Third order convergence theorem for a family of Newton like methods in Banach space
DOI:
https://doi.org/10.33993/jnaat431-996Keywords:
nonlinear equations in Banach space, third order Newton like methods, recurrence relations, error bounds, convergence domainAbstract
In this paper, we propose a family of Newton-like methods in Banach space which includes some well known third-order methods as particular cases. We establish the Newton-Kantorovich type convergence theorem for a proposed family and get an error estimate.Downloads
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D. K. R. Babajee, M.Z. Dauhoo, M.T. Darvishi, A. Karami and A. Barati, Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations, J. Comput. Appl. Math., 233 (2010), pp. 2002-2012, https://doi.org/10.1016/j.cam.2009.09.035 DOI: https://doi.org/10.1016/j.cam.2009.09.035
M. Frontini and E. Sormani, Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput., 149 (2004), pp. 771-782, https://doi.org/10.1016/s0096-3003(03)00178-4 DOI: https://doi.org/10.1016/S0096-3003(03)00178-4
M.A. Hernandez, Second derivative free variant of the Chebyshev method for nonlinear equations, J. Optimization theory and applications, 104 (2000), pp. 501-514,https://doi.org/10.1023/a:1004618223538 DOI: https://doi.org/10.1023/A:1004618223538
M.A. Hernandez, M.A. Salanova, Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method, J. Comput. Appl. Math, 126, pp. 131-143, 2000, https://doi.org/10.1016/s0377-0427(99)00347-7 DOI: https://doi.org/10.1016/S0377-0427(99)00347-7
H.H.H. Homeier, On Newton type methods with cubic convergence, J. Comput. Appl. Math., 176 (2005), pp. 425-432, https://doi.org/10.1016/j.cam.2004.07.027 DOI: https://doi.org/10.1016/j.cam.2004.07.027
J.L. Hueso, E. Martinez, J.R. Torregrosa, Third order iterative methods free from second derivative, Appl. Math. Comput., 215 (2009), pp. 58-65, https://doi.org/10.1016/j.amc.2009.04.046 DOI: https://doi.org/10.1016/j.amc.2009.04.046
Q. Wu, Y. Zhao, Third order convergence theorem by using majorizing function for a modified Newton method in Banach space, Appl. Math. Comput., 175 (2006), pp. 1515-1524, https://doi.org/10.1016/j.amc.2005.08.043 DOI: https://doi.org/10.1016/j.amc.2005.08.043
T. Zhanlav, Note on the cubic decreasing region of the Chebyshev method, J. Comput. Appl. Math, 235 (2010), pp. 341-344, https://doi.org/10.1016/j.cam.2010.05.034 DOI: https://doi.org/10.1016/j.cam.2010.05.034
T. Zhanlav, O. Chuluunbaatar, Higher order convergent iteration methods for solving nonlinear equations, Bulletin of People's Friendship University of Russia, 4 (2009), pp. 47-55.
T. Zhanlav, D. Khongorzul, Semilocal convergence with R-order three theorems for the Chebyshev method and its modifications, Optimization, Simulation and Control, Springer, pp. 331-345, 2012, https://doi.org/10.1007/978-1-4614-5131-0_21 DOI: https://doi.org/10.1007/978-1-4614-5131-0_21
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