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Dynamics of a new family of iterative processes for quadratic polynomials. (English) Zbl 1201.65071

A family of iterative methods is proposed for solving quadratic equations \(f(z)=0\) with \(f:{\mathbb C}\to{\mathbb C}\). These iterative methods include Newton and Chebyshev methods as special cases. The authors show convergence and dynamical behaviour of these iterative methods, particularly relating the coefficients of the iteration methods to the Catalan numbers, and the rational maps associated with these methods to the Catalan triangle. Computer graphs are used to illustrate the patterns of Julia sets of the methods.
Reviewer: Zhen Mei (Toronto)

MSC:

65H04 Numerical computation of roots of polynomial equations
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
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