Multi-step derivative-free preconditioned Newton method for solving systems of nonlinear equations

Abstract

Preconditioning of systems of nonlinear equations modifies the associated Jacobian and provides rapid convergence. The preconditioners are introduced in a way that they do not affect the convergence order of parent iterative method. The multi-step derivative-free iterative method consists of a base method and multi-step part. In the base method, the Jacobian of the system of nonlinear equation is approximated by finite difference operator and preconditioners add an extra term to modify it. The inversion of modified finite difference operator is avoided by computing LU factors. Once we have LU factors, we repeatedly use them to solve lower and upper triangular systems in the multi-step part to enhance the convergence order. The convergence order of m-step Newton iterative method is $$m+1$$ . The claimed convergence orders are verified by computing the computational order of convergence and numerical simulations clearly show that the good selection of preconditioning provides numerical stability, accuracy and rapid convergence.