Complex Roots-Finding Method of Nonlinear Equations

Abstract

Various researchers developed numerical methods to solve nonlinear equations. But still, there is a need to improve these methods to get accurate results. Some techniques exist where convergence is not assured. These methods increase the problem's cost using a second/higher-order derivative. So, cost efficiency is also considered an issue in many fast-converging ways. This paper proposes a new numerical method to find complex and real roots of nonlinear equations. By taking an initial guess w₀ ∈ C, the new approach contributes to a series of iterations converging to a real or complex solution. Moreover, the method evaluates complex roots even if a real number is taken as an initial guess. This innovation presents that the proposed method achieves second-order convergence. The number of iterations is also determined to check the performance of the new iterative scheme. Using Python 3.10.9 package, the authors have tested the method's efficacy on several numerical problems, presented the results in the tables, and illustrated them with the help of graphs.