Chebyshev-Halley methods for analytic functions
The Chebyshev-Halley iteration method \[ z^{n+1}=z^n - u_f(z^n)\left[1+\frac{L_f(z^n)}{2(1-\alpha L_f(z^n))}\right] \] is discussed for approximating zeros of an analytic function \(f(z)\). Here \(u_f(z)=\frac{f(z)}{f'(z)}\) and \(L_f(z)=\frac{f(z)f''(z)}{(f'(z))^2}\). \(\alpha\) is a real constant. This method is firstly modified for approximating a multiple zero of an analytic function. Locally cubic convergence is established if the multiplicity of the zero is known. Furthermore, the method is extended to a family of simultaneous methods for finding all distinct zeros of a polynomial. This family includes the Halley-like method, the Euler-Chebyshev-like method and the Schröder-like method. Numerical examples are used to illustrate efficiency of the proposed methods.
scientific article; zbMATH DE number 5290915
scientific article