Local Convergence Analysis of a Novel Kurchatov-Type Derivative-Free Methods with and Without Memory for Solving Nonlinear Systems

First, a Kurchatov-type derivative-free single-parameter method without memory for solving nonlinear systems is constructed, and the convergence order of the method is proved to be third order by using the higher-order Fréchet derivative. By using the Kurchatov-type difference operator to represent the variable parameter, the method without memory develops into two Kurchatov-type methods with memory, and the convergence order of the two methods with memory can reach 3.303 and 3.414, respectively. The new Kurchatov-type methods with and without memory use LU decomposition only once per iteration and require less computational cost when applying them to solve nonlinear systems. Second, in order to enlarge the application range of the novel Kurchatov-type method, the local convergence of the new Kurchatov-type method with and without memory is proved by using the [Formula: see text]-continuity condition of the first Fréchet derivative. In this way, the influence of higher-order derivatives on the proof of convergence is avoided, and the application range of the method is expanded. In addition, the radius of convergence around the solution is obtained by theoretical proof, and the uniqueness of the solution is proved. Finally, convergence radius is calculated by solving nonlinear systems with Kurchatov-type methods in numerical experiments, and the error and calculation time of different iteration methods are compared.