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Ostrowski-Kantorovich theorem and \(S\)-order of convergence of Halley method in Banach spaces. (English) Zbl 0786.65051
The author considers the numerical solution of a nonlinear operator equation \(P(x) = 0\) in a Banach space setting. The original Halley method for solving one-dimensional equations is generalized in operator form in Banach spaces. (It should be pointed out that the 3rd equation has a small error.)
The Ostrowski-Kantorovich theorem for the Halley method is proved and an error bound for the above generalized Halley method is also given under similar assumptions as in the Newton-Kantorovich theorem for the Newton method. Moreover, some properties of \(S\)-order of convergence and sufficient asymptotic error bounds are also discussed.
There is no numerical example to illustrate the implementation of the above generalized Halley algorithm.

65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
47J25 Iterative procedures involving nonlinear operators
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