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On the local convergence of a family of Euler-halley type iterations with a parameter. (English) Zbl 1181.65081
The authors are going to study the local convergence of a family of Euler–Halley type iterations depending on a parameter, for solving a nonlinear operator equation in a real or complex Banach space. The idea of introducing one real parameter $$\alpha$$ in the algorithm for finding the solution of equation is used also, by other authors, for example by W. Werner [“Some improvement of classical methods for the solution of nonlinear equations”, Lect. Notes Math. 878, 426–440 (1981; Zbl 0494.65033)]. For particular values of the parameter $$\alpha$$, are obtained some famous methods: $$\alpha=\frac12$$, the Halley method; $$\alpha=0$$, the Chebyshev-Euler method; $$\alpha=1$$, the super-Halley method.
Under the so called second-order generalized Lipshitz assumption of the nonlinear operator from the left-hand side of the equation, the local convergence of the family of Euler-Halley type iterations, is discussed and the radius of the optimal convergence ball is estimated, for each real value of the parameter. The convergence analysis is done separately for positive values and for negative values of the parameter. It is verified that for the local convergence there is no universal constant for the iterations, which is quite different from the semi-local behaviour of iterations.

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