On a family of iterative methods for simultaneous extraction of all roots of algebraic polynomial (Q1763292)

A family of iterative methods based on the frame: \[ x_{k+1}=x_k-u_k\left(1+{f(x_k-u_k)\over f(x_k)-2\lambda f(x_k-u_k)}\right) \] is discussed for simultaneous approximation of all the zeros of a polynomial \(f(x)\). Here \(u_k=f(x_k)/f'(x_k)\) and \(\lambda\) is a parameter. It is shown that Newton, Newton-secant, Traub and Traub-Ostrowski methods can be recovered by the appropriate value of the parameter \(\lambda\). It is proven this family of iterative methods has the convergence order of three. Numerical examples confirm convergence of the proposed method.