Dynamic Analysis of a Family of Iterative Methods with Fifth-Order Convergence

In this paper, a new class of fifth-order Chebyshev–Halley-type methods with a single parameter is proposed by using the polynomial interpolation method. The convergence order of the new method is proved. The dynamic behavior of the new method on quadratic polynomials P(x)=(x−a)(x−b) is analyzed, the strange fixed points and the critical points of the operator are obtained, the corresponding parameter planes and dynamic planes are drawn, the stability and convergence of the iterative method are visualized, and some parameter values with good properties are selected. The fractal results of the new method corresponding to different parameters about polynomial G(x) are plotted. Numerical results show that the new method has less computing and higher computational accuracy than the existing Chebyshev–Halley-type methods. The fractal results show the new method has good stability and convergence. The numerical results of different iteration methods are compared and agree with the results of dynamic analysis.