Search Advanced

AIMS Mathematics

2023, Volume 8, Issue 9: 22371-22384. doi: 10.3934/math.20231141
Previous Article Next Article
Research article

Semilocal convergence analysis of an eighth order iterative method for solving nonlinear systems

  • School of Mathematical Sciences, Bohai University, Jinzhou 121000, China
  • Received: 03 June 2023 Revised: 08 July 2023 Accepted: 11 July 2023 Published: 13 July 2023

  • MSC : 49M15, 65J15, 65G99

  • In this paper, the semilocal convergence of the eighth order iterative method is proved in Banach space by using the recursive relation, and the proof process does not need high order derivative. By selecting the appropriate initial point and applying the Lipschitz condition to the first order Fréchet derivative in the whole region, the existence and uniqueness domain are obtained. In addition, the theoretical results of semilocal convergence are applied to two nonlinear systems, and satisfactory results are obtained.

    Citation: Xiaofeng Wang, Yufan Yang, Yuping Qin. Semilocal convergence analysis of an eighth order iterative method for solving nonlinear systems[J]. AIMS Mathematics, 2023, 8(9): 22371-22384. doi: 10.3934/math.20231141

    Related Papers:

  • In this paper, the semilocal convergence of the eighth order iterative method is proved in Banach space by using the recursive relation, and the proof process does not need high order derivative. By selecting the appropriate initial point and applying the Lipschitz condition to the first order Fréchet derivative in the whole region, the existence and uniqueness domain are obtained. In addition, the theoretical results of semilocal convergence are applied to two nonlinear systems, and satisfactory results are obtained.


