Local convergence study of tenth-order iterative method in Banach spaces with basin of attraction

Kasmita Devi; Prashanth Maroju; Kasmita Devi; Prashanth Maroju

doi:10.3934/math.2024324

AIMS Mathematics

2024, Volume 9, Issue 3: 6648-6667. doi: 10.3934/math.2024324

Previous Article Next Article

Research article Special Issues

Local convergence study of tenth-order iterative method in Banach spaces with basin of attraction

Kasmita Devi ,
Prashanth Maroju ^,

Department of Mathematics, SAS, VIT-AP University, Amaravati, 522237, Andhra Pradesh, India

Received: 17 November 2023 Revised: 21 December 2023 Accepted: 10 January 2024 Published: 06 February 2024
MSC : 65Hxx, 65H05, 65H10

Many applications from computational mathematics can be identified for a system of non-linear equations in more generalized Banach spaces. Analytical methods do not exist for solving these type of equations, and so we solve these equations using iterative methods. We introduced a new numerical technique for finding the roots of non-linear equations in Banach space. The method is tenth-order and it is an extension of the fifth-order method which is developed by Arroyo et.al. ^[1]. We provided a convergence analysis to demonstrate that the method exhibits tenth-order convergence. Also, we discussed the local convergence properties of the suggested method which depends on the fundamental supposition that the first-order Fréchet derivative of the involved function $ \Upsilon $ satisfies the Lipschitz conditions. This new approach is not only an extension of prior research, but also establishes a theoretical concept of the radius of convergence. We validated the efficacy of our method through various numerical examples. Our method is comparable with the methods of Tao Y et al. ^[2]. We also compared it with higher-order iterative methods, and we observed that it either performs similarly or better for the numerical examples. We also gave the basin of attraction to demonstrate the behaviour in the complex plane.
- local convergence,
- Lipschitz continuity,
- Fréchet derivative,
- non-linear equations
Citation: Kasmita Devi, Prashanth Maroju. Local convergence study of tenth-order iterative method in Banach spaces with basin of attraction[J]. AIMS Mathematics, 2024, 9(3): 6648-6667. doi: 10.3934/math.2024324

Related Papers:

Abstract

Many applications from computational mathematics can be identified for a system of non-linear equations in more generalized Banach spaces. Analytical methods do not exist for solving these type of equations, and so we solve these equations using iterative methods. We introduced a new numerical technique for finding the roots of non-linear equations in Banach space. The method is tenth-order and it is an extension of the fifth-order method which is developed by Arroyo et.al. ^[1]. We provided a convergence analysis to demonstrate that the method exhibits tenth-order convergence. Also, we discussed the local convergence properties of the suggested method which depends on the fundamental supposition that the first-order Fréchet derivative of the involved function $ \Upsilon $ satisfies the Lipschitz conditions. This new approach is not only an extension of prior research, but also establishes a theoretical concept of the radius of convergence. We validated the efficacy of our method through various numerical examples. Our method is comparable with the methods of Tao Y et al. ^[2]. We also compared it with higher-order iterative methods, and we observed that it either performs similarly or better for the numerical examples. We also gave the basin of attraction to demonstrate the behaviour in the complex plane.

