Abstract
In this paper, the semilocal convergence for a class of multi-point modified Chebyshev-Halley methods in Banach spaces is studied. Different from the results in reference [11], these methods are more general and the convergence conditions are also relaxed. We derive a system of recurrence relations for these methods and based on this, we prove a convergence theorem to show the existence-uniqueness of the solution. A priori error bounds is also given. The R-order of these methods is proved to be 5+q with ω−conditioned third-order Fréchet derivative, where ω(μ) is a non-decreasing continuous real function for μ > 0 and satisfies ω(0) ≥ 0, ω(tμ) ≤ t q ω(μ) for μ > 0,t ∈ [0,1] and q ∈ [0,1]. Finally, we give some numerical results to show our approach.
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References
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equation in Several Variables. Academic Press, New York (1970)
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: The Halley method. Comput. 44, 169–184 (1990)
Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for the super-Halley method, Comput. Math. Applic. 36, 1-8 (1998)
Argyros, I.K., Chen, D.: Results on the Chebyshev method in banach spaces. Proyecciones 12(2), 119–128 (1993)
Hernández, M.A., Salanova, M.A.: Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method. J. Comput. Appl. Math. 126, 131–143 (2000)
Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput.Math. with Appl. 41, 433-445 (2001)
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II: The Chebyshev method. Comput. 45, 355-367 (1990)
Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math 157, 197–205 (2003)
Argyros, I.K.: Quadratic equations and applications to Chandrasekhar’s and related equations. Bull. Austral. Math. Soc 32, 275–292 (1985)
Hernández, M.A.: Second-derivative-free variant of the Chebyshev method for nonlinear equations. J. Optim. Theory Appl. 104(3), 501–515 (2000)
Wang, X., Kou, J.: Semilocal convergence and R-order for modified Chebyshev-Halley methods. Numer. Algoritm. 64, 105–126 (2012)
Gutiérrez, J.M., Hernández, M.A.: Third-order iterative methods for operators with bounded second derivative. J. Comput. Appl. Math. 82, 171–183 (1997)
Proinov, P.D.: New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complexity 26, 3–42 (2010)
Argyros, I.K.: Improved generalized differentiability conditions for Newton-like methods. J.Complexity 26, 316–333 (2010)
Gutiérrez, J.M., Magreñán, Á.A., Romero, N.: On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions. Appl. Math. Comput. 221, 79–88 (2013)
Amat, S., Magreñán, Á.A., Romero, N.: On a two-step relaxed Newton-type method. Appl. Math. Comput. 219(24), 11341–11347 (2013)
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Wang, X., Kou, J. Convergence for a class of multi-point modified Chebyshev-Halley methods under the relaxed conditions. Numer Algor 68, 569–583 (2015). https://doi.org/10.1007/s11075-014-9861-9
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DOI: https://doi.org/10.1007/s11075-014-9861-9