Dynamics of a new family of iterative processes for quadratic polynomials (Q847253)

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.