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A stable class of improved second-derivative free Chebyshev-Halley type methods with optimal eighth order convergence. (English) Zbl 1347.65097
Summary: In this paper, we present a uniparametric family of modified Chebyshev-Halley type methods with optimal eighth-order of convergence. In terms of computational cost, each member of the family requires only four functional evaluations per step, and hence is optimal in the sense of the Kung-Traub conjecture (cf. [H. T. Kung and J. F. Traub, J. Assoc. Comput. Mach. 21, 643–651 (1974; Zbl 0289.65023)]). Moreover, in order to have additional information to choose some elements of the class, in particular some stable enough, we use complex dynamics tools to analyze their stability. Then, some ranges of values of the parameter are found to be avoided but we show that the region of stable members of this family is vast. It is found by way of illustration that these proposed methods are very useful in high precision computations.

MSC:
65H05 Numerical computation of solutions to single equations
65Y20 Complexity and performance of numerical algorithms
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