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Bhavna; Bhatia, Saurabh

On some extension of Traub-Steffensen type methods in Banach spaces. (English) Zbl 07931029

Comput. Appl. Math. 43, No. 7, Paper No. 413, 25 p. (2024).
Summary: In the present work, we construct a sixth order derivative free family without memory for solving nonlinear operator equations in Banach spaces. We further modify this to a ninth order family with memory without any additional functional evaluation. Local convergence analysis of both these methods have been studied using assumptions only on the first derivative. Numerical computations validate the theoretical results and show the superiority of our methods over the existing ones. Basins of attraction have also been presented to see the dynamical behaviour of the proposed methods.

MSC:

65A05 Tables in numerical analysis
65G99 Error analysis and interval analysis
65J15 Numerical solutions to equations with nonlinear operators
65Z05 Applications to the sciences

Keywords:

local convergence; Banach space; nonlinear equations; radius of convergence; basin of attraction
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References:

[1] Ahmad, F.; Soleymani, F.; Haghani, FK; Serra-Capizzano, S., Higher order derivative-free iterative methods with and without memory for systems of nonlinear equations, Appl Math Comput, 314, 199-211, 2017 · Zbl 1426.65071
[2] Argyros, IK; Chui, CK; Wuytack, L., Computational theory of iterative methods, Series: studies in computational mathematics, 2007, New York: Elsevier Publ. Co., New York · Zbl 1147.65313
[3] Argyros, IK; George, S., Higher order derivative-free iterative methods with and without memory in Banach space under weak conditions, Bangmod Int J Math Comput Sci, 3, 25-34, 2017
[4] Argyros, IK; George, S., Local convergence for a Chebyshev-type method in Banach space free of derivatives, Adv Theory Nonlinear Anal Appl, 2, 62-69, 2018 · Zbl 1412.65010
[5] Argyros, IK; Kansal, M.; Kanwar, V.; Bajaj, S., Higher order derivative free families of Chebyshev-Halley type methods with or without memory for solving nonlinear equations, Appl Math Comput, 315, 224-245, 2017 · Zbl 1426.65064
[6] Argyros, IK; Shakhno, S.; Yarmola, H., Extending the applicability of iterative methods free of derivatives by means of memory, Ann Univ Sci Budapest Sect Comput, 51, 289-300, 2020 · Zbl 1474.65154
[7] Behl, R.; González, D.; Maroju, P.; Motsa, SS, An optimal and efficient general eighth order derivative free scheme for simple roots, J Comput Appl Math, 330, 666-675, 2018 · Zbl 1376.65079 · doi:10.1016/j.cam.2017.07.036
[8] Chicharro, FI; Cordero, A.; Garrido, N.; Torregrosa, JR, On the improvement of the order of convergence of iterative methods for solving nonlinear systems by means of memory, Appl Math Lett, 104, 106277, 2020 · Zbl 1439.65068 · doi:10.1016/j.aml.2020.106277
[9] Chun, C.; Neta, B., An efficient derivative-free method for the solution of systems of equations, Numer Funct Anal Optim, 42, 834-848, 2021 · Zbl 07379887 · doi:10.1080/01630563.2021.1931313
[10] Cordero A, Garrido N, Torregrosa J, Triguero-Navarro P (2022) Design of iterative methods with memory for solving nonlinear systems. doi:10.22541/au.166236702.25444373/v1 · Zbl 1530.65050
[11] Cordero, A.; Villalba, EG; Torregrosa, JR; Triguero-Navarro, P., Introducing memory to a family of multi-step multidimensional iterative methods with weight function, Expo Math, 41, 398-417, 2023 · Zbl 1515.65122 · doi:10.1016/j.exmath.2023.04.004
[12] Jain, P., Steffensen type methods for solving nonlinear equations, Appl Math Comput, 194, 527-533, 2007 · Zbl 1193.65063
[13] Lalehchini, MJ; Lotfi, T.; Mahdiani, K., Adaptive Steffensen-like methods with memory for solving nonlinear equations with the highest possible efficiency indices, Int J Ind Math, 11, 337-345, 2019
[14] Magrenan, AA; Argyros, IK, Contemporary study of iterative methods convergence, dynamics and applications, 2018, Elsevier · Zbl 1386.65007
[15] Narang, M.; Bhatia, S.; Kanwar, V., New efficient derivative free family of seventh order methods for solving systems of nonlinear equations, Numer Algorithms, 76, 283-307, 2017 · Zbl 1375.65072 · doi:10.1007/s11075-016-0254-0
[16] Narang, M.; Bhatia, S.; Kanwar, V., An efficient family of Steffensen-type methods with memory for solving systems of nonlinear equations, Comput Math Methods, 3, 6, e1192, 2021 · doi:10.1002/cmm4.1192
[17] Neta, B., A new derivative-free method to solve nonlinear equations, Mathematics, 9, 583, 2021 · doi:10.3390/math9060583
[18] Ortega, JM; Rheinboldt, WC, Iterative solution of nonlinear equations in several variables, 1970, New York: Academic Press, New York · Zbl 0241.65046
[19] Ostrowski, AM, Solution of equations and systems of equations, 1966, New York: Academic Press, New York · Zbl 0222.65070
[20] Saffari, H.; Mirzai, NM; Mansouri, I., An accelerated incremental algorithm to trace the nonlinear equilibrium path of structures, J Comput Phys, 9, 425-442, 2012
[21] Said Solaiman, O.; Abdul Karim, SA; Hashim, I., Optimal fourth and eighth-order of convergence derivative-free modifications of King’s method, J King Saud Univ Sci, 31, 4, 1499-1504, 2019 · doi:10.1016/j.jksus.2018.12.001
[22] Seaid, M.; Frank, M.; Klar, A.; Pinnau, R.; Thommes, G., Efficient numerical methods for radiation in gas turbines, J Comput Appl Math, 170, 217-239, 2004 · Zbl 1221.80023 · doi:10.1016/j.cam.2004.01.003
[23] Sharma, JR; Arora, H., Efficient higher order derivative-free multipoint methods with and without memory for systems of nonlinear equations, Int J Comput Math, 5, 920-938, 2018 · Zbl 1499.65192 · doi:10.1080/00207160.2017.1298747
[24] Sharma, H.; Kansal, M.; Behl, R., An efficient two-step iterative family adaptive with memory for solving nonlinear equations and their applications, Math Comput Appl, 27, 97, 2022
[25] Sharma, E.; Mittal, SK; Jaiswal, JP; Panday, S., An efficient Bi-parametric with memory iterative methods for solving nonlinear equations, Appl Math, 3, 1019-1033, 2023
[26] Traub, JF, Iterative methods for the solution of equations, 1964, Englewood Cliffs: Prentice-Hall, Englewood Cliffs · Zbl 0121.11204
[27] Zerah, G., An efficient Newton’s method for the numerical solution of fluid integral equations, J Comput Phys, 61, 280-285, 1985 · Zbl 0605.76050 · doi:10.1016/0021-9991(85)90087-7
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