    加载中


    [1] J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, New York: Academic Press, 1970. https://doi.org/10.1016/C2013-0-11263-9
    [2] R. Behl, S. Bhalla, Á. A. Magrenán, S. Kumar, An efficient high order iterative scheme for large nonlinear systems with dynamics, J. Comput. Appl. Math., 404 (2022), 113249. http://dx.doi.org/10.1016/j.cam.2020.113249 doi: 10.1016/j.cam.2020.113249
    [3] C. Chun, B. Neta, Developing high order methods for the solution of systems of nonlinear equations, Appl. Math. Comput., 342 (2019), 178–190. http://dx.doi.org/10.1016/j.amc.2018.09.032 doi: 10.1016/j.amc.2018.09.032
    [4] 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
    [5] X. Wang, W. Li, Stability analysis of simple root seeker for nonlinear equation, Axioms, 12 (2023), 215. https://doi.org/10.3390/axioms12020215 doi: 10.3390/axioms12020215
    [6] T. Zhanlav, K. Otgondorj, Higher order Jarratt-like iterations for solving systems of nonlinear equations, Appl. Math. Comput., 395 (2021), 125849. https://dx.doi.org/10.1016/j.amc.2020.125849 doi: 10.1016/j.amc.2020.125849
    [7] S. Regmi, Optimized iterative methods with applications in diverse disciplines, New York: Nova Science Publishers, Inc., 2021.
    [8] B. Neta, A new derivative-free method to solve nonlinear equations, Mathematics, 9 (2021), 583. http://dx.doi.org/10.3390/math9060583 doi: 10.3390/math9060583
    [9] C. Chun, B. Neta, An efficient derivative-free method for the solution of systems of equations, Numer. Func. Anal. Opt., 42 (2021), 834–848. http://dx.doi.org/10.1080/01630563.2021.1931313 doi: 10.1080/01630563.2021.1931313
    [10] R. Behl, A. Cordero, J. R. Torregrosa, A new higher-order optimal derivative free scheme for multiple roots, J. Comput. Appl. Math., 404 (2022), 113773. http://dx.doi.org/10.1016/j.cam.2021.113773 doi: 10.1016/j.cam.2021.113773
    [11] M. Kansal, A. S. Alshomrani, S. Bhalla, R. Behl, M. Salimi, One parameter optimal derivative-free family to find the multiple roots of algebraic nonlinear equations, Mathematics, 8 (2020), 2223. http://dx.doi.org/10.3390/math8122223 doi: 10.3390/math8122223
    [12] J. R. Sharma, S. Kumar, L. Jantschi, On derivative free multiple-root finders with optimal fourth order convergence, Mathematics, 8 (2020), 1091. http://dx.doi.org/10.3390/math8071091 doi: 10.3390/math8071091
    [13] X. Wang, Fixed-point iterative method with eighth-order constructed by undetermined parameter technique for solving nonlinear systems, Symmetry, 13 (2021), 863. http://dx.doi.org/10.3390/sym13050863 doi: 10.3390/sym13050863
    [14] I. K. Argyros, S. George, On the complexity of extending the convergence region for Traub's method, J. Complexity, 56 (2020), 101423. http://dx.doi.org/10.1016/j.jco.2019.101423 doi: 10.1016/j.jco.2019.101423
    [15] I. K. Argyros, S. George, Ball comparison between four fourth convergence order methods under the same set of hypotheses for solving equations, Int. J. Appl. Comput. Math., 7 (2021), 9. http://dx.doi.org/10.1007/S40819-020-00946-8 doi: 10.1007/S40819-020-00946-8
    [16] I. K. Argyros, A new convergence theorem for the Steffenssen method in Banach space and applications, Rev. Anal. Numér. Théor. Approx, 29 (2000), 119–127.
    [17] A. Cordero, E. G. Villalba, J. R. Torregrosa, P. Triguero-Navarro, Convergence and stability of a parametric class of iterative schemes for solving nonlinear systems, Mathematics, 9 (2021), 86. http://dx.doi.org/10.3390/math9010086 doi: 10.3390/math9010086
    [18] I. K. Argyros, S. George, S. Shakhno, H. Yarmola, Perturbed Newton methods for solving nonlinear equations with applications, Symmetry, 14 (2022), 2206. http://dx.doi.org/10.3390/sym14102206 doi: 10.3390/sym14102206
    [19] A. Cordero, J. G. Maimó, E. Martinez, J. R. Torregrosa, Semilocal convergence of the extension of Chun's method, Axioms, 10 (2021), 161. http://dx.doi.org/10.3390/axioms10030161 doi: 10.3390/axioms10030161
    [20] S. Amat, M. A. Hernández, N. Romero, Semilocal convergence of a sixth order iterative method for quadratic equations, Appl. Numer. Math., 62 (2012), 833–841. http://dx.doi.org/10.1016/j.apnum.2012.03.001 doi: 10.1016/j.apnum.2012.03.001
    [21] A. Cordero, M. A. Hernández-Verón, N. Romero, J. R. Torregrosa, Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces, J. Comput. Appl. Math., 273 (2015), 205–213. http://dx.doi.org/10.1016/j.cam.2014.06.008 doi: 10.1016/j.cam.2014.06.008
    [22] M. A. Hernández-Verón, E. Martinez, C. Teruel, Semilocal convergence of a k-step iterative process and its application for solving a special kind of conservative problems, Numer. Algor., 76 (2017), 309–331. http://dx.doi.org/10.1007/s11075-016-0255-z doi: 10.1007/s11075-016-0255-z
    [23] V. Candela, A. Marquina, Recurrence relations for rational cubic models Ⅱ: the Chebyshev method, Computing, 45 (1990), 355–367. http://dx.doi.org/10.1007/BF02238803 doi: 10.1007/BF02238803
    [24] J. A. Ezquerro, M. A. Hernández-Verón, Halley's method for operators with unbounded second derivative, Appl. Numer. Math., 57 (2007), 354–360. http://dx.doi.org/10.1016/j.apnum.2006.05.001 doi: 10.1016/j.apnum.2006.05.001
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(1078) PDF downloads(58) Cited by(3)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(3)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog