Modified Lagrange Interpolating Polynomial (MLIP) Method: A Straightforward Procedure to Improve Function Approximation
Abstract
:1. Introduction
2. Description of the Proposed Modified Lagrange Interpolating Polynomial (MLIP) Method
3. Applications of Modified Lagrange Interpolating Polynomial (MLIP) Method
3.1. Case Study 1
3.2. Case Study 2
4. Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Faires, J.D.; Burden, R.L. Numerical Methods, 3rd ed.; Brooks Cole: Pacific Grove, CA, USA, 2002. [Google Scholar]
- Liu, Y.P.; He, J.H. A fast and accurate estimation of amperometric current response in reaction kinetics. J. Electroanal. Chem. 2025, 978, 118884. [Google Scholar] [CrossRef]
- Hurtado, A.N.; Sánchez, F.C.D. Métodos Numéricos: Aplicados a la Ingeniería; Grupo Editorial Patria: Puebla, Mexico, 2014. [Google Scholar]
- Chapra, S.C.; Canale, R.P. Numerical Methods For Engineers, 5th ed.; McGraw-Hill: New York, NY, USA, 2006. [Google Scholar]
- Hamming, R. Numerical Methods for Scientists and Engineers, 2nd ed.; Courier Corporation: North Chelmsford, MA, USA, 2012. [Google Scholar]
- Epperson, J. An Introduction to Numerical Methods and Analysis; John Wiley and Sons: Hoboken, NJ, USA, 2021. [Google Scholar]
- Griffiths, D.; Higham, D. Numerical Methods for Ordinary Differential Equations: Initial Value Problems; Springer: Berlin/Heidelberg, Germany, 2010; Volume 5. [Google Scholar] [CrossRef]
- Zhao, H.; Sun, J.Z.; Wang, F.; Zhao, L. A finite equivalence of multisecret sharing based on Lagrange interpolating polynomial. In Security and Communication Networks; Wiley Online Library: Hoboken, NJ, USA, 2013. [Google Scholar] [CrossRef]
- Roy, S.K. Lagrange’s Interpolating Polynomial Approach to solve multi-choice transportation problem. Int. J. Appl. Comput. Math. 2015, 1, 639–649. [Google Scholar] [CrossRef]
- Adams, W.P. Use of Lagrange Interpolating Polynomials in the RLT. In Wiley Encyclopedia of Operations Research and Management Science; Cochran, J.J., Ed.; John Wiley and Sons, Inc.: Hoboken, NJ, USA, 2010. [Google Scholar]
- Errrachid, M.; Essanhaji, A.; Messaondi, A. RMVPIA: A new algorithm for computing the Lagrange multivariate polynomial interpolation. Numer. Algorithms 2020, 84, 1507–1534. [Google Scholar] [CrossRef]
- Wilson, L.; Vaughn, N.; Krasny, R. A GPU-accelerated fast multipole method based on barycentric Lagrange interpolation and dual tree traversal. Comput. Phys. Commun. 2021, 265, 108017. [Google Scholar] [CrossRef]
- Ide, N.A.D. Using Lagrange interpolation for solving nonlinear algebraic equations. Eng. Math. 2016, 1, 39–43. [Google Scholar]
- Yadav, O.P.; Sahu, A.K. Electrocardiogram estimation using Lagrange interpolation. Electron. Lett. 2021, 57, 169–172. [Google Scholar] [CrossRef]
- Zhou, S.P. Simultaneous Lagrange Interpolating Approximation Need Not Always Be Convergent. Constr. Approx. 1994, 10, 87–93. [Google Scholar] [CrossRef]
- Ibrahim, S. Application of Lagrange Interpolation Method to Solve First-Order Differential Equation Using Newton Interpolation Approach. Eurasian J. Sci. Eng. 2023, 9, 89–98. [Google Scholar] [CrossRef]
