Abstract
Some Kurchatov-type accelerating parameters are used to construct some derivative-free iterative methods with memory for solving nonlinear systems. New iterative methods are developed from an initial scheme without memory with order of convergence three. New methods have the convergence order 2+5≈4.236 and 5, respectively. The application of new methods can solve standard nonlinear systems and nonlinear ordinary differential equations (ODEs) in numerical experiments. Numerical results support the theoretical results.
Highlights
Many real-world problems that arise in various scientific fields are modeled by mathematically interesting nonlinear systems F(x) = 0
Iterative method is a kind of efficient method for solving nonlinear systems
We should note that the main objective of this paper was to develop a high-order method and prove the local convergence order of new methods
Summary
Using the same variable parameter B(j) as method (5), Ahmad et al [13] and Kansal et al [14] proposed some high order iterative methods with memory for solving nonlinear systems. Using the Kurchatov’s divided difference operator [15], Chicharro et al [16] designed two derivative-free methods with memory for solving nonlinear systems. They constructed the following third-order iterative method without memory s(j) = z(j) + BF(z(j)),.
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