On the Convergence, Dynamics and Applications of a New Class of Nonlinear System Solvers

Abstract

In this paper, a uni-parametric class of third-order iterative algorithms for solving systems of nonlinear equations is proposed. The local convergence of the suggested schemes is studied using generalized Lipschitz-type condition on the first-order Frechet derivative. Furthermore, we analyze the numerical stability of the new methods applying complex dynamics tool. The nonlinear systems related to the equation of molecular interaction, a boundary value problem, the integral equation from Chandrasekhar’s work, etc. are discussed. The most interesting fact about the proposed third-order family is that it generates a super convergent scheme (for $$\gamma =2$$ ) for solving quadratic nonlinear systems (QNS). This particular method produces much better results for QNS in comparison with other third-order schemes. Also, it is observed that the approximate computational order of convergence (ACOC) of this scheme (for $$\gamma =2$$ ) is approximately four (3.99–4.00) while solving QNS.