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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.

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