A New Optimal Numerical Root-Solver for Solving Systems of Nonlinear Equations Using Local, Semi-Local, and Stability Analysis

Abstract

This paper introduces an iterative method with a remarkable level of accuracy, namely fourth-order convergence. The method is specifically tailored to meet the optimality condition under the Kung–Traub conjecture by linear combination. This method, with an efficiency index of approximately 1.5874, employs a blend of localized and semi-localized analysis to improve both efficiency and convergence. This study aims to investigate semi-local convergence, dynamical analysis to assess stability and convergence rate, and the use of the proposed solver for systems of nonlinear equations. The results underscore the potential of the proposed method for several applications in polynomiography and other areas of mathematical research. The improved performance of the proposed optimal method is demonstrated with mathematical models taken from many domains, such as physics, mechanics, chemistry, and combustion, to name a few.

Optimal Combination of the Splitting–Linearizing Method to SSOR and SAOR for Solving the System of Nonlinear Equations

“…Recently, Al-Obaidi and Darvish [7] gave a comparative study on the qualification criteria for three categories nonlinear solvers for solving nonlinear equations. The multi-point and higher-order iterative algorithms based on the Newton technique for solving nonlinear equations can be seen in [8,9].…”