Abstract
We revisit a one-step intermediate Newton method for the iterative computation of a zero of the sum of two nonlinear operators that was analyzed by Uko and Velásquez (Rev. Colomb. Mat. 35:21–27, 2001). By utilizing weaker hypotheses of the Zabrejko-Nguen kind and a modified majorizing sequence we perform a semilocal convergence analysis which yields finer error bounds and more precise information on the location of the solution that the ones obtained in Rev. Colomb. Mat. 35:21–27, 2001. This error analysis is obtained at the same computational cost as the analogous results of Uko and Velásquez (Rev. Colomb. Mat. 35:21–27, 2001). We also give two generalizations of the well-known Kantorovich theorem on the solvability of nonlinear equations and the convergence of Newton’s method. Finally, we provide a numerical example to illustrate the predicted-by-theory performance of the Newton iterates involved in this paper.
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Argyros, I.K., Uko, L.U. An improved convergence analysis of a one-step intermediate Newton iterative scheme for nonlinear equations. J. Appl. Math. Comput. 38, 243–256 (2012). https://doi.org/10.1007/s12190-011-0476-1
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DOI: https://doi.org/10.1007/s12190-011-0476-1
Keywords
- Nonlinear equations
- Newton’s method
- Intermediate Newton scheme
- Zabrejko-Nguen conditions
- Kantorovich theorem
- Iterative solution
- Majorant method
- Majorizing sequence
- Lipschitz condition
- Center-Lipschitz condition