There are numerous applications for finding zero of derivatives in function optimization. In this paper, a two-step fourth-order method was presented for finding a zero of the derivative. In the research process of iterative methods, determining the ball of convergence was one of the important issues. This paper discussed the radii of the convergence ball, uniqueness of the solution, and the measurable error distances. In particular, in contrast to Wang's method under hypotheses up to the fourth derivative, the local convergence of the new method was only analyzed under hypotheses up to the second derivative, and the convergence order of the new method was increased to four. Furthermore, different radii of the convergence ball was determined according to different weaker hypotheses. Finally, the convergence criteria was verified by three numerical examples and the new method was compared with Wang's method and the same order method by numerical experiments. The experimental results showed that the convergence order of the new method is four and the new method has higher accuracy at the same cost, so the new method is finer.
Citation: Xiaofeng Wang, Dongdong Ruan. Convergence ball of a new fourth-order method for finding a zero of the derivative[J]. AIMS Mathematics, 2024, 9(3): 6073-6087. doi: 10.3934/math.2024297
There are numerous applications for finding zero of derivatives in function optimization. In this paper, a two-step fourth-order method was presented for finding a zero of the derivative. In the research process of iterative methods, determining the ball of convergence was one of the important issues. This paper discussed the radii of the convergence ball, uniqueness of the solution, and the measurable error distances. In particular, in contrast to Wang's method under hypotheses up to the fourth derivative, the local convergence of the new method was only analyzed under hypotheses up to the second derivative, and the convergence order of the new method was increased to four. Furthermore, different radii of the convergence ball was determined according to different weaker hypotheses. Finally, the convergence criteria was verified by three numerical examples and the new method was compared with Wang's method and the same order method by numerical experiments. The experimental results showed that the convergence order of the new method is four and the new method has higher accuracy at the same cost, so the new method is finer.
| [1] |
S. Amat, I. K. Argyros, S. Busquier, M. A. Hernández-Verón, E. Martinez, On the local convergence study for an efficient k-step iterative method, J. Comput. Appl. Math., 343 (2018), 753–761. http://dx.doi.org/10.1016/j.cam.2018.02.028 doi: 10.1016/j.cam.2018.02.028
|
| [2] |
I. K. Argyros, S. George, Enlarging the convergence ball of the method of parabola for finding zero of derivatives, Appl. Math. Comput., 256 (2015), 68–74. http://dx.doi.org/10.1016/j.amc.2015.01.030 doi: 10.1016/j.amc.2015.01.030
|
| [3] |
I. K. Argyros, S. George, On the complexity of extending the convergence region for traub's method, J. Complex., 56 (2020), 101423. http://dx.doi.org/10.1016/j.jco.2019.101423 doi: 10.1016/j.jco.2019.101423
|
| [4] |
I. K. Argyros, S. Hilout, Weaker conditions for the convergence of Newton's method, J. Complex., 28 (2012), 364–387. http://dx.doi.org/10.1016/j.jco.2011.12.003 doi: 10.1016/j.jco.2011.12.003
|
| [5] |
Z. D. Huang, The convergence ball of Newton's method and the uniqueness ball of equations under Hölder-type continuous derivatives, Comput. Math. Appl., 5 (2004), 247–251. http://dx.doi.org/10.1016/s0898-1221(04)90021-1 doi: 10.1016/s0898-1221(04)90021-1
|
| [6] | J. M. Ortega, W. C. Rheinbolt, Iterative Solution of Nonlinear Equations in Several Variables, New York: Academic Press, 1970. http://dx.doi.org/10.1137/1.9780898719468 |
