In this paper, we study the dynamical behaviour of the Chebyshev--Halley family applied on a family of degree n polynomials. For n=2 we bound the set of parameters for which the iterative methods have convergence regions which do not correspond to the basins of attraction of the roots. We also study the dynamics of indifferent fixed points on the boundary of the regions of parameters with bad behaviour. Finally, we provide a numerical study on the boundedness of the regions of parameters with bad behaviour for the family of degree n polynomials

Campos, Beatriz

Canela Sánchez, Jordi

Vindel, Pura

Communications in Nonlinear Science and Numerical Simulation

Diposit Digital de Documents de la UAB

Convergence regions for the Chebyshev--Halley family

In this paper we study the dynamical behavior of the Chebyshev–Halley methods on the family of degree n polynomials . 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  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

Canela, Jordi

Repositori Institucional de la Universitat Jaume I

Convergence regions for the Chebyshev–Halley family

http://repositori.uji.es/xmlui/bitstream/10234/174856/1/56079.pdf

Convergence regions for the Chebyshev--Halley family

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Diposit Digital de Documents de la UAB

Repositori Institucional de la Universitat Jaume I