On a characterization of some Newton-like methods of \(R\)-order at least three
For the nonlinear equation \(F(x) = 0\) in Banach space an \(R\)-order convergent (\(R\geq 3\)) Newton-like method is introduced with the help of the first and second derivatives. \textit{W. Gander}'s result on Halley's method [Am. Math. Mon. 92, 131--134 (1985; Zbl 0574.65041)] is extended to a Banach space with a multi-parametric family iteration process. The semi-local convergence with \(R\)-order at least three is proven. Numerical examples confirm the order of convergence.
nonlinear operator equation \(R\)-order of convergence
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