Investigation and Analysis of Newton-Cotes Integration Formulas and Errors from Two up to Eight Nodes Using Lagrange Interpolation

Authors

  • Nikzad Jamali Department of Mathematics, Sar-e-pul Higher Education Institute, Afghanistan

DOI:

https://doi.org/10.54536/ajise.v4i1.3885

Keywords:

Error Analysis, Integration, Interpolation, Newton-Cotes

Abstract

Much research has been done regarding the Newton Coates method and generally the results that are available are about the trapezoidal method and the Sampson method and not much attention is paid to the higher levels of the Newton Coates method. This article has been written to find out what is the relationship between the high-order Newton-Cotes method and study behavior of error in high order methods. In this research numerical integration preformed, Newton Coates method is examined in detail and formulate the two points, three points, four points up to eight points are will investigate. Furthermore, this study showed that, analyze the errors of each method and show the relation between two consecutive Newton Coates methods. On the other hand, the research showed that, odd point’s method would be better. It is worth mentioning that to obtain these methods, and in this study the researcher used the Lagrange interpolator polynomial.

Downloads

Download data is not yet available.

References

Alomari, M. W., & Dragomir, S. S. (2014). Various error estimations for several Newton-Cotes quadrature formulae in terms of at most first derivative and applications in numerical integration. Jordan J. Math. Stat, 7(2), 89-108.

AL-Sammarraie, O. A., & Bashir, M. A. (2015). Error analysis of the high order Newton Cotes formulas. International Journal of Scientific and Research Publications, 5, 1-6.

Chalpuri, M., Sucharitha, J., & Madhu, M. (2018). Advanced Family of Newton-Cotes Formulas. Journal of Informatics and Mathematical Sciences, 10(3), 1-14.

Cruz-Uribe, D., & Neugebauer, C. J. (2003). An elementary proof of error estimates for the trapezoidal rule. Mathematics Magazine, 76(4), 303-306.

Darkwah, K., Nortey, E., & Lotsi, C. (2016). A proposed numerical integration method using polynomial interpolation. British Journal of Mathematics & Computer Science, 16(2), 1-11.

Dehghan, M., Masjed-Jamei, M., & Eslahchi, M. R. (2005). The semi-open Newton–Cotes quadrature rule and its numerical improvement. Applied mathematics and computation, 171(2), 1129-1140.

Eslahchi, M. R., Dehghan, M., & Masjed-Jamei, M. (2005). The first kind Chebyshev–Newton–Cotes quadrature rules (closed type) and its numerical improvement. Applied mathematics and computation, 168(1), 479-495.

Fornberg, B. (2021). Improving the accuracy of the trapezoidal rule. SIAM Review, 63(1), 167-180.

Junior, P. A. A. M., & Magalhaes, C. A. (2010). Higher-order newton-cotes formulas. Journal of Mathematics and Statistics, 6(2), 193-204.

Khatri, A., Shaikh, A. A., & Abro, K. A. (2019). Closed Newton cotes quadrature rules with derivatives. Mathematical Theory and Modeling, 9(5), 65-72.

Kambo, N. S. (1970). Error of the Newton-Cotes and Gauss-Legendre quadrature formulas. Mathematics of Computation, 24(110), 261-269.

Magalhaes, P. A. A., Junior, P. A. A. M., Magalhaes, C. A., & Magalhaes, A. L. M. A. (2021). New formulas of numerical quadrature using spline interpolation. Archives of Computational Methods in Engineering, 28, 553-576.

Ramachandran, T., & Parimala, R. (2015). Open Newton cotes quadrature with midpoint derivative for integration of Algebraic functions. Int. J. Res. Eng. Technol, 4, 430-435.

Ramachandran, T., Udayakumar, D., & Parimala, R. (2016). Comparison of arithmetic mean, geometric mean and harmonic mean derivative-based closed Newton Cotes quadrature. Progress in Nonlinear dynamics and Chaos, 4, 35-43.

Sermutlu, E. (2005). Comparison of Newton–Cotes and Gaussian methods of quadrature. Applied mathematics and computation, 171(2), 1048-1057.

Sermutlu, E. (2019). A close look at Newton-Cotes integration rules. Results in Nonlinear Analysis, 2(2), 48-60.

Shaikh, M. M., Chandio, M. S., & Soomro, A. S. (2016). A modified four-point closed mid-point derivative based quadrature rule for numerical integration. Sindh University Research Journal (Science Series), 48(2), 389-392.

Uddin, M. J., Moheuddin, M. M., & Kowsher, M. (2019). A new study of trapezoidal, simpson’s 1/3 and simpson’s 3/8 rules of numerical integral problems. Applied Mathematics and Sciences: An International Journal, 6(4), 1-14.

Zadorin, A. I. (2015). Lagrange interpolation and Newton-Cotes formulas for functions with boundary layer components on piecewise-uniform grids. Numerical Analysis and Applications, 8, 235-247.

Zadorin, A. I. (2020, May). Optimization of nodes of Newton-Cotes formulas in the presence of an exponential boundary layer. In Journal of Physics: Conference Series, 1546(1), 012107. IOP Publishing.

Zhao, W., & Li, H. (2013). Midpoint Derivative-Based Closed Newton-Cotes Quadrature. In Abstract and Applied Analysis, 2013(1), 492507. Hindawi Publishing Corporation.

Published

2024-12-19

How to Cite

Jamali, N. (2024). Investigation and Analysis of Newton-Cotes Integration Formulas and Errors from Two up to Eight Nodes Using Lagrange Interpolation. American Journal of Innovation in Science and Engineering, 4(1), 1–8. https://doi.org/10.54536/ajise.v4i1.3885