Fourth-order convergent iterative method for nonlinear equation (Q858769)

To find a simple real zero \(r\) of an equation (1) \(f(x)= 0\) with sufficiently differentiable function \(f: \mathbb{R}\to\mathbb{R}\), the authors first propose a quadratically convergent iterative method \[ x_{k+1}= x_k-2f(x_k)/(f'(x_k)\pm\sqrt{f^{\prime 2}(x_k)+ 4p^3 f^3(x_k)}),\tag{2} \] where \(p\in\mathbb{R}\), \(|p|<\infty\), and sign is chosen such as to make the denominator largest in magnitude. For \(p= 0\) (2) coincides with the Newton method. Further, they suggest a predictor-corrector iterative method by combining (2) and the Newton method: \[ z_k= x_k- 2f(x_k)/(fz_k)\pm \sqrt{f^{\prime 2}_k(x_k)+ 4p^3 f^3(x_k)}),\tag{3} \] \[ x_{k+1}= z_k- f(z_k)/f'(z_k).\tag{4} \] It is proved that if \(x_0\) is sufficiently close to the root \(r\), then the iterative method defined by (3)--(4) has the fourth-order of convergence. The presented results of the numerical solution of 7 examples confirm the efficiency of the new developed predictor-corrector method.