A New Family of Optimal Eighth-Order Iterative Scheme for Solving Nonlinear Equations
Keywords:
Convergence order; Efficiency index; Iterative methods; Nonlinear equations; Optimal eighth-order.
Abstract
The objective of this manuscript is to introduce a new family of optimal eight-order iterative methods for computing the numerical zeros of a nonlinear univariate equation that is not dependent on the second derivative. The family was designed to enhance the order of convergence by merging Bawazir’s method and Newton’s method as a third step. To demonstrate the performance of the offered scheme, assorted numerical comparisons have been investigated. In addition, the efficiency index of the new family is 1.6818.
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References
[1] J. F. Traub, Iterative methods for the solution of equations, vol. 312. American Mathematical Soc., 1982.
[2] S. Journal, A. S. Al-Hazmi, and I. A. Al-Subaihi, “SCHOLARS SCITECH RESEARCH ORGANIZATION A Class of Eight Order Iterative Methods for Solving Nonlinear Equations”, [Online]. Available: www.scischolars.com
[3] T. G. Al-Harbi and I. A. Al-Subaihi, “An Optimal Class of Eighth-Order Iterative Methods Based on King’s Method,” Journal of Progressive Research in Mathematics (JPRM), 2018, [Online]. Available: www.scitecresearch.com/journals
[4] O. Said Solaiman, S. A. Abdul Karim, and I. Hashim, “Optimal fourth- and eighth-order of convergence derivative-free modifications of King’s method,” Journal of King Saud University - Science, vol. 31, no. 4, pp. 1499–1504, 2019, doi: https://doi.org/10.1016/j.jksus.2018.12.001.
[5] H. I. Siyyam, M. T. Shatnawi, and I. A. Al-Subaihi, “A new one parameter family of iterative methods with eighth-order of convergence for solving nonlinear equations,” International Journal of Pure and Applied Mathematics, vol. 84, no. 5, pp. 451–461, 2013.
[6] P. Sivakumar, K. Madhu, and J. Jayaraman, “Optimal eighth and sixteenth order iterative methods for solving nonlinear equation with basins of attraction,” Appl. Math. E-notes, vol. 21, pp. 320–343, 2021.
[7] H. S. Wilf, “Iterative Methods for the Solution of Equations (JF Traub),” SIAM Review, vol. 8, no. 4, p. 550, 1966.
[8] H. T. Kung and J. F. Traub, “Optimal Order of One-Point and Multipoint Iteration,” J. ACM, vol. 21, no. 4, pp. 643–651, Oct. 1974, doi: 10.1145/321850.321860.
[9] M. S. Petković and L. D. Petković, “Families of optimal multipoint methods for solving nonlinear equations: a survey,” Applicable Analysis and Discrete Mathematics, pp. 1–22, 2010.
[10] H. M. Bawazir, “Fifth and Eleventh-Order Iterative Methods for Roots of Nonlinear Equations,” Hadhramout University Journal of Natural & Applied Sciences, vol. 17, no. 2, p. 6, 2020.
[11] R. F. King, “A family of fourth order methods for nonlinear equations,” SIAM journal on numerical analysis, vol. 10, no. 5, pp. 876–879, 1973.
[12] O. S. Solaiman and I. Hashim, “Optimal eighth-order solver for nonlinear equations with applications in chemical engineering,” Intell. Autom. Soft Comput, vol. 13, pp. 87–93, 2020.
[13] L. Liu and X. Wang, “Eighth-Order Methods with High Efficiency Index for Solving Nonlinear Equations,” Appl. Math. Comput., vol. 215, no. 9, pp. 3449–3454, Jan. 2010, doi: 10.1016/j.amc.2009.10.040.
[14] J. R. Sharma and H. Arora, “An efficient family of weighted-Newton methods with optimal eighth order convergence,” Applied Mathematics Letters, vol. 29, pp. 1–6, 2014.
[15] A. A. Al-Harbi and I. A. Al-Subaihi, “Family of Optimal Eighth-Order of Convergence for Solving Nonlinear Equations,” Journal of Progressive Research in Mathematics (JPRM), vol. 4, 2015, [Online]. Available: www.scitecresearch.com/journals/index.php/jprm
[16] S. Parimala, K. Madhu, and J. Jayaraman, “A new class of optimal eighth order method with two weight functions for solving nonlinear equation,” J. Nonlinear Anal. Appl, vol. 2018, pp. 83–94, 2018.
