# Iterative method improving Newton's method with higher order

## Authors

• Shno Othman Ahmed Computer Department, College of Sciences, University of Salahaddin, Erbil, Kurdistan Region, Iraq. Author

## Keywords:

Iterative Methods, Convergence Order, Newton-method, nonlinear equations, Numerical Examples

## Abstract

In the present article, we have constructed and looked at improving two- and three-step iterative methods to locate simple roots of non-linear equations. It appears that the three-step iterative method converges to the eighth order. The whole aim of this research is to derive and present a new modified iterative method of higher order and to obtain less iteration than the classical Newton method for solving nonlinear equations of simple roots. The present technique has to evaluate three functions and two first derivatives in each iteration. The advantage of the proposed method has been observed to have at least better performance effective and more stability by comparing with the other methods for the same or less than order. Also, noted that our method gives better results in terms of the number of iterations. To demonstrate the efficacy and popularity, different numerical illustrations are provided for the recommended techniques.

## References

Abbasbandy, S. (2003). Improving Newton–Raphson method for nonlinear equations by modified DOI: https://doi.org/10.1016/S0096-3003(03)00282-0

Adomian decomposition method. Applied mathematics and computation, 145(2-3), 887-893.

Ostrowski, A. M. (2016). Solution of equations and systems of equations: Pure and applied mathematics:

A series of monographs and textbooks, vol. 9 (Vol. 9). Elsevier.

Traub, J. F. (1982). Iterative methods for the solution of equations (Vol. 312). American Mathematical

Soc.

Chun, C. (2005). Iterative methods improving Newton's method by the decomposition method. Computers DOI: https://doi.org/10.1016/j.camwa.2005.08.022

& Mathematics with Applications, 50(10-12), 1559-1568.

Noor, M. A., & Noor, K. I. (2007). Fifth-order iterative methods for solving nonlinear equations. Applied DOI: https://doi.org/10.1016/j.amc.2006.05.146

mathematics and computation, 188(1), 406-410.

Siyyam, H. I. (2009). An iterative method with fifth-order convergence for nonlinear equations. Appl.

Math. Sci, 3, 2041-2053.

Xiaojian, Z. (2008). Modified Chebyshev–Halley methods free from second derivative. Applied DOI: https://doi.org/10.1016/j.amc.2008.05.092

mathematics and computation, 203(2), 824-827.

Omran, H. H. (2013). Modified third order iterative method for solving nonlinear equations. Al-Nahrain DOI: https://doi.org/10.1155/2013/850365

Journal of Science, 16(3), 239-245.

Kou, J., Li, Y., & Wang, X. (2007). A composite fourth-order iterative method for solving non-linear DOI: https://doi.org/10.1016/j.amc.2006.05.181

equations. Applied Mathematics and Computation, 184(2), 471-475.

Jarratt, P. (1969). Some efficient fourth order multipoint methods for solving equations. BIT Numerical DOI: https://doi.org/10.1007/BF01933248

Mathematics, 9(2), 119-124.

Chun, C., & Neta, B. (2009). A third-order modification of Newton’s method for multiple roots. Applied DOI: https://doi.org/10.1016/j.amc.2009.01.087

Mathematics and Computation, 211(2), 474-479.

Bahgat, M. S. (2012). New two-step iterative methods for solving nonlinear equations. Journal of DOI: https://doi.org/10.5539/jmr.v4n3p128

Mathematics Research, 4(3), 128.

Ostrowski, A. M. (1973). Solution of equations in Euclidean and Banach spaces. Academic Press.

Gutierrez, J. M., & Hernández, M. A. (1997). A family of Chebyshev-Halley type methods in Banach DOI: https://doi.org/10.1017/S0004972700030586

spaces. Bulletin of the Australian Mathematical Society, 55(1), 113-130.

Argyros, I. K., Chen, D., & Qian, Q. (1994). The Jarratt method in Banach space setting. Journal of DOI: https://doi.org/10.1016/0377-0427(94)90093-0

Computational and Applied Mathematics, 51(1), 103-106.

Sharma, J. (2005). A composite third order Newton–Steffensen method for solving nonlinear DOI: https://doi.org/10.1016/j.amc.2004.10.040

equations. Applied Mathematics and Computation, 169(1), 242-246.

Gautschi, W. (1997). Numerical Analysis: An Introduction Birkhauser. Barton, Mass, USA.

2023-12-20

Articles

## How to Cite

Iterative method improving Newton’s method with higher order. (2023). Journal of Zankoy Sulaimani - Part A, 25(2), 9. https://doi.org/10.17656/jzs.10920