Ninth and Twelfth-Order Iterative Methods for Roots of Nonlinear Equations
Keywords:
Nonlinear equation, Iterative method, Convergence order.Abstract
This paper introduces two iterative methods for obtaining numerical solutions to nonlinear equations.
The proposed methods achieve convergence orders of nine and twelve, respectively. A detailed convergence analysis confirms their superior efficiency indices compared to several existing techniques. Numerical examples are presented to illustrate the performance and to validate the theoretical convergence orders of the proposed methods.
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Bawazir HMS (2020). Seventh and Eleventh-Order Iterative Methods for Solving Nonlinear
Equations. International Journal of Multidisciplinary Sciences and Advanced Technology.
(10): 56 - 63. https://www.ijmsat.com/archives/ijmsat-volume-1-issue-10
Bawazir HMS (2024). Fourth,Fifth and Seventh-Order Iterative Methods for Solving Nonlinear Equations. Hadhramout University Journal of Natural & Applied Sciences. accepted.
https://digitalcommons.aaru.edu.jo/huj nas/
Behl R, Kanwar V and Sharma KK (2014). New Modified Optimal Families of King’s
and Traub-Ostrowski’s Methods. Numerical Analysis and Applications. 7(1): 26-35.
https://link.springer.com/article/10.1134/S1995423914010030
Bhatti MM, Abbas T and Rashidi MM (2017). Entropy Generation as a Practical Tool
of Optimisation for Non-Newtonian Nanofluid Flow Through a Permeable Stretching
Surface Using SLM. Journal of Computational Design and Engineering. 4( 1): 21-28.
https://doi.org/10.1016/j.jcde.2016.08.004
Burden RL and Faires JD (2005). Numerical Analysis. Edition 8, Thomson Brooks/Cole.
Grau M and D´ıaz-Barrero JL (2006). An improvement to Ostrowski rootfinding method. Applied Mathematics and Computation. 173(1): 450-456.
https://doi.org/10.1016/j.amc.2005.04.043
Gautschi W (1997). Numerical Analysis: An Introduction. Birkh¨auser Boston Inc. Boston.
Hu Z, Guocai L and Tian L (2011). An Iterative Method with Ninth-Order Convergence for
Solving Nonlinear Equations. International Journal of Contemporary Mathematical Sciences.
(1): 17 - 23. https://www.m-hikari.com/ijcms-2011/1-4-2011/huzyIJCMS1-4-2011.pdf
Jackett DR, McDougall TJ, Feistel R, Wright DG and Griffies SM (2006). Algorithms for density, potential temperature, conservative temperature and freezing temperature of seawater. Journal of Atmospheric and Oceanic Technologyl. 23: 1709-1728.
https://doi.org/10.1175/JTECH1946.1
Jarratt P (1966). Some Fourth Order Multipoint Iterative Methods for Solving Equations. Mathematics of Computation. 20(95): 434-437. https://doi.org/10.1090/S0025-5718-
-99924-8
Jarratt P (1969). Some Efficient Fourth-Order Multipoint Methods for Solving Equations.
BIT Numerical Mathematics. 9(2): 119-124. https://doi.org/10.1007/BF01933248
Maleknejad K, Nouri K and Torkzadeh L (2016). Operational Matrix of Fractional Integration Based on the Shifted Second Kind Chebyshev Polynomials for Solving Fractional Differential Equations. Mediterranean Journal of Mathematics. 13(3): 1377-1390.
http://dx.doi.org/10.1007/s00009-015-0563-x
Niazkar M and Trkkan GE (2021). Application of Third-Order Schemes to Improve the Convergence of the Hardy Cross Method in Pipe Network Analysis. Advances in Mathematical
Physics.(ID 6692067.) http://dx.doi.org/10.1155/2021/6692067
Nori K, Ranjbar H and Torkzadeh L (2019). Two High Order Iterative Methods for
Roots of Nonlinear Equations. Punjab University. Journal of Mathematics. 51(3): 47-59.
https://pu.edu.pk/images/journal/maths/PDF/Paper-3 51 3 2019.pdf
Sharma JR (2015). Improved Chebyshev-Halley Methods with Sixth and Eighth
Order Convergence. Applied Mathematics and Computation. 256: 119-124.
http://dx.doi.org/10.1016/j.amc.2015.01.002
Sharma JR, Guha RK and Sharma R (2011). Some modified Newton’s methods
with fourth-order convergence. Advances in Applied Science Research. 2(1): 240-247.
https://www.primescholars.com/articles/some-modified-newtons-methods-withfourthorderconvergence.pdf
Trachoo K, Prathumwan D and Chaiya I (2022). An efficient two-step iterative method with
fifth-order convergence for solving non-linear equations. Journal of Analysis and Applications.
(1): 81 - 90. https://www.sasip.net/JAA March 2022/Din V5.pdf
Traub JF (1964). Iterative Methods for the Solution of Equation. Prentice-Hall, Englewood
Cliffs, NJ, USA.
Weerakoon S and Fernando TGI (2000). A Variant of Newton’s Method with
Accelerated Third-Order Convergence. Applied Mathematics Letters. 13(8): 87-93.
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Copyright (c) 1970 Hassan M. S. Bawazir
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