A splitting method to solve a single nonlinear equation with derivative-free iterative schemes

Abstract

In the paper, we convert a single nonlinear equation to a system consisting of two equations. While a quasi-linear term is added on the first equation, the nonlinear term in the second equation is decomposed at two sides through a weight parameter. After performing a linearization, an iterative scheme is derived, which is proven of third-order convergence for certain parameters. An affine quasi-linear transformation in the plane is established, and the condition for the spectral radius being smaller than one for the convergence of the iterative scheme is derived. By using the splitting method, we can further identify a sufficient condition for the convergence of the iterative scheme. Then, we develop a step-wisely quasi-linear transformation technique to solve nonlinear equations. Proper values of the parameters are qualified by the derived inequalities for both iterative schemes, which accelerate the convergence speed. The performances of the proposed iterative schemes are assessed by numerical tests, whose advantages are fast convergence, saving the function evaluation per iteration and without needing the differential of the given function.