zhengquan Cheng

Department of Civil and Environmental Engineering, University of Waterloo, Ontario, Canada

Contact Email

http://orcid.org/0009-0009-3560-1344
Wei-Chau Xie

Department of Civil and Environmental Engineering, University of Waterloo, Ontario, Canada

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http://orcid.org/0000-0002-9761-7968
Mahesh Pandey

Department of Civil and Environmental Engineering, University of Waterloo, Ontario, Canada

Contact Email

http://orcid.org/0000-0002-5083-2599
Binh-Le Ly

Atomic Energy of Canada, Mississauga, Canada

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Plain Text BibTeX RIS

APA Style

Cheng, Z., Xie, W., Pandey, M., Ly, B. (2026). Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form. American Journal of Applied Mathematics, 14(1), 14-26. https://doi.org/10.11648/j.ajam.20261401.13

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ACS Style

Cheng, Z.; Xie, W.; Pandey, M.; Ly, B. Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form. Am. J. Appl. Math. 2026, 14(1), 14-26. doi: 10.11648/j.ajam.20261401.13

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AMA Style

Cheng Z, Xie W, Pandey M, Ly B. Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form. Am J Appl Math. 2026;14(1):14-26. doi: 10.11648/j.ajam.20261401.13

Copy | Download

@article{10.11648/j.ajam.20261401.13,
  author = {Zhengquan Cheng and Wei-Chau Xie and Mahesh Pandey and Binh-Le Ly},
  title = {Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {1},
  pages = {14-26},
  doi = {10.11648/j.ajam.20261401.13},
  url = {https://doi.org/10.11648/j.ajam.20261401.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261401.13},
  abstract = {In this paper, a generalized Lagrange interpolation formula expressed in matrix form is developed to systematically expand a sampled function with enhanced flexibility and computational rigor. The proposed formulation employs appropriate coordinate functions that not only satisfy prescribed boundary conditions but also exploit the symmetry or anti-symmetry inherent in the function under consideration. When such conditions are absent, the coordinate functions naturally degenerate into polynomial bases, thereby reproducing the classical Lagrange interpolation as a special case. The expansion coefficients are efficiently obtained through the collocation method, ensuring numerical simplicity and stability. The matrix-based generalized Lagrange interpolation exhibits substantial versatility beyond traditional interpolation tasks. It can be readily applied to numerical differentiation and integration under both uniform and non-uniform sampling schemes. Moreover, the approach proves useful in solving ordinary differential equations with specified boundary constraints, as well as in problems involving root-finding and extremum detection of functions. Numerical experiments demonstrate the accuracy and robustness of the proposed method, revealing a marked reduction in the Runge phenomenon even when the number of sampling points is limited. The results further indicate that computational efficiency and precision improve progressively as the number of samples increases. Overall, the generalized interpolation framework developed herein provides a unified and reliable computational tool for interpolation, differentiation, integration, and boundary-value problems, thereby offering broad potential for applications in numerical analysis and scientific computing.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form
AU  - Zhengquan Cheng
AU  - Wei-Chau Xie
AU  - Mahesh Pandey
AU  - Binh-Le Ly
Y1  - 2026/01/20
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261401.13
DO  - 10.11648/j.ajam.20261401.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 14
EP  - 26
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261401.13
AB  - In this paper, a generalized Lagrange interpolation formula expressed in matrix form is developed to systematically expand a sampled function with enhanced flexibility and computational rigor. The proposed formulation employs appropriate coordinate functions that not only satisfy prescribed boundary conditions but also exploit the symmetry or anti-symmetry inherent in the function under consideration. When such conditions are absent, the coordinate functions naturally degenerate into polynomial bases, thereby reproducing the classical Lagrange interpolation as a special case. The expansion coefficients are efficiently obtained through the collocation method, ensuring numerical simplicity and stability. The matrix-based generalized Lagrange interpolation exhibits substantial versatility beyond traditional interpolation tasks. It can be readily applied to numerical differentiation and integration under both uniform and non-uniform sampling schemes. Moreover, the approach proves useful in solving ordinary differential equations with specified boundary constraints, as well as in problems involving root-finding and extremum detection of functions. Numerical experiments demonstrate the accuracy and robustness of the proposed method, revealing a marked reduction in the Runge phenomenon even when the number of sampling points is limited. The results further indicate that computational efficiency and precision improve progressively as the number of samples increases. Overall, the generalized interpolation framework developed herein provides a unified and reliable computational tool for interpolation, differentiation, integration, and boundary-value problems, thereby offering broad potential for applications in numerical analysis and scientific computing.
VL  - 14
IS  - 1
ER  -

Copy | Download