Abstract
In this paper, we study a variant of the super-Halley method with fourth-order convergence for nonlinear equations in Banach spaces. We make an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived and then an existence-uniqueness theorem is given to establish the R-order of the method to be four and a priori error bounds. Finally, some numerical applications are presented to demonstrate our approach.
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Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equation in Several Variables. Academic Press, New York (1970)
Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003)
Gutiérrez, J.M., Hernández, M.A.: An acceleration of Newton’s method: super-Halley method, Appl. Math. Comput. 117, 223-239 (2001)
Kou, J., Li, Y., Wang, X.: A variant of super-Halley method with accelerated fourth-order convergence. Appl. Math. Comput. 186, 535–539 (2007)
Rall, L.B.: Computational Solution of Nonlinear Operator Equations. Robert E. Krieger, New York (1979)
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: the Halley method. Computing 44, 169–184 (1990)
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II: the Halley method. Computing 45, 355–367 (1990)
Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for the super-Halley method. Comput. Math. Appl. 7(36), 1–8 (1998)
Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41, 433–445 (2001)
Ezquerro, J.A., Hernández, M.A.: Recurrence relations for Chebyshev-type methods. Appl. Math. Optim. 41, 227–236 (2000)
Ezquerro, J.A., Hernández, M.A.: New iterations of R-order four with reduced computational cost. BIT Numer. Math. 49, 325–342 (2009)
Parida, P.K., Gupta, D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206, 873–887 (2007)
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Wang, X., Gu, C. & Kou, J. Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer Algor 56, 497–516 (2011). https://doi.org/10.1007/s11075-010-9401-1
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DOI: https://doi.org/10.1007/s11075-010-9401-1