1. Convergence regions for the Chebyshev–Halley family
- Author
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B. Campos, Jordi Canela, P. Vindel, and The authors would like to thank the anonymous referees for their suggestions and comments, which improved this paper. The first and third authors were supported by the Spanish project MTM2014-52016-C02-2-P, the Generalitat Valenciana Project PROMETEO/2016/089 and UJI project P1.1B20115-16. The second author was supported by RedIUM and MINECO (Spain) through the research network MTM2014-55580-REDT and by the mathematics institute IMAC (Castellon, Spain).
- Subjects
roots ,Numerical Analysis ,Pure mathematics ,Chebyshev–Halley family ,Iterative method ,polynomials ,Applied Mathematics ,Chebyshev systems ,010102 general mathematics ,010103 numerical & computational mathematics ,dynamics ,Parameter space ,Fixed point ,01 natural sciences ,Attraction ,Chebyshev filter ,Critical point (mathematics) ,Combinatorics ,Quadratic equation ,Modeling and Simulation ,Polinomis ,0101 mathematics ,Sistemes de Chebyshev ,Mathematics - Abstract
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 = ∞ 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.
- Published
- 2018