AWeight-Function Generalization of Singh’s Fifth-Order Method for Systems of Nonlinear Equations, with Application to a Discretized Stationary Viscous Burgers Problem

We present and analyze a new weighted family of iterative methods for solving systems of nonlinear equations. The proposed schemes are constructed as a generalization of the fifth-order method of Singh et al. by incorporating appropriate weight functions into the correction step, thereby generating a flexible class of methods that includes the original scheme as a special case. Sufficient conditions on the weight functions are established to guarantee fifth-order local convergence. Several admissible choices are presented to illustrate the versatility of the family. The practical performance of the proposed variants is investigated on a collection of large-scale nonlinear systems. Furthermore, the family is applied to the nonlinear algebraic system obtained from the finite-difference discretization of a stationary one-dimensional viscous Burgers problem. Numerical experiments indicate that the proposed methods provide a competitive and accurate alternative for solving nonlinear systems of this type.