- Wang, M.F.; Au, F. Precise integrations methods based on Lagrange Piecewise Interpolation Polynomials. Int. J. Numer. Methods Eng. 2009, 77, 998–1014. [Google Scholar] [CrossRef]
- Marzaban, H.; Hoseini, S.; Razzaghi, M. Solution of Volterra’s populations model via block-pulse functions and Lagrange interpolating polynomials. Math. Methods Appl. Sci. 2009, 32, 127–134. [Google Scholar] [CrossRef]
- Chen, G.; Liu, J.; Wang, L. Color Image Sharing Method Based on Lagrange’s Interpolating Polynomial. In Health Information Science; He, J., Liu, X., Krupinski, E.A., Xu, G., Eds.; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2012; Volume 7231, pp. 63–75. [Google Scholar] [CrossRef]
- Senapati, M.K.; Pradhan, C.; Nayak, P.K.; Samantaray, S.R. Lagrange Interpolating Polynomial based deloading control scheme for variable speed wind turbines. Int. Trans. Electr. Energy Syst. 2019, 29, e2824. [Google Scholar] [CrossRef]
- Vazquez-Leal, H.; Sandoval-Hernandez, M.A.; Filobello-Nino, U. The novel family of transcendental Leal-functions with applications to science and engineering. Heliyon 2020, 6, e05418. [Google Scholar] [CrossRef] [PubMed]
- Mohsen, A. A simple solution of the Bratu problem. Comput. Math. Appl. 2014, 67, 26–33. [Google Scholar] [CrossRef]
- Ascher, U.M.; Mattheij, R.M.M.; Russell, R.D. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations; Classics in Applied Mathematics; SIAM: Philadelphia, PA, USA, 1995; p. xxv×593. [Google Scholar] [CrossRef]
- Ascher, U.M.; Petzold, L.R. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1998. [Google Scholar] [CrossRef]
- Vazquez-Leal, H.; Sandoval-Hernandez, M.; Castaneda-Sheissa, R.; Filobello-Nino, U.; Sarmiento-Reyes, A. Modified Taylor solution of equation of oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics. Int. J. Appl. Math. Res. 2015, 4, 253–258. [Google Scholar] [CrossRef]
- He, J.H. Variational iteration method—A kind of non-linear analytical technique: Some examples. Int. J.-Non-Linear Mech. 1999, 34, 699–708. [Google Scholar] [CrossRef]
- He, J.H. An Old Babylonian Algorithm and Its Modern Applications. Symmetry 2024, 16, 1467. [Google Scholar] [CrossRef]
x | Ln(x) | (20) | (23) | LIP Passing Through Five Nodes | Thiele Interpolation | Least Squares Method | Splines | (26) | Leal Polynomial [21] |
---|---|---|---|---|---|---|---|---|---|
1.2 | 0.182321556 | 0.157113360 | 0.181465970 | 0.180683490 | 0.182224332 | 0.128851000 | 0.171459184 | 0.182323063 | 0.182326443 |
1.7 | 0.530628251 | 0.509463177 | 0.530871113 | 0.531175597 | 0.530647608 | 0.582791313 | 0.53361834 | 0.530628426 | 0.530628954 |
2.2 | 0.788457360 | 0.804050729 | 0.788256876 | 0.787939614 | 0.788445392 | 0.659409851 | 0.787723445 | 0.788457525 | 0.788458127 |
2.7 | 0.993251773 | 1.040876016 | 0.993866911 | 0.995029632 | 0.993280323 | 2.034778342 | 0.993296370 | 0.993253767 | 0.993262355 |
