A modification of the super-Halley method under mild differentiability conditions
The authors consider the problem of solving a nonlinear equation \(F(x) = 0\) where \(F: \Omega \subseteq X \to Y\) is a nonlinear Fréchet differentiable operator in an open domain \(\Omega\) of a Banach space \(X\) with value in a Banach space Y. The authors derive a new family of two-step methods from a one-point iteration of order three, called the convex acceleration of Newton's method or the super-Halley method, see \textit{D. Chen, I. K. Argyros} and \textit{Q. Qian} [Appl. Math. Lett. 7(5), 49-52 (1995; Zbl 0811.65043)]; \textit{J. M. Gutierrez} and \textit{M. A. Hernandez} [Bull. Aust. Math. Soc. 55, 113-130 (1997; Zbl 0893.47043]; \textit{J. M. Gutierrez} and \textit{M. A. Hernandez} [Comput. Math. Appl. 36, No. 7, 1-8 (1998; Zbl 0933.65063)]. A theorem on the existence and uniqueness of the solution as well as on the convergence of the method, assuming only that the first derivative of \(F\) satisfies a Lipschitz condition, is proved. Finally an example to illustrate the power of the new convergence result, is given.
scientific article; zbMATH DE number 1419830