Extended ball convergence for a seventh order derivative free class of algorithms for nonlinear equations
In the earlier work, expensive Taylor formula and conditions on derivatives up to the eighth
order have been utilized to establish the convergence of a derivative free class of seventh order
iterative algorithms. Moreover, no error distances or results on uniqueness of the solution were
given. In this study, extended ball convergence analysis is derived for this class by imposing
conditions on the first derivative. Additionally, we offer error distances and convergence radius
together with the region of uniqueness for the solution. Therefore, we enlarge the practical
utility of these algorithms. Also, convergence regions of a specific member of this class are displayed
for solving complex polynomial equations. At the end, standard numerical applications
are provided to illustrate the efficacy of our theoretical findings.
S. Amat, S. Busquier, Advances in iterative methods for nonlinear equations, Cham, Springer, 2016.
I. Argyros, Computational theory of iterative methods, New York, CRC Press, 2007.
I.K. Argyros, Convergence and application of Newton-type iterations, Berlin, Springer, 2008.
I.K. Argyros, Polynomial operator equations in abstract spaces and applications, New York, CRC Press, 1998.
I.K. Argyros, S. George, Local convergence for an almost sixth order method for solving equations under weak conditions, SeMA J., 75 (2017), №2, 163–171.
I.K. Argyros, S. George, Mathematical modeling for the solution of equations and systems of equations with applications, Vol. IV, New York, Nova Publisher, 2020.
I.K. Argyros, S. Hilout, Computational methods in nonlinear analysis, New Jersey, World Scientific Publishing House, 2013.
I.K. Argyros, ´A.A. Magre˜n´an, Iterative methods and their dynamics with applications: A Contemporary Study, New York, CRC Press, 2017.
I.K. Argyros, ´A.A. Magre˜n´an, A contemporary study of iterative methods, New York, Elsevier (Academic Press), 2018.
I.K. Argyros, D. Sharma, S.K. Parhi, S.K. Sunanda, On the convergence, dynamics and applications of a new class of nonlinear system solvers, Int. J. Appl. Comput. Math., 6 (2020), №5, Article Number: 142. DOI: https://doi.org/10.1007/s40819-020-00893-4
I.K. Argyros, S. Shakhno, Extended two-step-Kurchatov method for solving Banach space valued nondifferentiable equations, Int. J. Appl. Comput. Math., 6 (2020), no.2, Article Number: 32.
A. Cordero, J.R. Torregrosa, Variants of Newtons method using fifth-order quadrature formulas, Appl. Math. Comput., 190 (2007), 686–698.
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), №1, Article Number: 86, 1–19. DOI: https://doi.org/10.3390/math9010086
M. Grau-S´anchez, M. Noguera, S. Amat, On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, J. Comput. Appl. Math., 237 (2013), 363–372.
J.L. Hueso, E. Mart´inez, J.R. Torregrosa, Third and fourth order iterative methods free from second derivative for nonlinear systems, Appl. Math. Comput., 211 (2009), 190–197.
Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensen’s method of fourth-order convergence and its applications, Appl. Math. Comput., 216 (2010), 1978–1983.
´A.A. Magre˜n´an, Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput., 233 (2014), 29–38.
M. Narang, S. Bhatia, V. Kanwar, New efficient derivative free family of seventh-order methods for solving systems of nonlinear equations, Numer. Algor., 76 (2017), №1, 283–307.
M.S. Petkovi´c, B. Neta, L. Petkovi´c, D. D˜zuni´c, Multipoint methods for solving nonlinear equations, Amsterdam, Elsevier, 2013.
L.B. Rall, Computational solution of nonlinear operator equations, New York, Robert E. Krieger, 1979.
W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Mathematical Models and Numerical Methods (A.N. Tikhonov et al. eds.), 3 (1978), 129–142.
D. Sharma, S.K. Parhi, S.K. Sunanda, Extending the convergence domain of deformed Halley method under ω condition in Banach spaces, Bol. Soc. Mat. Mex. 27, 32 (2021). https://doi.org/10.1007/s40590-021-00318-2
J.R. Sharma, H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algor., 67 (2014), 917–933.
J.R. Sharma, P. Gupta, Efficient family of Traub-Steffensen-type methods for solving systems of nonlinear equations, Adv. Numer. Anal., 2014 (2014), Article ID: 152187, 1–11.
S. Shakhno, Gauss-Newton-Kurchatov method for the solution of nonlinear least-squares problems, J. Math. Sci., 247 (2020), №1, 58–72.
J.F. Traub, Iterative methods for solution of equations, New Jersey, Prentice-Hall, 1964.
X. Wang, T. Zhang, A family of Steffensen type methods with seventh-order convergence, Numer. Algor., 62 (2013), 429–444.
X. Wang, T. Zhang, W. Qian, M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems, Numer. Algor., 70 (2015), 545–558.
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