References

[1]	A. Cordero, J. A. Ezquerro, M. A. Hernández-Verón, J. R. Torregrosa, On the local convergence of a fifth-order iterative method in Banach spaces, Appl. Math. Comput., 251 (2015), 396–403. https://doi.org/10.1016/j.amc.2014.11.084 doi: 10.1016/j.amc.2014.11.084
[2]	Y. Tao, K. Madhu, Optimal fourth, eighth and sixteenth order methods by using divided difference techniques and their basins of attraction and its application, Mathematics, 7 (2019), 322. https://doi.org/10.3390/math7040322 doi: 10.3390/math7040322
[3]	I. K. Argyros, S. K. Khattri, S. George, Local convergence of an at least sixth-order method in Banach spaces, J. Fixed Point Theory Appl., 21 (2019), 23. https://doi.org/10.1007/s11784-019-0662-6 doi: 10.1007/s11784-019-0662-6
[4]	S. Amat, I. K. Argyros, S. Busquier, M. A. Hernández-Verón, E. Martínez, On the local convergence study for an efficient k-step iterative method, J. Comput. Appl. Math., 343 (2018), 753–761. https://doi.org/10.1016/j.cam.2018.02.028 doi: 10.1016/j.cam.2018.02.028
[5]	T. M. Pavkov, V. G. Kabadzhov, I. K. Ivanov, S. I. Ivanov, Local convergence analysis of a one parameter family of simultaneous methods with applications to real-world problems, Algorithms, 16 (2023), 103. https://doi.org/10.3390/a16020103 doi: 10.3390/a16020103
[6]	P. Maroju, Á. A. Magreñán, Í. Sarría, A. Kumar, Local convergence of fourth and fifth order parametric family of iterative methods in Banach spaces, J. Math. Chem., 58 (2020), 686–705. https://doi.org/10.1007/s10910-019-01097-y doi: 10.1007/s10910-019-01097-y
[7]	A. Kumar, P. Maroju, R. Behl, D. K. Gupta, S. S. Motsa, A family of higher order iterations free from second derivative for nonlinear equations in R, J. Comput. Appl. Math., 330 (2018), 676–694. https://doi.org/10.1016/j.cam.2017.07.005 doi: 10.1016/j.cam.2017.07.005
[8]	F. Soleimani, F. Soleymani, S. Shateyi, Some iterative methods free from derivatives and their basins of attraction for nonlinear equations, Discrete Dyn. Nat. Soc., 2013 (2013), 301718. https://doi.org/10.1155/2013/301718 doi: 10.1155/2013/301718
[9]	S. Singh, D. K. Gupta, Iterative methods of higher order for nonlinear equations, Vietnam J. Math., 44 (2016), 387–398. https://link.springer.com/article/10.1007/s10013-015-0135-1
[10]	S. Sutherland, Finding roots of complex polynomials with Newton’s method, Boston University, 1989.
[11]	H. Singh, J. R. Sharma, Simple yet highly efficient numerical techniques for systems of nonlinear equations, Comput. Appl. Math., 42 (2023), 22. https://doi.org/10.1007/s40314-022-02159-9 doi: 10.1007/s40314-022-02159-9
[12]	A. A. Magreñán-Ruiz, I. K. Argyros, Two-step Newton methods, J. Complexity, 30 (2014), 533–553. https://doi.org/10.1016/j.jco.2013.10.002 doi: 10.1016/j.jco.2013.10.002
[13]	G. A. Nadeem, W. Aslam, F. Ali, An optimal fourth-order second derivative free iterative method for nonlinear scientific equations, Kuwait J. Sci., 50 (2023), 1–15. https://doi.org/10.48129/kjs.18253 doi: 10.48129/kjs.18253
[14]	P. Maroju, R. Behl, S. S. Motsa, Some novel and optimal families of King's method with eighth and sixteenth-order of convergence, J. Comput. Appl. Math., 318 (2017), 136–148. https://doi.org/10.1016/j.cam.2016.11.018 doi: 10.1016/j.cam.2016.11.018
[15]	A. K. Maheshwari, A fourth order iterative method for solving nonlinear equations, Appl. Math. Comput., 211 (2009), 383–391. https://doi.org/10.1016/j.amc.2009.01.047 doi: 10.1016/j.amc.2009.01.047
[16]	V. Arroy, A. Cordero, J. R. Torregrosa, M. P. Vassileva, Artificial satellites preliminary orbit determination by the modified high-order Gauss method, Int. J. Comput. Math., 89 (2012), 347–356. https://doi.org/10.1080/00207160.2011.560266 doi: 10.1080/00207160.2011.560266
[17]	N. Y. Abdul-Hassan, A. H. Ali, C. Park, A new fifth-order iterative method free from second derivative for solving nonlinear equations, J. Appl. Math. Comput., 68 (2022), 2877–2886. https://doi.org/10.1007/s12190-021-01647-1 doi: 10.1007/s12190-021-01647-1
[18]	A. S. Alshomrani, R. Behl, P. Maroju, Local convergence of parameter based method with six and eighth order of convergence, J. Math. Chem., 58 (2020), 841–853. https://doi.org/10.1007/s10910-020-01113-6 doi: 10.1007/s10910-020-01113-6
[19]	O. S. Solaiman, I. Hashim, Two new efficient sixth order iterative methods for solving nonlinear equations, J. King Saud Univ. Sci., 31 (2019), 701–705, https://doi.org/10.1016/j.jksus.2018.03.021 doi: 10.1016/j.jksus.2018.03.021
[20]	R. Behl, P. Maroju, S. S. Motsa, A family of second derivative free fourth order continuation method for solving nonlinear equations, J. Comput. Appl. Math., 318 (2017), 38–46. https://doi.org/10.1016/j.cam.2016.12.008 doi: 10.1016/j.cam.2016.12.008
[21]	A. G. Wiersma, The complex dynamics of Newton's method, University of Groningen, 2016.

Reader Comments

Your name:*

Email:*
© 2024 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)