| [7] | F. A. Potra, V. Ptak, Nondiscrete Induction and Iterative Processes, Boston: Research Notes in Mathematics, 1984. http://dx.doi.org/10.1137/1029105 |
| [8] |
P. D. Proinov, General local convergence theory for a class of iterative processes and its applications to Newton's method, J. Complex., 25 (2009), 38–62. http://dx.doi.org/10.1016/j.jco.2008.05.006 doi: 10.1016/j.jco.2008.05.006
|
| [9] |
L. B. Rall, A note on the convergence of Newton's method, SIAM J. Numer. Anal., 11 (1974), 34–36. http://dx.doi.org/10.1137/0711004 doi: 10.1137/0711004
|
| [10] |
H. M. Ren, Q. B. Wu, The convergence ball of the secant method under Hölder continuous divided differences, J. Comput. Appl. Math., 194 (2006), 284–293. http://dx.doi.org/10.1016/j.cam.2005.07.008 doi: 10.1016/j.cam.2005.07.008
|
| [11] |
H. Ren, I. K. Argyros, On the complexity of extending the convergence ball of Wang's method for finding a zero of a derivative, J. Complex., 64 (2021), 101526. http://dx.doi.org/10.1016/j.jco.2020.101526 doi: 10.1016/j.jco.2020.101526
|
| [12] |
W. C. Rheinboldt, A unified convergence theory for a class of iterative processes, SIAM J. Numer. Anal., 5 (1968), 42–63. http://dx.doi.org/10.1137/0705003 doi: 10.1137/0705003
|
| [13] |
J. R. Sharma, I. K. Argyros, Local convergence of a Newton-Traub composition in banach spaces, SeMA J., 75 (2017), 57–68. http://dx.doi.org/10.1007/s40324-017-0113-5 doi: 10.1007/s40324-017-0113-5
|
| [14] | J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, New York: Springer-Verlag, 1980. http://dx.doi.org/10.1017/CBO9780511801181 |
| [15] |
X. H. Wang, C. Li, Convergence of Newton's method and uniqueness of the solution of equations in Banach spaces II, Acta Math. Sinica, 19 (2003), 405–412. https://doi.org/10.1007/s10114-002-0238-y doi: 10.1007/s10114-002-0238-y
|
| [16] |
X. H. Wang, C. Li, On the convergent iteration method of order two for finding zeros of the derivative, in Chinese, Math. Numer. Sin., 23 (2001), 121–128. http://dx.doi.org/10.3321/j.issn:0254-7791.2001.01.013. doi: 10.3321/j.issn:0254-7791.2001.01.013
|
| [17] |
X. Wang, Y. Cao, A numerically stable high-order Chebyshev-Halley type multipoint iterative method for calculating matrix sign function, AIMS Math., 8 (2023), 12456–12471. http://dx.doi.org/10.3934/math.2023625 doi: 10.3934/math.2023625
|
| [18] |
X. Wang, Y. Yang, Y. Qin, Semilocal convergence analysis of an eighth order iterative method for solving nonlinear systems, AIMS Math., 8 (2023), 22371–22384. http://dx.doi.org/10.3934/math.20231141 doi: 10.3934/math.20231141
|
| [19] |
X. Wang, T. Zhang, W. Qian, M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems, Numer. Algor., 70 (2015), 545–558. https://dx.doi.org/10.1007/s11075-015-9960-2 doi: 10.1007/s11075-015-9960-2
|
| [20] | X. Wang, X. Chen, W. Li, Dynamical behavior analysis of an eighth-order Sharma's method, Int. J. Biomath., 2023, 2350068. https://dx.doi.org/10.1142/S1793524523500687 |
| [21] | X. Wang, J. Xu, Conformable vector Traub's method for solving nonlinear systems, Numer. Alor., 2024. https://dx.doi.org/10.1007/s11075-024-01762-7 |
| [22] |
Q. B. Wu, H. M. Ren, Convergence ball of a modified secant method for finding zero of derivatives, Appl. Math. Comput., 174 (2006), 24–33. http://dx.doi.org/10.1016/j.amc.2005.05.007 doi: 10.1016/j.amc.2005.05.007
|
| [23] |
Q. B. Wu, H. M. Ren, W. H. Bi, The convergence ball of wang's method for finding a zero of a derivative, Math. Comput. Model., 49 (2009), 740–744. http://dx.doi.org/10.1016/j.mcm.2008.04.002 doi: 10.1016/j.mcm.2008.04.002
|
Xiaofeng Wang, Dongdong Ruan. Convergence ball of a new fourth-order method for finding a zero of the derivative[J]. AIMS Mathematics, 2024, 9(3): 6073-6087. doi: 10.3934/math.2024297
DownLoad: