On the Derivation of a Fast Solver for Nonlinear Systems of Equations Utilizing Frozen Substeps with Applications

In this manuscript, we propose a multi-step framework for solving nonlinear systems of algebraic equations. To improve the solver’s efficiency, the Jacobian matrix is held constant during the second sub-step, while a specialized strategy is applied in the third sub-step to maximize convergence speed without necessitating additional Jacobian evaluations. The proposed method achieves fifth-order convergence for simple roots, with its theoretical convergence established. Finally, computational experiments are conducted to illustrate the performance of the proposed solver in addressing nonlinear equation systems.