Interpolation-Based Iterative Methods for Nonlinear Problems

Nonlinear problems pose significant challenges in the development of iterative solvers due to costly derivative evaluations, nonexistent or non-smooth derivatives, and singularities. To address these obstacles, this study presents a novel class of Newton-Interpolation-Based Iterative methods (NIBM) that circumvent derivative evaluations for nonlinear problems. A comprehensive convergence analysis framework is established to systematically investigate the convergence rates of NIBM. Numerical experiments demonstrate the efficacy and robustness of NIBM methods in solving nonlinear equations, highlighting their potential for applications in scientific and engineering disciplines. The results underscore the efficiency and reliability of NIBM methods.