3.0 | 1.098612288 | 1.155245300 | 1.100754996 | 1.105164383 | 1.098698751 | 7.224870552 | 1.097578265 | 1.098639716 | 1.098768749 |
3.5 | 1.252762968 | 1.299650963 | 1.257094249 | 1.267060235 | 1.252902950 | 28.544006237 | 1.249322210 | 1.252896335 | 1.253608595 |
x | (37) | (38) | |
---|---|---|---|
0 | 0 | 0 | 0 |
0.1 | 0.29858379 | 0.298590087 | 0.297190666 |
0.2 | 0.55122088 | 0.551213018 | 0.548501319 |
0.3 | 0.74489845 | 0.74489284 | 0.741018925 |
0.4 | 0.86723883 | 0.867239748 | 0.862540412 |
0.5 | 0.90914265 | 0.909144265 | 0.904447458 |
0.6 | 0.86723883 | 0.867240001 | 0.864771632 |
0.7 | 0.74489845 | 0.744892447 | 0.749705070 |
0.8 | 0.55122088 | 0.55121261 | 0.570507225 |
0.9 | 0.29858378 | 0.298588354 | 0.331136834 |
1.0 | 0 | 0 | 0 |
x | Derivative (37) | Derivative of (38) | |
---|---|---|---|
0 | 3.174730842 | 3.174682878 | 3.160707044 |
0.1 | 2.777204768 | 2.777110353 | 2.763472447 |
0.2 | 2.253529923 | 2.253446744 | 2.240896836 |
0.3 | 1.598927344 | 1.599014466 | 1.588620933 |
0.4 | 0.832272053 | 0.832298764 | 0.826893790 |
0.5 | 0 | 0.000030120 | 0.007643290 |
0.6 | −0.832272053 | −0.832301972 | −0.790433990 |
0.7 | −1.598927344 | −1.599020127 | −1.490226272 |
0.8 | −2.253529923 | −2.253444516 | −2.082633859 |
0.9 | −2.777204768 | −2.777145209 | −2.748868191 |
1.0 | −3.174730842 | −3.174479705 | −4.059588223 |
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Filobello-Nino, U.A.; Vazquez-Leal, H.; Sandoval-Hernandez, M.A.; Dominguez-Chavez, J.A.; Salinas-Castro, A.; Jimenez-Fernandez, V.M.; Huerta-Chua, J.; Hoyos-Reyes, C.; Carrillo-Ramon, N.; Flores-Mendez, J. Modified Lagrange Interpolating Polynomial (MLIP) Method: A Straightforward Procedure to Improve Function Approximation. AppliedMath 2025, 5, 34. https://doi.org/10.3390/appliedmath5020034
Filobello-Nino UA, Vazquez-Leal H, Sandoval-Hernandez MA, Dominguez-Chavez JA, Salinas-Castro A, Jimenez-Fernandez VM, Huerta-Chua J, Hoyos-Reyes C, Carrillo-Ramon N, Flores-Mendez J. Modified Lagrange Interpolating Polynomial (MLIP) Method: A Straightforward Procedure to Improve Function Approximation. AppliedMath. 2025; 5(2):34. https://doi.org/10.3390/appliedmath5020034
Chicago/Turabian StyleFilobello-Nino, Uriel A., Hector Vazquez-Leal, Mario A. Sandoval-Hernandez, Jose A. Dominguez-Chavez, Alejandro Salinas-Castro, Victor M. Jimenez-Fernandez, Jesus Huerta-Chua, Claudio Hoyos-Reyes, Norberto Carrillo-Ramon, and Javier Flores-Mendez. 2025. "Modified Lagrange Interpolating Polynomial (MLIP) Method: A Straightforward Procedure to Improve Function Approximation" AppliedMath 5, no. 2: 34. https://doi.org/10.3390/appliedmath5020034
APA StyleFilobello-Nino, U. A., Vazquez-Leal, H., Sandoval-Hernandez, M. A., Dominguez-Chavez, J. A., Salinas-Castro, A., Jimenez-Fernandez, V. M., Huerta-Chua, J., Hoyos-Reyes, C., Carrillo-Ramon, N., & Flores-Mendez, J. (2025). Modified Lagrange Interpolating Polynomial (MLIP) Method: A Straightforward Procedure to Improve Function Approximation. AppliedMath, 5(2), 34. https://doi.org/10.3390/appliedmath5020034