A family of third-order multipoint methods for solving nonlinear equations (Q2495960)

This paper is concerned with third-order multipoint iterative methods for finding simple zeros of a nonlinear equation \(f(x)=0\). The author considers an iteration scheme of the type \[ x_{n+1}=x_n-\frac {\alpha f(x_n)}{f^\prime (x_n) + f^\prime \{ x_n+\beta u(x_n)\} },\quad n\geq 0, \] where \(u(x_n)=f(x_n)/\{ f^\prime (x_n) \pm p f (x_n)\},\) \(p\in \mathbb{R}\) and \(\alpha, \beta\) are the disposal parameters. The values \(2\) and \(-1\) are used for the parameters \(\alpha\) and \(\beta\) to ensure the iteration will be cubically convergent for simple and real roots of nonlinear equations. Further, a new family with cubic convergence is obtained by discrete modification and the experiments show that the method is suitable in the case where Steffensen or Newton-Steffensen fail.