A note on the Kantorovich theorem for Newton iteration
Journal of Computational and Applied Mathematics
A new semilocal convergence theorem for Newton's method
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. IV: optimization and nonlinear equations
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
On the convergence of Newton's method for a class of nonsmooth operators
Journal of Computational and Applied Mathematics
International Journal of Computer Mathematics
Journal of Computational and Applied Mathematics
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We study the problem of approximating a locally unique solution of an operator equation using Newton's method. The well-known convergence theorem of L.V. Kantorovich involves a bound on the second Fréchet-derivative or the Lipschitz-Fréchet-differentiability of the operator involved on some neighborhood of the starting point. Here we provide local and semilocal convergence theorems for Newton's method assuming the Fréchet-differentiability only at a point which is a weaker assumption. A numerical example is provided to show that our result can apply to solve a scalar equation where the above-mentioned ones may not.