A New Family of Methods for Solving Systems of Nonlinear Equations

A new family of iterative methods for solving systems of nonlinear equations is introduced, extending and unifying several existing approaches within a general framework. In the single-variable case, Newton’s method was accelerated using Padé approximations. This strategy is then extended to systems of nonlinear equations, leading to the construction of higher-order iteration functions with convergence orders ranging from two to four. By varying the order (p, q) of the Padé approximation, several known methods and their modifications emerge as special cases of this generalized approach. Theoretical convergence rates are established, and numerical experiments, conducted using high-precision arithmetic, confirm both the predicted behavior and improved computational efficiency across a variety of test problems.