A Weight Function Generalization of Singh–Sharma Fifth-Order Method for Systems of Nonlinear Equations, with Application to a Discretized Stationary Viscous Burgers Problem

We present and analyze a weighted family of iterative methods for solving systems of nonlinear equations. The proposed schemes are constructed as a generalization of the Singh–Sharma fifth-order method by incorporating suitable 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, and the resulting error equation shows how the weights influence the leading error term. 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.