The Third-order Iterative Method for Solving Nonlinear Equations
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Published
Dec 28, 2022
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Abstract
In this paper, we present a new iterative method for solving nonlinear equations, which were developed from the concept of Rafiq et al. The new method is based on Newton’s method and using Taylor’s Series to prove the convergence of the method. This iterative method requires three evaluations of the function, and only use the first derivative. Analysis of its convergence shows that the order of convergence of the new iterative method is third. Numerical comparisons are made with other methods to show the efficiency of the proposed method.
Keywords: Nonlinear equations, Newton’s method, Oder of convergence, Iterative method
References
Chun, C. (2007). A method for obtaining iterative formulas of order three. Applied Mathematics Letters, 20, 1103-1109. https://doi.org/10.1016/j.aml.2006.11.010
Chun, C., Bac, H. J., & Neta, B. (2009). New families of nonlinear third-order solvers for finding multiple roots. Computers & Mathematics with Applications, 57, 1574-1582. https://doi.org/10.1016/j.camwa.
2008.10.070
Chun, C., & Ham, Y. (2007). A one-parameter fourth-order family of iterative method for nonlinear equations. Applied Mathematics and Computation, 189, 610-614. https://doi.10.1016/j.amc.2006.11.113
Chun, C., & Kim, Y. (2010). Several New Third-Order Iterative Method for Solving Nonlinear Equations. Acta Applicandae Mathematicae, 109, 1053-1063. https://doi.org/10.1007/s10440-008-9359-3
Gautschi, W. (2012). Numerical analysis (2nd ed.). Boston: Birkhauser,
Kang, S. M., Ali, F., & Rafiq, A. (2016). Iterative method for solving scalar equations. Journal of Nonlinear Sciences and Applications, 9, 1035-1042.
Nappassanan, S., & Montri, T. (2017). A new third-order Iterative method for solving nonlinear equations. Journal of Science and Technology, Ubon Ratchathani University, 19(2), 173-177.
Rostam, K. S., & Fuad, W. K. (2011). New Third-order Method for Solving Nonlinear Equations. Journal of Applied Sciences Research, 7(6), 916-921.
Schröder, E. (1870). Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Mathematische Annalen, 2, 317-365. https://doi.org/10.1007/BF01444024
Chun, C., Bac, H. J., & Neta, B. (2009). New families of nonlinear third-order solvers for finding multiple roots. Computers & Mathematics with Applications, 57, 1574-1582. https://doi.org/10.1016/j.camwa.
2008.10.070
Chun, C., & Ham, Y. (2007). A one-parameter fourth-order family of iterative method for nonlinear equations. Applied Mathematics and Computation, 189, 610-614. https://doi.10.1016/j.amc.2006.11.113
Chun, C., & Kim, Y. (2010). Several New Third-Order Iterative Method for Solving Nonlinear Equations. Acta Applicandae Mathematicae, 109, 1053-1063. https://doi.org/10.1007/s10440-008-9359-3
Gautschi, W. (2012). Numerical analysis (2nd ed.). Boston: Birkhauser,
Kang, S. M., Ali, F., & Rafiq, A. (2016). Iterative method for solving scalar equations. Journal of Nonlinear Sciences and Applications, 9, 1035-1042.
Nappassanan, S., & Montri, T. (2017). A new third-order Iterative method for solving nonlinear equations. Journal of Science and Technology, Ubon Ratchathani University, 19(2), 173-177.
Rostam, K. S., & Fuad, W. K. (2011). New Third-order Method for Solving Nonlinear Equations. Journal of Applied Sciences Research, 7(6), 916-921.
Schröder, E. (1870). Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Mathematische Annalen, 2, 317-365. https://doi.org/10.1007/BF01444024
Section
Research Articles
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How to Cite
COMEMUANG, Chalermwut; WISESRAM, Naratip; PONGSAWAT, Natthapoom.
The Third-order Iterative Method for Solving Nonlinear Equations.
Naresuan University Journal: Science and Technology (NUJST), [S.l.], v. 31, n. 1, p. 29-36, dec. 2022.
ISSN 2539-553X.
Available at: <https://www.journal.nu.ac.th/NUJST/article/view/Vol-31-No-1-2023-29-36>. Date accessed: 29 sep. 2025.
doi: https://doi.org/10.14456/nujst.2023.4.