Basins of Attraction of an Iterative Scheme and Their Applications
DOI:
https://doi.org/10.26713/cma.v14i5.2292Keywords:
Nonlinear equations, Iterative method, Efficiency index, Convergence order, Basins of attractionAbstract
This work presents a seventh-order iterative scheme with the help of a variational iteration technique for finding the root of the nonlinear equations. The convergence analysis of the method is discussed and shows that it has seventh-order convergence with four functional evaluations per iteration. Therefore, the efficiency index is 1.6265. The computational performance of the suggested scheme is compared with some well-existing methods of the same order and is tested on various nonlinear equations, including real-world problems. Furthermore, we analyzed the dynamics of the proposed method using basins of attraction in the complex domain by taking some polynomial functions and compared our results with other known methods.
Downloads
References
I.A. Al-Subaihi and A.J. Al-Qarni, Higher-order iterative methods for solving nonlinear equations, Life Science Journal 11(12) (2014), 85 – 91, URL: https://www.lifesciencesite.com/lsj/life1112/015_26086life111214_85_91.pdf.
P.B. Chand, F.I. Chicharro and P. Jain, On the design and analysis of High-Order Weerakoon-Fernando methods based on weight functions, Computational and Mathematical Methods 2(5) (2020), e1114, 17 pages, DOI: 10.1002/cmm4.1114.
P.B. Chand, F.I. Chicharro, N. Garrido and P. Jain, Design and complex dynamics of Potra–Ptáktype optimal methods for solving nonlinear equations and its applications, Mathematics 7(10) (2019), 942, 21 pages, DOI: 10.3390/math7100942.
F.I. Chicharro, A. Cordero, N. Garrido and J.R. Torregrosa, Wide stability in a new family of optimal fourth-order iterative methods, Computational and Mathematical Methods 1(2) (2019), e1023, 14 pages, DOI: 10.1002/cmm4.1023.
N. Kakarlapudi, M.S.K. Mylapalli, R. Sri and S. Marapaga, Applications of an efficient iterative scheme for finding zeros of nonlinear equations and its basins of attraction, Communications in Mathematics and Applications 14(1) (2023), 67 – 79, DOI: 10.26713/cma.v14i1.2113.
M.M.S. Kumar, R.K. Palli, P. Chaganti and R. Sri, An optimal fourth order iterative method for solving non-linear equations, IAENG International Journal of Applied Mathematics 52(3) (2022), 732 – 741, URL: https://www.iaeng.org/IJAM/issues_v52/issue_3/IJAM_52_3_25.pdf.
B.H. Mohamed, Seventh and twelfth-order iterative methods for roots of nonlinear equations, Hadhramout University Journal of Natural & Applied Sciences 18(1) (2021), 9 – 15, URL: https://digitalcommons.aaru.edu.jo/huj_nas/vol18/iss1/2.
A. Naseem, M.A. Rehman and J. Younis, Some real-life applications of a newly designed algorithm for nonlinear equations and its dynamics via computer tools, Complexity 2021 (2021), Article ID 9234932, 9 pages, DOI: 10.1155/2021/9234932.
M. Shams, N. Rafiq and N. Kausar, Inverse family of numerical methods for approximating all simple and roots with multiplicity of nonlinear polynomial equations with engineering applications, Mathematical Problems in Engineering 2021 (2021), Article ID 3124615, 9 pages, DOI: 10.1155/2021/3124615.
P. Sivakumar and J. Jayaraman, Some new higher order weighted newton methods for solving nonlinear equation with applications, Mathematical and Computational Applications 24(2) (2019), 59, 16 pages, DOI: 10.3390/mca24020059.
O.S. Solaiman and I. Hashim, Optimal eighth-order solver for nonlinear equations with applications in chemical engineering, Intelligent Automation & Soft Computing 27(2) (2021), 379 – 390, DOI: 10.32604/iasc.2021.015285.
N. Srisarakham and M. Thongmoon, A note on three-step iterative methodwith seventh order of convergence for solving nonlinear equations, Thai Journal of Mathematics 14(3) (2016), 565 – 573, URL: https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/619.
J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, New York (1977).
Wartono, Rahmawati and R. Agustin, New modification of third-order iterative method with optimal fourth-order convergence for solving nonlinear equations, International Journal of Scientific Research in Mathematical and Statistical Sciences 6(1) (2019), 155 – 161.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.