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Article

Argyros, Ioannis K. ; Hilout, Saïd
Extending the applicability of Newton's method using nondiscrete induction. (English). Czechoslovak Mathematical Journal, vol. 63 (2013), issue 1, pp. 115-141
MSC: 47J25, 49M15, 65G99, 65H10, 65J15 | MR 3035501 | Zbl 1274.65163 | DOI: 10.1007/s10587-013-0008-2
Keywords:
Newton's method; Banach space; rate of convergence; semilocal convergence; nondiscrete mathematical induction; estimate function
Summary:
We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.
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