[17] S. Weerakoon and T. Fernando, “A variant of Newton’s method with accelerated third-order convergence,” Applied Mathematics Letters, vol. 13, no. 8, pp. 87–93, 2000.
[2] S. Journal, A. S. Al-Hazmi, and I. A. Al-Subaihi, “SCHOLARS SCITECH RESEARCH ORGANIZATION A Class of Eight Order Iterative Methods for Solving Nonlinear Equations”, [Online]. Available: www.scischolars.com
[3] T. G. Al-Harbi and I. A. Al-Subaihi, “An Optimal Class of Eighth-Order Iterative Methods Based on King’s Method,” Journal of Progressive Research in Mathematics (JPRM), 2018, [Online]. Available: www.scitecresearch.com/journals
[4] O. Said Solaiman, S. A. Abdul Karim, and I. Hashim, “Optimal fourth- and eighth-order of convergence derivative-free modifications of King’s method,” Journal of King Saud University - Science, vol. 31, no. 4, pp. 1499–1504, 2019, doi: https://doi.org/10.1016/j.jksus.2018.12.001.
[5] H. I. Siyyam, M. T. Shatnawi, and I. A. Al-Subaihi, “A new one parameter family of iterative methods with eighth-order of convergence for solving nonlinear equations,” International Journal of Pure and Applied Mathematics, vol. 84, no. 5, pp. 451–461, 2013.
[6] P. Sivakumar, K. Madhu, and J. Jayaraman, “Optimal eighth and sixteenth order iterative methods for solving nonlinear equation with basins of attraction,” Appl. Math. E-notes, vol. 21, pp. 320–343, 2021.
[7] H. S. Wilf, “Iterative Methods for the Solution of Equations (JF Traub),” SIAM Review, vol. 8, no. 4, p. 550, 1966.
[8] H. T. Kung and J. F. Traub, “Optimal Order of One-Point and Multipoint Iteration,” J. ACM, vol. 21, no. 4, pp. 643–651, Oct. 1974, doi: 10.1145/321850.321860.
[9] M. S. Petković and L. D. Petković, “Families of optimal multipoint methods for solving nonlinear equations: a survey,” Applicable Analysis and Discrete Mathematics, pp. 1–22, 2010.
[10] H. M. Bawazir, “Fifth and Eleventh-Order Iterative Methods for Roots of Nonlinear Equations,” Hadhramout University Journal of Natural & Applied Sciences, vol. 17, no. 2, p. 6, 2020.
[11] R. F. King, “A family of fourth order methods for nonlinear equations,” SIAM journal on numerical analysis, vol. 10, no. 5, pp. 876–879, 1973.
[12] O. S. Solaiman and I. Hashim, “Optimal eighth-order solver for nonlinear equations with applications in chemical engineering,” Intell. Autom. Soft Comput, vol. 13, pp. 87–93, 2020.
[13] L. Liu and X. Wang, “Eighth-Order Methods with High Efficiency Index for Solving Nonlinear Equations,” Appl. Math. Comput., vol. 215, no. 9, pp. 3449–3454, Jan. 2010, doi: 10.1016/j.amc.2009.10.040.
[14] J. R. Sharma and H. Arora, “An efficient family of weighted-Newton methods with optimal eighth order convergence,” Applied Mathematics Letters, vol. 29, pp. 1–6, 2014.
[15] A. A. Al-Harbi and I. A. Al-Subaihi, “Family of Optimal Eighth-Order of Convergence for Solving Nonlinear Equations,” Journal of Progressive Research in Mathematics (JPRM), vol. 4, 2015, [Online]. Available: www.scitecresearch.com/journals/index.php/jprm
[16] S. Parimala, K. Madhu, and J. Jayaraman, “A new class of optimal eighth order method with two weight functions for solving nonlinear equation,” J. Nonlinear Anal. Appl, vol. 2018, pp. 83–94, 2018.
[17] S. Weerakoon and T. Fernando, “A variant of Newton’s method with accelerated third-order convergence,” Applied Mathematics Letters, vol. 13, no. 8, pp. 87–93, 2000.
Published
2022-04-12
How to Cite
Al-Turkmani, Z., & Al-Subaihi, I. (2022). A New Family of Optimal Eighth-Order Iterative Scheme for Solving Nonlinear Equations. Journal of Progressive Research in Mathematics, 19(1), 44-53. Retrieved from http://www.scitecresearch.com/journals/index.php/jprm/article/view/2131
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