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Convergence regions for the Chebyshev-Halley family

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In this paper we study the dynamical behavior of the Chebyshev-Halley methods on the family of degree $n$ polynomials $z^{n}+c$. We prove that, despite increasing the degree, it is still possible to draw the parameter space by using the orbit of a single critical point. For the methods having $z=\infty $ as an attracting fixed point, we show how the basins of attraction of the roots become smaller as the value of $n$ grows. We also demonstrate that, although the convergence order of the Chebyshev-Halley family is 3, there is a member of order 4 for each value of $n$. In the case of quadratic polynomials, we bound the set of parameters which correspond to iterative methods with stable behaviour other than the basins of attraction of the roots.

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Communications In Nonlinear Science And Numerical Simulation, 2018, vol. 56, p. 508-525

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Articles publicats en revistes (Matemàtiques i Informàtica)

# metadata.dc.title[].value
Convergence regions for the Chebyshev-Halley family